Review Topical Sections

Autoimmune liver disease and the enteric microbiome

  • Received: 04 March 2018 Accepted: 08 May 2018 Published: 14 May 2018
  • The human enteric microbiome is highly complex and has more than 150 times more genes within it than its host. The host and the microbiome have a commensurate relationship that can evolve over time. The typically symbiotic relationship between the two can become pathogenic. The microbiome composition in adults reflects their history of exposure to bacteria and environmental factors during early life, their genetic background, age, interactions with the immune system, geographical location, and, most especially, their diet. Similarly, these factors are thought to contribute to the development of autoimmune disease. It is possible that alterations in the intestinal microbiome could lead to liver disease. There is emerging data for the contribution of the microbiome in development of primary sclerosing cholangitis, primary biliary cholangitis, and autoimmune hepatitis; liver disorders associated with aberrant immune function in genetically susceptible individuals.

    Citation: Kerri Glassner, Eamonn MM Quigley, Lissa Franco, David W Victor III. Autoimmune liver disease and the enteric microbiome[J]. AIMS Microbiology, 2018, 4(2): 334-346. doi: 10.3934/microbiol.2018.2.334

    Related Papers:

    [1] Yue Dong, Xinzhu Meng . Stochastic dynamic analysis of a chemostat model of intestinal microbes with migratory effect. AIMS Mathematics, 2023, 8(3): 6356-6374. doi: 10.3934/math.2023321
    [2] Jean Luc Dimi, Texance Mbaya . Dynamics analysis of stochastic tuberculosis model transmission withimmune response. AIMS Mathematics, 2018, 3(3): 391-408. doi: 10.3934/Math.2018.3.391
    [3] Ruoyun Lang, Yuanshun Tan, Yu Mu . Stationary distribution and extinction of a stochastic Alzheimer's disease model. AIMS Mathematics, 2023, 8(10): 23313-23335. doi: 10.3934/math.20231185
    [4] Yuanfu Shao . Dynamics and optimal harvesting of a stochastic predator-prey system with regime switching, S-type distributed time delays and Lévy jumps. AIMS Mathematics, 2022, 7(3): 4068-4093. doi: 10.3934/math.2022225
    [5] Lin Xu, Linlin Wang, Hao Wang, Liming Zhang . Optimal investment game for two regulated players with regime switching. AIMS Mathematics, 2024, 9(12): 34674-34704. doi: 10.3934/math.20241651
    [6] Xiaodong Wang, Kai Wang, Zhidong Teng . Global dynamics and density function in a class of stochastic SVI epidemic models with Lévy jumps and nonlinear incidence. AIMS Mathematics, 2023, 8(2): 2829-2855. doi: 10.3934/math.2023148
    [7] Hong Qiu, Yanzhang Huo, Tianhui Ma . Dynamical analysis of a stochastic hybrid predator-prey model with Beddington-DeAngelis functional response and Lévy jumps. AIMS Mathematics, 2022, 7(8): 14492-14512. doi: 10.3934/math.2022799
    [8] Chuangliang Qin, Jinji Du, Yuanxian Hui . Dynamical behavior of a stochastic predator-prey model with Holling-type III functional response and infectious predator. AIMS Mathematics, 2022, 7(5): 7403-7418. doi: 10.3934/math.2022413
    [9] Ahmed Ghezal, Mohamed balegh, Imane Zemmouri . Markov-switching threshold stochastic volatility models with regime changes. AIMS Mathematics, 2024, 9(2): 3895-3910. doi: 10.3934/math.2024192
    [10] Yuhuai Zhang, Xinsheng Ma, Anwarud Din . Stationary distribution and extinction of a stochastic SEIQ epidemic model with a general incidence function and temporary immunity. AIMS Mathematics, 2021, 6(11): 12359-12378. doi: 10.3934/math.2021715
  • The human enteric microbiome is highly complex and has more than 150 times more genes within it than its host. The host and the microbiome have a commensurate relationship that can evolve over time. The typically symbiotic relationship between the two can become pathogenic. The microbiome composition in adults reflects their history of exposure to bacteria and environmental factors during early life, their genetic background, age, interactions with the immune system, geographical location, and, most especially, their diet. Similarly, these factors are thought to contribute to the development of autoimmune disease. It is possible that alterations in the intestinal microbiome could lead to liver disease. There is emerging data for the contribution of the microbiome in development of primary sclerosing cholangitis, primary biliary cholangitis, and autoimmune hepatitis; liver disorders associated with aberrant immune function in genetically susceptible individuals.


    Over the past three decades, there have been expanded attempts to develop more flexible distributions for modeling data in different applied sciences including economics, engineering, biological studies, environmental sciences, medical sciences, and finance. One such attempt is to add one or more parameters to the baseline model to construct a new extended distribution.

    Some well-known families are the Marshall-Olkin-G [1], beta-G [2], transmuted-G [3], Kumaraswamy-G [4], Weibull-G [5], transmuted exponentiated generalized-G [6], Kumaraswamy transmuted-G family [7], generalized transmuted-G [8], generalized odd log-logistic-G [9], log-logistic tan-G [10], Marshall-Olkin-Weibull-H [11], and modified generalized-G families [12], among others.

    Recently, Kavya and Manoharan [13] introduced a new transformation called the Kavya-Manoharan-G (KM-G) class. Many extended forms of classical distributions have been proposed based on the KM-G family. For example, the KM inverse length biased exponential [14], KM Kumaraswamy exponential [15], KM log-logistic [16], KM power-Lomax [17], KM Burr X [18], and KM Kumaraswamy distributions [19].

    In this paper, we propose a new family of distributions by adding one extra shape parameter in the KM-G class to construct the so-called generalized Kavya-Manoharan-G (GKM-G) family, which provides greater flexibility to the generated models. The GKM-G family is constructed based on the exponentiated-H (exp-H) family [20] as one of the most widely used generalization techniques. Using this technique, the cumulative distribution function (CDF) of the exp-H class takes the form

    $ F\left(x;\alpha , \boldsymbol{\varphi }\right) = {\left[H\left(x;\boldsymbol{\varphi }\right)\right]}^{\alpha }, x\in \mathfrak{R}, \alpha > 0, $ (1.1)

    where $ H\left(x; \boldsymbol{\varphi }\right) $ is the baseline CDF, which depends on the parameter vector $ \boldsymbol{\varphi }. $

    The GKM-G family can be considered as a proportional reversed hazard (PRH) family. The PRH models are very important in reliability theory and survival analysis, especially in the analysis of left censored lifetime data and in the study of parallel systems [21]. More information about the PRH models can be explored in [22] and [23].

    In fact, the technique for generating exp-H models can be traced back to Lehmann [20]. This generalization method received a great deal of attention in the last three decades, and more than fifty exp-H distributions have already been published. Some notable exponentiated distributions include the exponentiated Weibull [24], exponentiated Weibull-Pareto [25], and exponentiated Weibull family [26], among others. It has been illustrated that the exp-H distributions provide greater flexibility and have useful applications in many applied fields such as biomedical sciences, environmental studies, and reliability analysis. Since the exponentiated distribution is more appealing than its baseline counterpart, we provide the same approach for a family of distributions studied by [13].

    We study a comprehensive description of some of its mathematical properties. The new family may attract wider applications in reliability, engineering, environmental, and medicine fields due to its simple analytical forms and its flexibility. The special sub-models of the GKM-G family can provide right-skewed, reversed-J shaped, and unimodal densities, as well as increasing, bathtub, decreasing, unimodal, and modified bathtub hazard rate (HR) shapes. These flexible shapes are important for modeling several real-life data encountered in many applied fields.

    A special sub-model based on the exponential (E) distribution called the generalized Kavya-Manoharan exponential (GKME) is studied. The GKME distribution provides greater flexibility for modeling real-life data in several applied fields such as reliability, environmental, and medicine, as illustrated in Section 8. Five real data applications show that the GKME distribution provides consistently better fits as compared to other competing extended forms of the E model, namely, the generalized exponential (GE) [27], generalized inverted exponential (GIE) [28], Marshall-Olkin exponential (MOE) [1], alpha-power exponential (APE) [29], generalized Dinesh-Umesh-Sanjay exponential (GDUSE) [30], Kavya-Manoharan exponential (KME) [13], and E distributions.

    Additionally, the behavior of the unknown parameters of the GKME distribution for several sample sizes and parameter combinations is investigated using eight different estimation procedures. A guideline for selecting the optimum estimation method to estimate the GKME parameters is developed, which we believe applied statisticians and reliability engineers would find useful. Also, comprehensive simulations are performed to evaluate and compare the performance of various estimators.

    The rest of the paper is organized as follows. In Section 2, the GKM-G family is defined. Four special sub-models of the GKM-G family are presented in Section 3. In Section 4, some mathematical properties of the GKM-G class are obtained. In Section 5, we derive the $ r $th moment of the GKME distribution and present some numerical results for it. Estimation methods of the GKME parameters are presented in Section 6. In Section 7, we provide a detailed simulation study. Section 8 provides five real-life data applications to show empirically the flexibility of the GKME distribution. Finally, some remarks and future perspectives are presented in Section 9.

    Let $ G\left(x; \boldsymbol{\varphi }\right) $ and $ g\left(x; \boldsymbol{\varphi }\right) $ denote the CDF and probability density function (PDF) of a baseline model with parameter vector $ \boldsymbol{\varphi } $, then the CDF of the KM family [13] is defined as

    $ H\left(x;\boldsymbol{\varphi }\right) = \frac{e}{e-1}\left[1-{e}^{-G\left(x;\boldsymbol{\varphi }\right)}\right], x\in \mathfrak{R}. $ (2.1)

    The corresponding PDF of (2.1) is defined by

    $ h\left(x;\boldsymbol{\varphi }\right) = \frac{e}{e-1}g\left(x;\boldsymbol{\varphi }\right){e}^{-G\left(x;\boldsymbol{\varphi }\right)}. $

    The HR function (HRF) reduces to

    $ \pi \left(x;\boldsymbol{\varphi }\right) = \frac{g\left(x;\boldsymbol{\varphi }\right){e}^{1-G\left(x;\boldsymbol{\varphi }\right)}}{{e}^{1-G\left(x;\boldsymbol{\varphi }\right)}-1}. $

    By inserting (2.1) in Eq (1.1), the CDF of the GKM-G family is defined by

    $ F\left(x;\alpha , \boldsymbol{\varphi }\right) = {\left(\frac{e}{e-1}\right)}^{\alpha }{\left[1-{e}^{-G\left(x;\boldsymbol{\varphi }\right)}\right]}^{\alpha }, x\in \mathfrak{R}, \alpha > 0 . $ (2.2)

    The PDF of the GKM-G family reduces to

    $ f\left(x;\alpha , \boldsymbol{\varphi }\right) = {\left(\frac{e}{e-1}\right)}^{\alpha }\alpha g\left(x;\boldsymbol{\varphi }\right){e}^{-G\left(x;\boldsymbol{\varphi }\right)}{\left[1-{e}^{-G\left(x;\boldsymbol{\varphi }\right)}\right]}^{\alpha -1}. $ (2.3)

    The HRF of the GKM-G family follows as

    $ \varphi \left(\boldsymbol{x};\alpha , \boldsymbol{\varphi }\right) = \frac{\alpha g\left(x;\boldsymbol{\varphi }\right){e}^{1-G\left(x;\boldsymbol{\varphi }\right)}{\left[e-{e}^{1-G\left(x;\boldsymbol{\varphi }\right)}\right]}^{\alpha -1}}{{\left(e-1\right)}^{\alpha }-{\left[e-{e}^{1-G\left(x;\boldsymbol{\varphi }\right)}\right]}^{\alpha }}. $

    The extra shape parameter $ \alpha $ may allow us to study the tail behavior of the PDF (2.3) with more flexibility. Additionally, the GKM-G family is considered an important class for modeling different real-life data due to its flexibility in accommodating all important forms of the HRF. A random variable $ X $ having the PDF (2.3) is denoted by $ X\sim $GKM-G ($ \alpha, \boldsymbol{\varphi } $). Simply, the proposed GKM-G family reduces to the KM-G family [13] when $ \alpha = 1 $.

    In this section, we provide four special sub-models of the GKM-G family, namely, the GKME, GKM-Burr X (GKMBX), GKM-Burr XII (GKMBXII), and GKM-log logistic (GKMLL) distributions. These sub-models are capable of modeling monotone and non-monotone failure rates including increasing, reversed J shaped, decreasing, bathtub, modified bathtub, and upside-down bathtub. They also can have right-skewed, symmetrical, and reversed-J shaped densities. Figures 14 display all these shapes.

    Figure 1.  Plots of the PDF and HRF of the GKME distribution for some parameter values.
    Figure 2.  Plots of the PDF and HRF of the GKMBX distribution for some parameter values.
    Figure 3.  Plots of the PDF and HRF of the GKMBXII distribution for some parameter values.
    Figure 4.  Plots of the PDF and HRF of the GKMLL distribution for some parameter values.

    The CDF and PDF of the E distribution are defined, respectively, by $ G\left(x\right) = 1-{e}^{-\lambda x} $ and $ g\left(x\right) = \lambda {e}^{-\lambda x} $, where $ x > 0, \lambda > 0 $. By inserting the CDF of the E distribution in (2.2), the CDF of the GKME distribution follows as

    $ F\left(x;\alpha , \lambda \right) = \frac{1}{{\left(e-1\right)}^{\alpha }}{\left(e-{e}^{{e}^{-\lambda x}}\right)}^{\alpha }, \qquad x > 0, \qquad \alpha , \lambda > 0. $ (3.1)

    The corresponding PDF of the GKME distribution reduces to

    $ f\left(x;\alpha , \lambda \right) = \frac{\alpha \lambda }{{\left(e-1\right)}^{\alpha }}{e}^{{-\lambda x+e}^{-\lambda x}}{\left(e-{e}^{{e}^{-\lambda x}}\right)}^{\alpha -1}. $ (3.2)

    Therefore, the random variable with PDF (3.2) is denoted by $ X\sim\mathrm{G}\mathrm{K}\mathrm{M}\mathrm{E}\left(\alpha, \lambda \right) $. The GKME distribution reduces to the KME distribution [13] for $ \alpha = 1 $.

    The HRF of the GKME distribution is given by

    $ h\left(x;\alpha , \lambda \right) = \frac{\alpha \lambda {e}^{{-\lambda x+e}^{-\lambda x}}{\left(e-{e}^{{e}^{-\lambda x}}\right)}^{\alpha -1}}{{\left(e-1\right)}^{\alpha }-{\left(e-{e}^{{e}^{-\lambda x}}\right)}^{\alpha }}. $

    Figure 1 displays some possible shapes of the PDF and HRF of the GKME distribution.

    The CDF of the Burr-X (BX) distribution is $ G\left(x\right) = {\left[1-{e}^{-{\left(\lambda x\right)}^{2}}\right]}^{\beta } $, where $ x > 0, \lambda, \beta > 0 $. By inserting the CDF of the BX distribution in (2.2), the CDF of the GKMBX distribution follows as

    $ F\left(x;\alpha , \lambda , \beta \right) = {\left(\frac{e}{e-1}\right)}^{\alpha }{\left[1-{e}^{-{\left(1-{e}^{-{\left(\lambda x\right)}^{2}}\right)}^{\beta }}\right]}^{\alpha },\qquad x > 0,\qquad \alpha , \lambda , \beta > 0. $ (3.3)

    The corresponding PDF follows as

    $ f\left(x;\alpha , \lambda , \beta \right) = \frac{2{e}^{\alpha }\alpha \beta {\lambda }^{2}}{{\left(e-1\right)}^{\alpha }}x{e}^{-{\left(\lambda x\right)}^{2}}{e}^{-{\left[1-{e}^{-{\left(\lambda x\right)}^{2}}\right]}^{\beta }}{\left[1-{e}^{-{\left(\lambda x\right)}^{2}}\right]}^{\beta -1}{\left\{1-{e}^{-{\left[1-{e}^{-{\left(\lambda x\right)}^{2}}\right]}^{\beta }}\right\}}^{\alpha -1}. $ (3.4)

    The GKMBX distribution reduces to the KMBX distribution for $ \alpha = 1 $. The GKM-Rayleigh distribution follows as a special case for $ \text{β} = 1 $. The KM-Rayleigh distribution follows for $ \alpha = \beta = 1. $ Figure 2 displays some possible shapes of the PDF and HRF of the GKMBX distribution.

    The CDF of the BXII distribution is $ G\left(x\right) = 1-{\left(1+{x}^{\lambda }\right)}^{-\beta } $, where $ x > 0, \beta, \lambda > 0 $. By inserting the CDF of the BXII distribution in (2.2), the CDF of the GKMBXII distribution is obtained as

    $ F\left(x;\alpha , \beta , \lambda \right) = {{\left(\frac{e}{e-1}\right)}^{\alpha }\left[1-{e}^{-1+{\left(1+{x}^{\lambda }\right)}^{-\beta }}\right]}^{\alpha }, x > 0, \alpha , \beta , \lambda > 0. $ (3.5)

    The PDF of the GKMBXII distribution reduces to

    $ f\left(x;\alpha , \beta , \lambda \right) = \frac{\alpha \beta \lambda }{{\left(e-1\right)}^{\alpha }}{x}^{\lambda -1}{\left(1+{x}^{\lambda }\right)}^{-1-\beta }{e}^{-1+\alpha +{\left(1+{x}^{\lambda }\right)}^{-\beta }}{\left[1-{e}^{-1+{\left(1+{x}^{\lambda }\right)}^{-\beta }}\right]}^{\alpha -1}. $ (3.6)

    Figure 3 shows some possible shapes of the PDF and HRF of the GKMBXII distribution.

    Consider the CDF of the log-logistic (LL) distribution, say, $ G\left(x\right) = 1-{\left[1+{\left(\frac{x}{\lambda }\right)}^{\beta }\right]}^{-1} $, where $ x > 0, \beta, \lambda > 0 $, then the CDF of the GKMLL distribution takes the form

    $ F\left(x;\alpha , \beta , \lambda \right) = {{\left(\frac{e}{e-1}\right)}^{\alpha }\left[1-{e}^{-1+{\left[1+{\left(\frac{x}{\lambda }\right)}^{\beta }\right]}^{-1}}\right]}^{\alpha }, x > 0, \alpha , \beta , \lambda > 0. $ (3.7)

    The corresponding PDF of the GKMLL distribution reduces to

    $ f\left(x;\alpha , \beta , \lambda \right) = \frac{\alpha \beta {\lambda }^{-\beta }}{{\left(e-1\right)}^{\alpha }}{x}^{\beta -1}{\left[1+{\left(\frac{x}{\lambda }\right)}^{\beta }\right]}^{-2}{e}^{-1+\alpha +{\left[1+{\left(\frac{x}{\lambda }\right)}^{\beta }\right]}^{-1}}{\left[1-{e}^{-1+{\left[1+{\left(\frac{x}{\lambda }\right)}^{\beta }\right]}^{-1}}\right]}^{\alpha -1}. $ (3.8)

    Figure 4 displays some possible shapes of the PDF and HRF of the GKMLL distribution.

    This section provides some mathematical properties of the GKM-G family.

    In this section, we provide a useful representation of the CDF and PDF of the GKM-G family in terms of exp-G density. Consider the following generalized binomial series

    $ {\left[1-z\right]}^{\alpha } = \sum\limits_{j = 0}^{\infty }{\left(-1\right)}^{j}\left(αj
    \right){z}^{j}. $
    (4.1)

    Applying (4.1) to (2.2), we obtain

    $ {\left[1-{e}^{-G\left(x\right)}\right]}^{\alpha } = \sum\limits_{j = 0}^{\infty }{\left(-1\right)}^{j}\left(αj
    \right){e}^{-jG\left(x\right)}. $
    (4.2)

    Using the exponential series, we can write

    $ {e}^{-jG\left(x\right)} = \sum\limits_{k = 0}^{\infty }{\left(-1\right)}^{k}\frac{{j}^{k}G{\left(x\right)}^{k}}{k!}. $ (4.3)

    Substituting (4.2) and (4.3) in (2.2), the CDF of the GKM-G takes the form

    $ F\left(x\right) = {\left(\frac{e}{e-1}\right)}^{\alpha }\sum\limits_{k, j = 0}^{\infty }\frac{{{\left(-1\right)}^{j+k}j}^{k}}{k!}\left(αj
    \right)G{\left(x\right)}^{k}. $

    Hence, the CDF of the GKM-G family can be expressed as

    $ F\left(x\right) = \sum\limits_{k = 0}^{\infty }{{a}_{k}H}_{k}\left(x\right), $ (4.4)

    where

    $ {a}_{k} = \sum\limits_{j = 0}^{\infty }{\left(\frac{e}{e-1}\right)}^{\alpha }\frac{{{\left(-1\right)}^{j+k}j}^{k}}{k!}\left(αj
    \right) $

    and $ {H}_{k}\left(x\right) = G{\left(x\right)}^{k} $ is the CDF of the exp-G family with power parameter $ k > 0 $. By differentiating the above equation, the PDF of the GKM-G family follows as

    $ f\left(x\right) = \sum\limits_{k = 0}^{\infty }{{a}_{k}h}_{k}\left(x\right), $ (4.5)

    where $ {h}_{k}\left(x\right) = kg\left(x\right)G{\left(x\right)}^{k-1} $ is the exp-G density with power parameter $ k. $ Thus, several mathematical properties of the GKM-G family can be obtained simply from those properties of the exp-G family.

    The quantile function (QF) of $ X $, say, $ Q\left(u\right) = {F}^{-1}\left(x\right) $, can be obtained by inverting (2.2), then the QF of the GKM-G family follows as

    $ Q\left(u\right) = {Q}_{G}\left(u\right)\left\{-\mathrm{l}\mathrm{o}\mathrm{g}\left[1-{\left(u{\varphi }^{-\alpha }\right)}^{\frac{1}{\alpha }}\right]\right\}, $

    where $ \varphi = e/(e-1) $ and $ {Q}_{G}\left(u\right) = {G}^{-1}\left(u\right) $ is the QF of the baseline $ G $ distribution and $ u\in \left(\mathrm{0, 1}\right) $.

    Henceforth, $ {T}_{k} $ denotes the exp-G random variable with power parameter $ k $. The $ r $th moment of $ X $ follows from (4.5) as

    $ {\mu }_{r}^{\text{'}} = {\rm E}\left({X}^{r}\right) = \sum\limits_{k = 0}^{\infty }{a}_{k}E\left({T}_{k}^{r}\right). $ (4.6)

    The moment generating function (MGF), $ {M}_{X}\left(t\right) = E\left({e}^{tX}\right) $, of $ X $ can be derived from (4.5) in two different formulas. The first one is given by

    $ {M}_{X}\left(t\right) = \sum\limits_{k = 0}^{\infty }{{a}_{k}M}_{k}\left(t\right), $

    where $ {M}_{k}\left(t\right) $ is the MGF of $ {T}_{k}\left(x\right) $. Hence, $ {M}_{X}\left(t\right) $ can be determined from the exp-G generating function.

    The second formula for $ {M}_{X}\left(t\right) $ follows from (4.5) as

    $ {M}_{X}\left(t\right) = \sum\limits_{k = 0}^{\infty }k{a}_{k}\tau \left(t, k-1\right), $

    where $ \tau \left(t, k-1\right) = \int_0^1\mathrm{e}\mathrm{x}\mathrm{p}\left[t{Q}_{G}\left(u\right)\right]{u}^{k-1}du $.

    The $ s $th incomplete moment of $ X $ can be expressed from (4.5) as

    $ {\varphi }_{s}\left(t\right) = \underset{-\infty }{\overset{t}{\int }}{x}^{s}f\left(x\right)dx = \sum\limits_{k = 0}^{\infty }{a}_{k}\underset{-\infty }{\overset{t}{\int }}{x}^{s}{h}_{k}\left(x\right)dx. $ (4.7)

    The first incomplete moment follows from (4.7) when $ s = 1 $. It can be applied to construct Bonferroni and Lorenz curves, which are defined, for a given probability $ \pi $, by $ B\left(\pi \right) = {\varphi }_{1}\left(q\right)/\left(\pi {\mu }_{1}^{\text{'}}\right) $ and $ L\left(\pi \right) = {\varphi }_{1}\left(q\right)/{\mu }_{1}^{\text{'}} $, respectively, where $ {\mu }_{1}^{\text{'}} $ is given by (4.6) with $ r = 1 $ and $ q = Q\left(\pi \right) $ is the QF of $ X $ at $ \pi $. These curves are very useful in economics, reliability, demography, insurance, and medicine.

    Now, $ {\varphi }_{1}\left(t\right) $ can be determined in two expressions. The first expression for $ {\varphi }_{1}\left(t\right) $ is derived from (4.7) as

    $ {\varphi }_{1}\left(t\right) = \sum\limits_{k = 0}^{\infty }{a}_{k}{l}_{k}\left(t\right). $ (4.8)

    Where $ {l}_{k}\left(t\right) = \int_{-\infty}^{t} x{h}_{k}\left(x\right)dx $ is the first incomplete moment of the exp-G family. A second expression for $ {\varphi }_{1}\left(t\right) $ takes the form

    $ {\varphi }_{1}\left(t\right) = \sum\limits_{k = 0}^{\infty }{a}_{k}{\upsilon }_{k}\left(t\right), $

    where $ {\upsilon }_{k}\left(t\right) = k \int_0^{G\left(t\right)}{Q}_{G}\left(u\right){u}^{k-1}du $, which can be computed numerically, and $ {Q}_{G}\left(u\right) $ is the QF corresponding to $ G\left(x\right) $, i.e., $ {Q}_{G}\left(u\right) = {G}^{-1}\left(u\right) $.

    The mean deviations about the mean $ \left[{\delta }_{1} = E\left(\left|X-{\mu }_{1}^{\text{'}}\right|\right)\right] $ and about the median $ \left[{\delta }_{2} = E\left(\left|X-M\right|\right)\right] $ of $ X $ are given by $ {\delta }_{1} = 2{\mu }_{1}^{\text{'}}F\left({\mu }_{1}^{\text{'}}\right)-2{\varphi }_{1}\left({\mu }_{1}^{\text{'}}\right) $ and $ {\delta }_{2} = {\mu }_{1}^{\text{'}}-2{\varphi }_{1}\left(M\right) $, respectively, where $ {\mu }_{1}^{\text{'}} = E\left(X\right), M = Q\left(0.5\right) $ is the median. $ F\left({\mu }_{1}^{\text{'}}\right) $ is easily evaluated from (2.2).

    The mean residual life (MRL) represents the expected additional life length for a unit, which is alive at age $ t $, and it is defined by $ {MRL}_{x}\left(t\right) = E\left(X-t|X > t\right), $ for $ t > 0 $. The MRL of $ X $ is

    $ {MRL}_{X}\left(t\right) = \frac{\left[1-{\varphi }_{1}\left(t\right)\right]}{S\left(t\right)}-t, $ (4.9)

    where $ S\left(t\right) $ is the survival function (SF) of the GKM-G family. Inserting (4.8) in (4.9), we obtain

    $ {MRL}_{X}\left(t\right) = \frac{1}{S\left(t\right)}\left[1-\sum\limits_{k = 0}^{\infty }{a}_{k}{l}_{k}\left(t\right)\right]-t. $

    The mean inactivity time (MIT) represents the waiting time elapsed since the failure of an item, on condition that this failure occurred in $ (0, t) $. The MIT is defined by $ {MIT}_{X}\left(t\right) = E\left(t-X|X\le t\right), $ for $ t > 0 $. The MIT of $ X $ reduces to

    $ {MIT}_{X}\left(t\right) = t-\frac{{\varphi }_{1}\left(t\right)}{F\left(t\right)}. $ (4.10)

    Combining Eqs (4.8) and (4.10), the MIT of $ X $ follows as

    $ {MIT}_{X}\left(t\right) = t-\frac{1}{F\left(t\right)}\sum\limits_{k = 0}^{\infty }{a}_{k}{l}_{k}\left(t\right). $

    The Rényi entropy of a random variable $ X $ represents a measure of variation of the uncertainty. The Rényi entropy is defined by

    $ {I}_{\theta } = \frac{1}{1-\theta }\mathrm{log}\left({\int }_{-\infty }^{\infty }f{\left(x\right)}^{\theta }dx\right), \theta > 0 \;{\rm{and}}\; \theta \ne 1 .$

    Using the GKM-G density (2.3), we can write

    $ f{\left(x\right)}^{\theta } = {\varphi }^{\theta \alpha }{\alpha }^{\theta }g{\left(x\right)}^{\theta }{e}^{-\theta G\left(x\right)}{\left[1-{e}^{-G\left(x\right)}\right]}^{\theta \left(\alpha -1\right)}, $

    where $ \varphi = e/(e-1) $. Applying the power series (4.2) to the last term, we obtain

    $ {\left[1-{e}^{-G\left(x\right)}\right]}^{\theta \left(\alpha -1\right)} = \sum\limits_{j = 0}^{\infty }{\left(-1\right)}^{j}\left(θ(α1)j
    \right){e}^{-jG\left(x\right)}. $

    Hence,

    $ f{\left(x\right)}^{\theta } = {\varphi }^{\theta \alpha }{\alpha }^{\theta }g{\left(x\right)}^{\theta }\sum\limits_{j = 0}^{\infty }{\left(-1\right)}^{j}\left(θ(α1)j
    \right){e}^{-\left(\theta +j\right)G\left(x\right)}. $

    Applying the exponential series, we obtain

    $ {e}^{-\left(\theta +j\right)G\left(x\right)} = \sum\limits_{k = 0}^{\infty }{\left(-1\right)}^{k}\frac{{\left(\theta +j\right)}^{k}G{\left(x\right)}^{k}}{k!}. $

    Thus, $ f{\left(x\right)}^{\theta } $ reduces to

    $ f{\left(x\right)}^{\theta } = \sum\limits_{k = 0}^{\infty }{\eta }_{k}g{\left(x\right)}^{\theta }G{\left(x\right)}^{k}, $

    where

    $ {\eta }_{k} = \sum\limits_{j = 0}^{\infty }\frac{{{\left(\alpha {\varphi }^{\alpha }\right)}^{\theta }\left(\theta +j\right)}^{k}}{k!}{\left(-1\right)}^{k+j}\left(θ(α1)j
    \right). $

    The Rényi entropy of the GKM-G family reduces to

    $ {I}_{\theta } = \frac{1}{1-\theta }\mathrm{log}\left[\sum\limits_{k = 0}^{\infty }{\eta }_{k}{\int }_{-\infty }^{\infty }g{\left(x\right)}^{\theta }G{\left(x\right)}^{k}dx\right]. $

    The Shannon entropy ($ SI $) follows as a special case of the Rényi entropy when $ \theta $ tends to 1.

    Let $ {X}_{1}, \dots, {X}_{n} $ be a random sample from the GKM-G family. The PDF of $ {X}_{i:n} $ can be written as

    $ {f}_{i:n}\left(x\right) = \frac{f\left(x\right)}{B\left(i, n-i+1\right)}\sum\limits_{j = 0}^{n-i}{\left(-1\right)}^{j}\left(n1j
    \right)F{\left(x\right)}^{j+i-1}, $
    (4.11)

    where $ B\left(., .\right) $ is the beta function. Using (2.2) and (2.3), we can write

    $ f\left(x\right)F{\left(x\right)}^{j+i-1} = {\varphi }^{\alpha \left(j+i\right)}\alpha g\left(x\right){e}^{-G\left(x\right)}{\left[1-{e}^{-G\left(x\right)}\right]}^{\alpha \left(j+i\right)-1}. $ (4.12)

    Applying the generalized binomial series to (4.12), we have

    $ f\left(x\right)F{\left(x\right)}^{j+i-1} = {\varphi }^{\alpha \left(j+i\right)}\sum\limits_{k, l = 0}^{\infty }\frac{{\alpha \left(-1\right)}^{l+k}{\left(1+l\right)}^{k}}{k!}\left(α(j+i)1l
    \right)g\left(x\right)G{\left(x\right)}^{k}. $
    (4.13)

    Inserting (4.13) in Eq (4.11), the PDF of $ {X}_{i:n} $ reduces to

    $ {f}_{i:n}\left(x\right) = \sum\limits_{k = 0}^{\infty }{b}_{k}{h}_{k+1}\left(x\right), $ (4.14)

    where $ {h}_{k+1}\left(x\right) = \left(k+1\right)g\left(x\right)G{\left(x\right)}^{k} $ is the exp-G density with power parameter $ k+1 $ and

    $ {b}_{k} = \sum\limits_{j = 0}^{n-i}\sum\limits_{l = 0}^{\infty }\frac{{\alpha \left(-1\right)}^{j+l+k}{\left(1+l\right)}^{k}}{B\left(i, n-i+1\right)}\left(n1j
    \right)\frac{{\varphi }^{\alpha \left(j+i\right)}}{(k+1) !}\left(α(j+i)1l
    \right). $

    Hence, the PDF of the GKM-G order statistics is a linear combination of exp-G densities. Based on (4.15), we can derive the properties of $ {X}_{i:n} $ from those properties of $ {T}_{k+1} $. For example, the $ q $th moment of $ {X}_{i:n} $ is given by

    $ E\left({X}_{i:n}^{q}\right) = \sum\limits_{k = 0}^{\infty }{b}_{k}E\left({T}_{k+1}^{q}\right). $ (4.16)

    The probability weighted moments (PWMs) are proposed by Greenwood et al. [31] as a special class of moments. This class is used to derive estimates of the parameters and quantiles of distributions, which can be expressed in inverse form. They also have moderate biases, low variance, and comparable with the maximum likelihood (ML) estimators. Let $ X $ be a random variable with PDF $ f\left(x\right) $ and CDF $ F\left(x\right) $, then the $ \left(j, i\right) $th PWM of $ X $, denoted by $ {\rho }_{j, i} $, is defined by

    $ {\rho }_{j, i} = E\left\{{X}^{j}F{\left(X\right)}^{i}\right\} = {\int }_{-\infty }^{\infty }{x}^{j}f\left(x\right)F{\left(x\right)}^{i}dx, $

    where $ j $and $ i $ are nonnegative integers.

    Using the CDF and PDF of the GKM-G family and Eq (4.14), we can write

    $ f\left(x\right)F{\left(x\right)}^{i} = {\varphi }^{\alpha \left(1+i\right)}\sum\limits_{k, l = 0}^{\infty }\frac{{\alpha \left(-1\right)}^{l+k}{\left(1+l\right)}^{k}}{k!}\left(α(1+i)1l
    \right)g\left(x\right)G{\left(x\right)}^{k}. $

    The last equation can be expressed as

    $ f\left(x\right)F{\left(x\right)}^{i} = \sum\limits_{k = 0}^{\infty }{m}_{k}{h}_{k+1}\left(x\right), $

    where

    $ {m}_{k} = \sum\limits_{l = 0}^{\infty }{\varphi }^{\alpha \left(1+i\right)}\frac{{\alpha \left(-1\right)}^{l+k}{\left(1+l\right)}^{k}}{(k+1) !}\left(α(1+i)1l
    \right). $

    Thus, the PWM of $ X $ is given by

    $ {\rho }_{j, i} = \sum\limits_{k = 0}^{\infty }{m}_{k}{\int }_{-\infty }^{\infty }{x}^{j}{h}_{k+1}\left(x\right)dx = \sum\limits_{k = 0}^{\infty }{m}_{k}E\left({T}_{k+1}^{j}\right). $

    In this section, we derive a simple formula for the $ r $th moment of the GKME distribution. Based on Eq (4.5), the PDF of the GKME distribution can be expressed as follows:

    $ f\left(x\right) = \sum\limits_{k = 0}^{\infty }{a}_{k}k\lambda {e}^{-\lambda x}{\left(1-{e}^{-\lambda x}\right)}^{k-1}. $

    Applying binomial expansion to $ {\left(1-{e}^{-\lambda x}\right)}^{k-1} $, the last equation follows as

    $ f\left(x\right) = \sum\limits_{k, m = 0}^{\infty }{a}_{k}k\lambda {\left(-1\right)}^{m}\left(k1m
    \right)exp\left(-\left(m+1\right)\lambda x\right). $

    The GKME PDF reduces to

    $ f\left(x\right) = \sum\limits_{m = 0}^{\infty }{v}_{m}{g}_{m+1}\left(x\right), $ (5.1)

    where

    $ {v}_{m} = \sum\limits_{j, k = 0}^{\infty }{\left(\frac{e}{e-1}\right)}^{\alpha }\frac{{{\left(-1\right)}^{j+k+m}j}^{k}}{\left(m+1\right)(k-1) !}\left(αj
    \right)\left(k1m
    \right) $

    and$ {g}_{m+1}\left(x\right) = \left(m+1\right)\lambda \mathrm{e}\mathrm{x}\mathrm{p}\left(-\left(m+1\right)\lambda x\right) $ denotes the E density with rate parameter $ \left(m+1\right)\lambda $. Equation (5.1) means that the GKME density is expressed as a single linear combination of E densities.

    Hence, the $ r $th moment of GKME distribution follows from (5.1) as

    $ {\mu }_{r}^{\text{'}} = \mathrm{{\rm E}}\left({X}^{r}\right) = r!\sum\limits_{m = 0}^{\infty }{v}_{m}{\left[\left(m+1\right)\lambda \right]}^{-r}. $ (5.2)

    Clearly, the mean of the GKME distribution, say, $ {\mu }_{1}^{\text{'}} = {\mu }_{X} $, follows from the above equation when $ r = 1. $ It is given by

    $ {\mu }_{X} = \frac{1}{\lambda }\sum\limits_{j, k, m = 0}^{\infty }{\left(\frac{e}{e-1}\right)}^{\alpha }\frac{{{\left(-1\right)}^{j+k+m}kj}^{k}}{{\left(m+1\right)}^{2}k!}\left(αj
    \right)\left(k1m
    \right). $

    Table 1 lists the numerical integration and the summation (SUM) formula of $ {\mu }_{X} $ for different values of $ \lambda $ and $ \alpha $ at truncated $ L $ terms. These numerical values are computed by R statistical software. Table 1 shows that the summation (5.2) converges to the numerical integral (NUI) of $ {\mu }_{X} $ for different values of $ \lambda $ and $ \alpha $ when $ L $ becomes very large, where $ L $ is the truncated terms from this summation.

    Table 1.  The generated values of $ {\mu }_{X} $ based on the SUM formula and NUI for different values of $ \lambda $ and $ \alpha $ at truncated $ L $ terms.
    $ \lambda $ $ \alpha $ $ L $ SUM NUI
    0.5 2 10 2.35866
    15 2.35800 2.35800
    25 2.35800
    50 2.35800
    3 10 2.84841
    15 2.92946 2.92941
    25 2.92941
    50 2.92941
    4 10 5.79089
    15 3.36228 3.36898
    25 3.36898
    50 3.36898
    0.9 2 10 1.31037
    15 1.31000 1.31000
    25 1.31000
    50 1.31000
    3 10 1.58245
    15 1.62748 1.62745
    25 1.62745
    50 1.62745
    4 10 3.21716
    15 1.86793 1.87166
    25 1.87166
    50 1.87166
    1.9 2 10 0.62070
    15 0.62053 0.62053
    25 0.62053
    50 0.62053
    3 10 0.74958
    15 0.77091 0.77090
    25 0.77090
    50 0.77090
    4 10 1.52392
    15 0.88481 0.88657
    25 0.88657
    50 0.88657

     | Show Table
    DownLoad: CSV

    Furthermore, the $ {\mu }_{X} $, variance ($ {\sigma }_{X}^{2} $), skewness ($ {\psi }_{1} $), and kurtosis ($ {\psi }_{2} $) of the GKME distribution are computed numerically for some values of $ \alpha $ and $ \lambda $ using the R software. The numerical values of the four measures are displayed in Table 2. This table indicates that $ {\mu }_{X} $ and $ {\sigma }_{X}^{2} $ are increasing functions of $ \alpha $, whereas $ {\psi }_{1} $ and $ {\psi }_{2} $ are decreasing functions of $ \alpha $. It is also noted that $ {\psi }_{1} $ can range in the interval (1.2643, 3.1841). The spread of $ {\psi }_{2} $ is much larger, ranging from 5.7854 to 18.1205.

    Table 2.  Mean, variance, skewness, and kurtosis of the GKME distribution for different values of $ \alpha $ and $ \lambda $.
    $ \lambda $ $ \alpha $ $ {\mu }_{x} $ $ {\sigma }_{x}^{2} $ $ {\psi }_{1} $ $ {\psi }_{2} $
    0.5 0.5 0.9244 2.0009 3.1841 18.1205
    1.5 1.9911 3.5881 2.1145 9.8657
    5 3.7270 5.1307 1.5402 6.8509
    10 4.9162 5.7234 1.3677 6.1525
    20 6.1899 6.1038 1.2643 5.7854
    0.75 0.5 0.6163 0.8893 3.1841 18.1205
    1.5 1.3274 1.5947 2.1145 9.8657
    5 2.4847 2.2803 1.5402 6.8509
    10 3.2775 2.5437 1.3677 6.1525
    20 4.1266 2.7128 1.2643 5.7854
    1 0.5 0.4622 0.5002 3.1841 18.1205
    1.5 0.9955 0.8970 2.1145 9.8657
    5 1.8635 1.2827 1.5402 6.8509
    10 2.4581 1.4308 1.3677 6.1525
    20 3.0950 1.5259 1.2643 5.7854
    1.5 0.5 0.3081 0.2223 3.1841 18.1205
    1.5 0.6637 0.3987 2.1145 9.8657
    5 1.2423 0.5701 1.5402 6.8509
    10 1.6387 0.6359 1.3677 6.1525
    20 2.0633 0.6782 1.2643 5.7854
    2.5 0.5 0.1849 0.0800 3.1841 18.1205
    1.5 0.3982 0.1435 2.1145 9.8657
    5 0.7454 0.2052 1.5402 6.8509
    10 0.9832 0.2289 1.3677 6.1525
    20 1.2380 0.2442 1.2643 5.7854

     | Show Table
    DownLoad: CSV

    The QF of the GKME distribution is given by

    $ Q\left(u\right) = \frac{-1}{\lambda }\mathrm{l}\mathrm{o}\mathrm{g}\left\{1+\mathrm{l}\mathrm{o}\mathrm{g}\left[1-{\left(u{\varphi }^{-\alpha }\right)}^{\frac{1}{\alpha }}\right]\right\}. $

    The QF can be used to study the relationships between the parameters ($ \lambda $, $ \alpha $) and the skewness and kurtosis, and the Galton´s skewness and Moors´ kurtosis depend on the QF. Figure 5 displays the Galton´s skewness and the Moors´ kurtosis for the GKME distribution for some parametric values of $ \lambda $ and $ \alpha $.

    Figure 5.  Galton´s skewness and Moors´ kurtosis for the GKME distribution.

    In this section, we use eight methods to estimate the GKME parameters, namely: the maximum likelihood estimators (MLEs), least-squares estimators (LSEs), weighted least-squares estimators (WLSEs), maximum product of spacing estimators (MPSEs), percentiles estimators (PCEs), Cramér-von Mises estimators (CVMEs), Anderson-Darling estimators (ADEs), and right-tail Anderson-Darling estimators (RTADEs).

    Let $ {x}_{1}, \dots, {x}_{n} $ be a random sample from the GKME distribution with parameters $ \alpha $ and $ \lambda $. Let $ {{x}_{1:n} < x}_{2:n} < \dots < {x}_{n:n} $ be the associated order statistics, then, the log-likelihood function has the form

    $ \mathcal{l} = n\mathrm{log}\alpha +n\mathrm{log}\lambda -\lambda \sum\limits_{i = 1}^{n}{x}_{i}+\sum\limits_{i = 1}^{n}{e}^{-\lambda {x}_{i}}+\left(\alpha -1\right)\sum\limits_{i = 1}^{n}\mathrm{log}\left({k}_{i}\right)-n\alpha \mathrm{log}\left(e-1\right), $

    where $ {k}_{i} = e-{e}^{{e}^{-\lambda {x}_{i}}} $. The MLEs of $ \text{α} $ and $ \lambda $ can be obtained by maximizing the last equation with respect to $ \alpha $ and $ \lambda $, or by solving the following nonlinear equations:

    $ \frac{\partial \mathcal{l}}{\partial \alpha } = \frac{n}{\alpha }+\sum\limits_{i = 1}^{n}\mathrm{log}\left({k}_{i}\right)-n\mathrm{log}\left(e-1\right) = 0 $

    and

    $ \frac{\partial \mathcal{l}}{\partial \lambda } = \frac{n}{\lambda }-\sum\limits_{i = 1}^{n}{x}_{i}-\sum\limits_{i = 1}^{n}{x}_{i}{e}^{-\lambda {x}_{i}}+\left(\alpha -1\right)\sum\limits_{i = 1}^{n}\frac{{x}_{i}{e}^{-\lambda {x}_{i}+{e}^{-\lambda {x}_{i}}}}{{k}_{i}} = 0. $

    The MLEs can also be obtained by using different programs such as R (optim function), Mathematica, and SAS (PROC NLMIXED).

    The LS and WLS methods are used to estimate the parameters of the beta distribution [32]. The LSEs and WLSEs of the GKME parameters $ \alpha $ and $ \lambda $ can be obtained by minimizing

    $ V\left(\alpha , \lambda \right) = \sum\limits_{i = 1}^{n}{\upsilon }_{i}{\left[{\left(\frac{e-{e}^{{e}^{-\lambda {x}_{i:n}}}}{e-1}\right)}^{\alpha }-\frac{i}{n+1}\right]}^{2}, $

    with respect to $ \alpha $ and $ \lambda $, where $ {\upsilon }_{i} = 1 $ in case of the LS approach and $ {\upsilon }_{i} = {\left(n+1\right)}^{2}\left(n+2\right)/\left[i\left(n-i+1\right)\right] $ in case of the WLS approach. Furthermore, the LSEs and WLSEs follow by solving the nonlinear equations

    $ \sum\limits_{i = 1}^{n}{\upsilon }_{i}\left[{\left(\frac{e-{e}^{{e}^{-\lambda {x}_{i:n}}}}{e-1}\right)}^{\alpha }-\frac{i}{n+1}\right]{\Delta }_{s}\left({x}_{i:n}⎸\alpha , \lambda \right) = 0, s = 1, 2, $

    where

    $ {\Delta }_{1}\left({x}_{i:n}⎸\alpha , \lambda \right) = \frac{\partial \mathcal{l}}{\partial \alpha }F\left({x}_{i:n}⎸\alpha , \lambda \right) = {\left(\frac{e-{e}^{{e}^{-\lambda {x}_{i:n}}}}{e-1}\right)}^{\alpha }\mathrm{log}\left(\frac{e-{e}^{{e}^{-\lambda {x}_{i:n}}}}{e-1}\right) $ (6.1)

    and

    $ {\Delta }_{2}\left({x}_{i:n}⎸\alpha , \lambda \right) = \frac{\partial \mathcal{l}}{\partial \lambda }F\left({x}_{i:n}⎸\alpha , \lambda \right) = \frac{\alpha {x}_{i:n}{e}^{-\lambda {x}_{i:n}}{e}^{{e}^{-\lambda {x}_{i:n}}}}{e-1}{\left(\frac{e-{e}^{{e}^{-\lambda {x}_{i:n}}}}{e-1}\right)}^{\alpha -1}. $ (6.2)

    The MPS method is used to estimate the parameters of continuous univariate models as an alternative to the ML method [33,34]. The uniform spacings of a random sample of size $ n $ from the GKME distribution can be defined by

    $ {D}_{i} = F\left({x}_{i:n}⎸\alpha , \lambda \right)-F\left({x}_{i-1:n}⎸\alpha , \lambda \right), $

    where $ {D}_{i} $ denotes the uniform spacings, where $ F\left({x}_{0:n}⎸\alpha, \lambda \right) = 0, F\left({x}_{n+1:n}⎸\alpha, \lambda \right) = 1 $, and $ \sum_{i = 1}^{n+1}{D}_{i}\left(\alpha, \lambda \right) = 1 $. The MPSEs of the GKME parameters can be obtained by maximizing

    $ G\left(\alpha , \lambda \right) = \frac{1}{n+1}\sum\limits_{i = 1}^{n+1}\mathrm{log}{D}_{i}\left(\alpha , \lambda \right), $

    with respect to $ \alpha $ and $ \lambda $. Further, the MPSEs of the GKME parameters can also be obtained by solving

    $ \frac{1}{n+1}\sum\limits_{i = 1}^{n+1}\frac{1}{{D}_{i}\left(\alpha , \lambda \right)}\left[{\Delta }_{s}\left({x}_{i:n}⎸\alpha , \lambda \right)-{\Delta }_{s}\left({x}_{i-1:n}⎸\alpha , \lambda \right)\right] = 0, s = 1, 2, $

    where $ {\Delta }_{s}\left({x}_{i:n}⎸\alpha, \lambda \right) = 0 $ are defined in (5.1) and (5.2) for $ s = 1, 2 $.

    The percentile method [35] is used to estimate the unknown parameters of the GKME distribution by equating the sample percentile points with the population percentile points. Let $ {u}_{i} = i/(n+1) $ be an unbiased estimator of $ F\left({x}_{i:n}⎸\alpha, \lambda \right) $, then the PCEs of the GKME parameters are obtained by minimizing the following function:

    $ P\left(\alpha , \lambda \right) = \sum\limits_{i = 1}^{n}{\left({x}_{i:n}-\frac{-1}{\lambda }\mathrm{l}\mathrm{o}\mathrm{g}\left\{1+\mathrm{l}\mathrm{o}\mathrm{g}\left[1-{\left({u}_{i}{\varphi }^{-\alpha }\right)}^{\frac{1}{\alpha }}\right]\right\}\right)}^{2}, $

    with respect to $ \alpha $ and $ \lambda $.

    The CVMEs [36,37] can be obtained based on the difference between the estimates of the CDF and the empirical CDF. The CVMEs of the GKME parameters are obtained by minimizing the following function

    $ C\left(\alpha , \lambda \right) = \frac{1}{12n}+\sum\limits_{i = 1}^{n}{\left[{\left(\frac{e-{e}^{{e}^{-\lambda {x}_{i:n}}}}{e-1}\right)}^{\alpha }-\frac{2i-1}{2n}\right]}^{2}. $

    Further, the CVMEs follow by solving the nonlinear equations

    $ \sum\limits_{i = 1}^{n}\left[{\left(\frac{e-{e}^{{e}^{-\lambda {x}_{i:n}}}}{e-1}\right)}^{\alpha }-\frac{2i-1}{2n}\right]\Delta_{s} \left({x}_{i:n}⎸\alpha , \lambda \right) = 0, $

    where $ {\Delta }_{s}\left({x}_{i:n}⎸\alpha, \lambda \right) = 0 $are defined in (5.1) and (5.2) for $ s = 1, 2 $.

    The ADEs are another type of minimum distance estimators. The ADEs of the GKME parameters are obtained by minimizing

    $ A\left(\alpha , \lambda \right) = -n-\frac{1}{n}\sum\limits_{i = 1}^{n}\left(2i-1\right)\left[\mathrm{log}F\left({x}_{i:n}⎸\alpha , \lambda \right)+\mathrm{log}\stackrel{-}{F}\left({x}_{n+1-i:n}⎸\alpha , \lambda \right)\right], $

    with respect to $ \alpha $ and $ \lambda $. The ADEs can also be determined by solving the nonlinear equations

    $ \sum\limits_{i = 1}^{n}\left(2i-1\right)\left[\frac{{\Delta }_{s}\left({x}_{i:n}⎸\alpha , \lambda \right)}{F\left({x}_{i:n}⎸\alpha , \lambda \right)}-\frac{{\Delta }_{j}\left({x}_{n+1-i:n}⎸\alpha , \lambda \right)}{S\left({x}_{n+1-i:n}⎸\alpha , \lambda \right)}\right] = 0. $

    The RTADEs of the GKME parameters $ \alpha $ and $ \lambda $ are obtained by minimizing the following function, with respect to $ \alpha $ and $ \lambda $,

    $ R\left(\alpha , \lambda \right) = \frac{n}{2}-2\sum\limits_{i = 1}^{n}F\left({x}_{i:n}⎸\alpha , \lambda \right)-\frac{1}{n}\sum\limits_{i = 1}^{n}\left(2i-1\right)\mathrm{log}\stackrel{-}{F}\left({x}_{n+1-i:n}⎸\alpha , \lambda \right). $

    In this section, we assess the performance of all estimation methods of the GKME parameters using a simulation study. We generate 2000 samples from the GKME distribution for different sample sizes of $ n = \left\{20, 50,100,250\right\} $ and different parametric values of $ \alpha = \left(0.5, 0.75, 1.5, 2\right) $ and $ \lambda = \left(0.5, 1.3, 1.5\right) $. We obtain the average values of the estimates (AEs) and mean square errors (MSEs) for each estimate.

    The performance of different estimators is evaluated in terms of MSEs, i.e., the most efficient estimation method will be the one whose MSEs values decay toward zero as the sample size increases. Tables 36 show the AEs and MSEs (in parentheses) of the MLEs, LSEs, WLSEs, MPSEs, PCEs, CVMEs, ADEs, and RTADEs. It is noted that, as the sample size increases, the AEs tend to the true parameter values. Furthermore, the values of the MSEs decay toward zero, indicating that all estimators are asymptotically unbiased. According to the values in these tables, all eight estimation methods perform very well in terms of MSEs.

    Table 3.  The AEs and MSEs of the GKME parameters for $ n = 20 $.
    Par. MLEs LSEs WLSEs MPSEs PCEs CVMEs ADEs RTADEs
    $ \alpha =0.5 $ 0.530(0.009) 0.470(0.012) 0.485(0.01) 0.44(0.01) 0.452(0.045) 0.529(0.012) 0.498(0.009) 0.517(0.012)
    $ \lambda =0.5 $ 0.545(0.019) 0.456(0.03) 0.48(0.026) 0.405(0.023) 0.409(0.046) 0.554(0.031) 0.496(0.022) 0.514(0.023)
    $ \alpha =0.5 $ 0.536(0.009) 0.476(0.012) 0.48(0.01) 0.444(0.01) 0.426(0.05) 0.534(0.012) 0.488(0.009) 0.507(0.012)
    $ \lambda =1.3 $ 1.459(0.144) 1.195(0.201) 1.208(0.18) 1.043(0.171) 1.03(0.337) 1.442(0.197) 1.274(0.156) 1.337(0.17)
    $ \alpha =0.5 $ 0.537(0.01) 0.472(0.012) 0.479(0.01) 0.432(0.01) 0.455(0.045) 0.529(0.013) 0.502(0.009) 0.509(0.012)
    $ \lambda =1.5 $ 1.658(0.188) 1.355(0.267) 1.407(0.24) 1.163(0.23) 1.226(0.431) 1.616(0.26) 1.535(0.213) 1.498(0.22)
    $ \alpha =0.75 $ 0.803(0.024) 0.694(0.029) 0.71(0.026) 0.643(0.028) 0.669(0.084) 0.81(0.035) 0.747(0.026) 0.778(0.031)
    $ \lambda =0.5 $ 0.552(0.017) 0.455(0.026) 0.467(0.019) 0.407(0.02) 0.413(0.038) 0.546(0.024) 0.498(0.019) 0.515(0.02)
    $ \alpha =0.75 $ 0.808(0.023) 0.702(0.031) 0.731(0.027) 0.653(0.027) 0.67(0.085) 0.793(0.032) 0.745(0.027) 0.754(0.031)
    $ \lambda =1.3 $ 1.429(0.115) 1.200(0.161) 1.255(0.141) 1.075(0.127) 1.094(0.238) 1.393(0.158) 1.267(0.113) 1.327(0.129)
    $ \alpha =0.75 $ 0.811(0.022) 0.708(0.029) 0.727(0.028) 0.644(0.027) 0.651(0.089) 0.782(0.032) 0.748(0.023) 0.759(0.032)
    $ \lambda =1.5 $ 1.633(0.147) 1.377(0.217) 1.431(0.201) 1.243(0.166) 1.205(0.345) 1.633(0.225) 1.498(0.153) 1.527(0.181)
    $ \alpha =1.5 $ 1.618(0.121) 1.413(0.157) 1.417(0.152) 1.25(0.138) 1.274(0.314) 1.628(0.18) 1.493(0.12) 1.532(0.181)
    $ \lambda =0.5 $ 0.537(0.012) 0.475(0.017) 0.484(0.013) 0.421(0.016) 0.423(0.024) 0.535(0.016) 0.494(0.011) 0.507(0.015)
    $ \alpha =1.5 $ 1.642(0.122) 1.388(0.17) 1.432(0.144) 1.258(0.142) 1.287(0.294) 1.61(0.184) 1.498(0.121) 1.519(0.18)
    $ \lambda =1.3 $ 1.398(0.082) 1.218(0.118) 1.232(0.106) 1.101(0.094) 1.101(0.146) 1.403(0.124) 1.297(0.089) 1.313(0.095)
    $ \alpha =1.5 $ 1.639(0.119) 1.365(0.177) 1.423(0.154) 1.252(0.134) 1.321(0.284) 1.605(0.18) 1.512(0.125) 1.53(0.189)
    $ \lambda =1.5 $ 1.605(0.103) 1.371(0.155) 1.419(0.14) 1.266(0.137) 1.308(0.188) 1.576(0.154) 1.494(0.121) 1.513(0.14)
    $ \alpha =2 $ 2.197(0.233) 1.824(0.311) 1.868(0.3) 1.633(0.302) 1.756(0.516) 2.214(0.374) 1.973(0.267) 2.133(0.348)
    $ \lambda =0.5 $ 0.536(0.011) 0.463(0.014) 0.474(0.013) 0.422(0.014) 0.438(0.017) 0.533(0.015) 0.493(0.012) 0.516(0.013)
    $ \alpha =2 $ 2.246(0.265) 1.872(0.347) 1.899(0.299) 1.639(0.287) 1.683(0.539) 2.188(0.339) 1.984(0.256) 2.076(0.351)
    $ \lambda =1.3 $ 1.395(0.072) 1.215(0.103) 1.253(0.083) 1.112(0.092) 1.116(0.128) 1.394(0.103) 1.292(0.076) 1.346(0.09)
    $ \alpha =2 $ 2.196(0.233) 1.854(0.327) 1.891(0.298) 1.631(0.27) 1.702(0.554) 2.131(0.353) 2.000(0.27) 2.042(0.336)
    $ \lambda =1.5 $ 1.609(0.099) 1.409(0.133) 1.445(0.115) 1.271(0.119) 1.277(0.194) 1.567(0.143) 1.480(0.1) 1.518(0.118)

     | Show Table
    DownLoad: CSV
    Table 4.  The AEs and MSEs of the GKME parameters for $ n = 50 $.
    Par. MLEs LSEs WLSEs MPSEs PCEs CVMEs ADEs RTADEs
    $ \alpha =0.5 $ 0.511(0.003) 0.489(0.005) 0.493(0.004) 0.466(0.004) 0.461(0.025) 0.51(0.005) 0.497(0.004) 0.505(0.005)
    $ \lambda =0.5 $ 0.519(0.008) 0.491(0.012) 0.487(0.009) 0.444(0.009) 0.431(0.02) 0.519(0.013) 0.496(0.009) 0.499(0.01)
    $ \alpha =0.5 $ 0.517(0.004) 0.486(0.004) 0.496(0.004) 0.467(0.004) 0.449(0.024) 0.511(0.005) 0.501(0.004) 0.505(0.005)
    $ \lambda =1.3 $ 1.344(0.051) 1.24(0.077) 1.265(0.059) 1.156(0.071) 1.107(0.159) 1.359(0.082) 1.285(0.06) 1.309(0.067)
    $ \alpha =0.5 $ 0.516(0.003) 0.488(0.005) 0.498(0.004) 0.466(0.004) 0.434(0.023) 0.512(0.005) 0.504(0.003) 0.502(0.005)
    $ \lambda =1.5 $ 1.56(0.066) 1.451(0.116) 1.479(0.086) 1.309(0.096) 1.24(0.215) 1.536(0.105) 1.487(0.08) 1.518(0.093)
    $ \alpha =0.75 $ 0.77(0.009) 0.731(0.013) 0.744(0.01) 0.685(0.01) 0.679(0.046) 0.77(0.013) 0.749(0.01) 0.759(0.013)
    $ \lambda =0.5 $ 0.521(0.007) 0.483(0.011) 0.502(0.008) 0.442(0.008) 0.439(0.016) 0.516(0.009) 0.503(0.007) 0.501(0.007)
    $ \alpha =0.75 $ 0.773(0.009) 0.728(0.013) 0.74(0.011) 0.686(0.011) 0.665(0.045) 0.767(0.012) 0.746(0.01) 0.76(0.012)
    $ \lambda =1.3 $ 1.348(0.041) 1.264(0.058) 1.283(0.051) 1.158(0.054) 1.126(0.107) 1.34(0.059) 1.293(0.048) 1.33(0.055)
    $ \alpha =0.75 $ 0.769(0.008) 0.721(0.013) 0.742(0.01) 0.696(0.01) 0.671(0.045) 0.766(0.013) 0.75(0.009) 0.762(0.015)
    $ \lambda =1.5 $ 1.542(0.051) 1.424(0.085) 1.48(0.073) 1.339(0.071) 1.304(0.141) 1.536(0.083) 1.492(0.06) 1.516(0.075)
    $ \alpha =1.5 $ 1.539(0.043) 1.464(0.07) 1.459(0.056) 1.354(0.056) 1.347(0.149) 1.542(0.072) 1.488(0.048) 1.533(0.07)
    $ \lambda =0.5 $ 0.514(0.004) 0.489(0.007) 0.49(0.006) 0.456(0.006) 0.455(0.01) 0.508(0.007) 0.493(0.005) 0.508(0.005)
    $ \alpha =1.5 $ 1.549(0.045) 1.452(0.065) 1.477(0.053) 1.34(0.059) 1.337(0.16) 1.538(0.071) 1.498(0.049) 1.514(0.078)
    $ \lambda =1.3 $ 1.328(0.027) 1.273(0.042) 1.283(0.038) 1.171(0.035) 1.165(0.07) 1.341(0.045) 1.299(0.032) 1.306(0.043)
    $ \alpha =1.5 $ 1.558(0.048) 1.437(0.067) 1.475(0.054) 1.361(0.052) 1.335(0.157) 1.546(0.08) 1.489(0.047) 1.521(0.076)
    $ \lambda =1.5 $ 1.557(0.04) 1.448(0.062) 1.48(0.049) 1.36(0.048) 1.346(0.091) 1.528(0.063) 1.485(0.044) 1.526(0.055)
    $ \alpha =2 $ 2.082(0.096) 1.934(0.128) 1.969(0.104) 1.803(0.108) 1.804(0.245) 2.072(0.129) 1.983(0.103) 2.041(0.142)
    $ \lambda =0.5 $ 0.516(0.004) 0.493(0.006) 0.497(0.004) 0.46(0.005) 0.454(0.008) 0.516(0.006) 0.5(0.005) 0.508(0.006)
    $ \alpha =2 $ 2.117(0.096) 1.928(0.138) 1.968(0.113) 1.797(0.104) 1.763(0.279) 2.066(0.141) 1.997(0.106) 2.029(0.148)
    $ \lambda =1.3 $ 1.357(0.028) 1.264(0.041) 1.277(0.033) 1.194(0.031) 1.188(0.058) 1.329(0.044) 1.303(0.033) 1.312(0.037)
    $ \alpha =2 $ 2.082(0.093) 1.921(0.137) 1.962(0.115) 1.792(0.108) 1.752(0.27) 2.051(0.133) 1.984(0.102) 2.009(0.149)
    $ \lambda =1.5 $ 1.551(0.041) 1.452(0.056) 1.483(0.042) 1.364(0.046) 1.354(0.078) 1.55(0.054) 1.502(0.042) 1.512(0.045)

     | Show Table
    DownLoad: CSV
    Table 5.  The AEs and MSEs of the GKME parameters for $ n = 100 $.
    Par. MLEs LSEs WLSEs MPSEs PCEs CVMEs ADEs RTADEs
    $ \alpha =0.5 $ 0.506(0.002) 0.496(0.002) 0.501(0.002) 0.481(0.002) 0.451(0.014) 0.504(0.002) 0.501(0.002) 0.504(0.003)
    $ \lambda =0.5 $ 0.507(0.004) 0.493(0.006) 0.502(0.005) 0.466(0.005) 0.446(0.012) 0.505(0.006) 0.506(0.005) 0.502(0.005)
    $ \alpha =0.5 $ 0.509(0.002) 0.493(0.002) 0.501(0.002) 0.477(0.002) 0.456(0.014) 0.508(0.003) 0.502(0.002) 0.5(0.003)
    $ \lambda =1.3 $ 1.331(0.029) 1.278(0.036) 1.296(0.031) 1.207(0.035) 1.161(0.081) 1.319(0.042) 1.304(0.033) 1.306(0.029)
    $ \alpha =0.5 $ 0.508(0.002) 0.495(0.002) 0.499(0.002) 0.475(0.002) 0.457(0.014) 0.503(0.002) 0.5(0.002) 0.502(0.003)
    $ \lambda =1.5 $ 1.531(0.039) 1.462(0.052) 1.489(0.044) 1.386(0.043) 1.349(0.099) 1.513(0.054) 1.514(0.04) 1.498(0.04)
    $ \alpha =0.75 $ 0.76(0.004) 0.743(0.006) 0.749(0.005) 0.714(0.005) 0.685(0.026) 0.757(0.006) 0.747(0.005) 0.753(0.007)
    $ \lambda =0.5 $ 0.507(0.003) 0.491(0.005) 0.499(0.004) 0.468(0.003) 0.454(0.008) 0.509(0.005) 0.497(0.004) 0.5(0.004)
    $ \alpha =0.75 $ 0.759(0.004) 0.736(0.006) 0.746(0.005) 0.716(0.005) 0.678(0.027) 0.751(0.006) 0.751(0.005) 0.753(0.007)
    $ \lambda =1.3 $ 1.325(0.019) 1.273(0.033) 1.288(0.024) 1.217(0.025) 1.173(0.061) 1.323(0.029) 1.291(0.023) 1.308(0.026)
    $ \alpha =0.75 $ 0.760(0.004) 0.737(0.007) 0.749(0.005) 0.712(0.005) 0.683(0.024) 0.758(0.006) 0.752(0.005) 0.754(0.007)
    $ \lambda =1.5 $ 1.522(0.028) 1.466(0.041) 1.489(0.037) 1.408(0.034) 1.354(0.075) 1.533(0.041) 1.512(0.032) 1.506(0.037)
    $ \alpha =1.5 $ 1.527(0.022) 1.479(0.03) 1.493(0.027) 1.403(0.027) 1.368(0.078) 1.516(0.035) 1.502(0.024) 1.508(0.036)
    $ \lambda =0.5 $ 0.508(0.002) 0.493(0.003) 0.499(0.003) 0.47(0.003) 0.464(0.005) 0.503(0.003) 0.498(0.002) 0.501(0.003)
    $ \alpha =1.5 $ 1.52(0.024) 1.46(0.032) 1.492(0.027) 1.411(0.026) 1.368(0.094) 1.509(0.035) 1.493(0.026) 1.521(0.039)
    $ \lambda =1.3 $ 1.323(0.016) 1.272(0.019) 1.292(0.019) 1.229(0.017) 1.212(0.035) 1.316(0.024) 1.295(0.018) 1.309(0.021)
    $ \alpha =1.5 $ 1.532(0.023) 1.485(0.035) 1.499(0.028) 1.405(0.027) 1.368(0.093) 1.518(0.034) 1.487(0.025) 1.512(0.042)
    $ \lambda =1.5 $ 1.525(0.019) 1.492(0.029) 1.498(0.023) 1.411(0.025) 1.392(0.049) 1.519(0.03) 1.492(0.024) 1.493(0.025)
    $ \alpha =2 $ 2.03(0.045) 1.957(0.065) 1.999(0.051) 1.875(0.06) 1.81(0.147) 2.049(0.073) 2.0(0.054) 2.018(0.064)
    $ \lambda =0.5 $ 0.508(0.002) 0.492(0.003) 0.499(0.002) 0.473(0.003) 0.467(0.004) 0.509(0.003) 0.499(0.002) 0.505(0.002)
    $ \alpha =2 $ 2.027(0.043) 1.962(0.066) 1.993(0.057) 1.874(0.052) 1.817(0.141) 2.039(0.064) 1.992(0.047) 2.015(0.072)
    $ \lambda =1.3 $ 1.316(0.013) 1.288(0.022) 1.299(0.017) 1.231(0.016) 1.221(0.026) 1.313(0.02) 1.294(0.015) 1.311(0.018)
    $ \alpha =2 $ 2.023(0.042) 1.965(0.072) 1.98(0.059) 1.856(0.054) 1.821(0.149) 2.026(0.066) 1.984(0.047) 2.033(0.077)
    $ \lambda =1.5 $ 1.519(0.018) 1.477(0.027) 1.493(0.021) 1.411(0.022) 1.397(0.038) 1.523(0.023) 1.49(0.018) 1.518(0.024)

     | Show Table
    DownLoad: CSV
    Table 6.  The AEs and MSEs of the GKME parameters for $ n = 250 $.
    Par. MLEs LSEs WLSEs MPSEs PCEs CVMEs ADEs RTADEs
    $ \alpha =0.5 $ 0.502(0.001) 0.499(0.001) 0.499(0.001) 0.49(0.001) 0.466(0.006) 0.506(0.001) 0.499(0.001) 0.501(0.001)
    $ \lambda =0.5 $ 0.506(0.002) 0.495(0.003) 0.498(0.002) 0.485(0.002) 0.468(0.005) 0.505(0.002) 0.501(0.002) 0.499(0.002)
    $ \alpha =0.5 $ 0.503(0.001) 0.497(0.001) 0.501(0.001) 0.487(0.001) 0.468(0.006) 0.503(0.001) 0.5(0.001) 0.502(0.001)
    $ \lambda =1.3 $ 1.312(0.011) 1.298(0.016) 1.305(0.013) 1.25(0.012) 1.205(0.034) 1.313(0.016) 1.302(0.013) 1.304(0.014)
    $ \alpha =0.5 $ 0.504(0.001) 0.499(0.001) 0.499(0.001) 0.488(0.001) 0.468(0.006) 0.503(0.001) 0.499(0.001) 0.500(0.001)
    $ \lambda =1.5 $ 1.508(0.014) 1.497(0.022) 1.499(0.017) 1.442(0.015) 1.402(0.04) 1.514(0.021) 1.5(0.017) 1.496(0.017)
    $ \alpha =0.75 $ 0.753(0.002) 0.745(0.003) 0.751(0.002) 0.729(0.002) 0.703(0.012) 0.756(0.003) 0.749(0.002) 0.751(0.003)
    $ \lambda =0.5 $ 0.503(0.001) 0.499(0.002) 0.502(0.001) 0.481(0.001) 0.47(0.003) 0.506(0.002) 0.499(0.001) 0.501(0.002)
    $ \alpha =0.75 $ 0.754(0.002) 0.745(0.003) 0.753(0.002) 0.729(0.002) 0.709(0.012) 0.753(0.003) 0.751(0.002) 0.750(0.003)
    $ \lambda =1.3 $ 1.307(0.008) 1.294(0.013) 1.307(0.01) 1.254(0.009) 1.233(0.022) 1.314(0.013) 1.301(0.01) 1.303(0.011)
    $ \alpha =0.75 $ 0.753(0.002) 0.746(0.002) 0.749(0.002) 0.729(0.002) 0.708(0.011) 0.75(0.003) 0.752(0.002) 0.754(0.003)
    $ \lambda =1.5 $ 1.508(0.011) 1.497(0.018) 1.501(0.012) 1.441(0.013) 1.422(0.032) 1.51(0.018) 1.505(0.014) 1.506(0.016)
    $ \alpha =1.5 $ 1.518(0.009) 1.497(0.016) 1.499(0.01) 1.444(0.011) 1.418(0.039) 1.508(0.013) 1.497(0.01) 1.513(0.015)
    $ \lambda =0.5 $ 0.505(0.001) 0.497(0.001) 0.5(0.001) 0.482(0.001) 0.479(0.002) 0.501(0.001) 0.5(0.001) 0.502(0.001)
    $ \alpha =1.5 $ 1.504(0.009) 1.5(0.014) 1.504(0.01) 1.455(0.01) 1.424(0.04) 1.513(0.014) 1.496(0.01) 1.502(0.015)
    $ \lambda =1.3 $ 1.301(0.006) 1.298(0.009) 1.302(0.006) 1.269(0.007) 1.252(0.014) 1.31(0.008) 1.297(0.007) 1.296(0.008)
    $ \alpha =1.5 $ 1.515(0.009) 1.498(0.013) 1.497(0.011) 1.46(0.01) 1.416(0.041) 1.502(0.014) 1.501(0.01) 1.508(0.014)
    $ \lambda =1.5 $ 1.509(0.007) 1.495(0.011) 1.502(0.009) 1.456(0.009) 1.433(0.018) 1.507(0.012) 1.501(0.01) 1.502(0.01)
    $ \alpha =2 $ 2.016(0.018) 1.991(0.03) 1.994(0.021) 1.929(0.02) 1.901(0.063) 2.003(0.029) 1.993(0.022) 2.005(0.03)
    $ \lambda =0.5 $ 0.503(0.001) 0.499(0.001) 0.499(0.001) 0.487(0.001) 0.483(0.002) 0.501(0.001) 0.499(0.001) 0.5(0.001)
    $ \alpha =2 $ 2.024(0.019) 1.979(0.027) 1.997(0.021) 1.941(0.018) 1.89(0.063) 2.022(0.03) 1.997(0.02) 2.005(0.03)
    $ \lambda =1.3 $ 1.307(0.005) 1.293(0.008) 1.301(0.006) 1.261(0.006) 1.249(0.011) 1.31(0.008) 1.299(0.006) 1.3(0.007)
    $ \alpha =2 $ 2.02(0.019) 1.981(0.028) 2.011(0.02) 1.938(0.021) 1.89(0.071) 2.010(0.026) 2.007(0.022) 2.002(0.029)
    $ \lambda =1.5 $ 1.513(0.007) 1.485(0.011) 1.503(0.008) 1.457(0.008) 1.445(0.015) 1.511(0.01) 1.500(0.008) 1.499(0.009)

     | Show Table
    DownLoad: CSV

    In this section, we present five applications of real-life data from medicine, environmental, and reliability fields to illustrate the flexibility of the GKME model. The ML method is used to estimate the parameters of each model and R statistical software is used for computations. We compare the fitting performance of the GKME with other competing E models, namely: The GE [27], GIE [28], MOE [1], APE [29], GDUSE [30], KME [13], and E distributions.

    To compare the competing distributions with the proposed GKME model, we calculate some goodness-of-fit statistics, including the Cramér-von Mises $ \left({W}^{\mathrm{*}}\right) $, Anderson-Darling $ \left({A}^{\mathrm{*}}\right) $, and Kolmogorov-Smirnov (KS) statistics with its $ p $-value.

    The first set of data is studied by Murthy et al. [38], and it represents the time between failures for a repairable item. The data observations are: 1.43, 0.11, 0.71, 0.77, 2.63, 1.49, 3.46, 2.46, 0.59, 0.74, 1.23, 0.94, 4.36, 0.40, 1.74, 4.73, 2.23, 0.45, 0.70, 1.06, 1.46, 0.30, 1.82, 2.37, 0.63, 1.23, 1.24, 1.97, 1.86, 1.17.

    The second set of data represents the waiting times (in minutes) before the service of 100 bank customers. This data set is previously studied by Ghitany et al. [39]. The data observations are: 0.8, 0.8, 3.2, 3.3, 4.6, 4.7, 6.2, 6.2, 7.7, 8, 9.7, 9.8, 12.5, 12.9, 17.3, 18.1, 27, 31.6, 1.3, 3.5, 4.7, 6.2, 8.2, 10.7, 13, 18.2, 33.1, 1.5, 1.8, 1.9, 3.6, 4, 4.1, 4.8, 4.9, 4.9, 6.3, 6.7, 6.9, 8.6, 8.6, 8.6, 10.9, 11, 11, 13, 13.3, 13.6, 18.4, 18.9, 19, 38.5, 1.9, 2.1, 2.6, 4.2, 4.2, 4.3, 5, 5.3, 5.5, 7.1, 7.1, 7.1, 8.8, 8.8, 8.9, 11.1, 11.2, 11.2, 13.7, 13.9, 14.1, 19.9, 20.6, 21.3, 2.7, 2.9, 3.1, 4.3, 4.4, 4.4, 5.7, 5.7, 6.1, 7.1, 7.4, 7.6, 8.9, 9.5, 9.6, 11.5, 11.9, 12.4, 15.4, 15.4, 17.3, 21.4, 21.9, 23.

    The third set of data is studied by Dumonceaux and Antle [40] and it consists of annual maximum flood levels (in millions of cubic feet per second) over a 20-year period of the Susquehanna River at Harrisburg, Pennsylvania. The data observations are: 0.654, 0.613, 0.315, 0.449, 0.297, 0.402, 0.379, 0.423, 0.379, 0.324, 0.269, 0.740, 0.418, 0.412, 0.494, 0.416, 0.338, 0.392, 0.484, 0.265.

    The fourth set of the data is analyzed by Mann [41], and it represents the number of vehicle fatalities for 39 counties in South Carolina in 2012. The data observations are: 22, 26, 17, 4, 48, 9, 9, 31, 27, 20, 12, 6, 5, 14, 9, 16, 3, 33, 9, 20, 68, 13, 51, 13, 2, 4, 17, 16, 6, 52, 50, 48, 23, 12, 13, 10, 15, 8, 1.

    The fifth set of data is given by Lee and Wang [42], and it represents the remission times (in months) of a random sample of 128 bladder cancer patients. The data observations are: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 6.97, 9.02, 13.29, 0.40, 2.26, 3.57, 5.06, 7.09, 9.22, 13.80, 25.74, 0.50, 2.46, 3.64, 5.09, 7.26, 9.47, 14.24, 25.82, 0.51, 2.54, 3.70, 5.17, 7.28, 9.74, 14.76, 26.31, 0.81, 2.62, 3.82, 5.32, 7.32, 10.06, 14.77, 32.15, 2.64, 3.88, 5.32, 7.39, 10.34, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 1.05, 2.69, 4.23, 5.41, 7.62, 10.75, 16.62, 43.01, 1.19, 2.75, 4.26, 5.41, 7.63, 17.12, 46.12, 1.26, 2.83, 4.33, 5.49, 7.66, 11.25, 17.14, 79.05, 1.35, 2.87, 5.62, 7.87, 11.64, 17.36, 1.40, 3.02, 4.34, 5.71, 7.93, 11.79, 18.10, 1.46, 4.40, 5.85, 8.26, 11.98, 19.13, 1.76, 3.25, 4.50, 6.25, 8.37, 12.02, 2.02, 3.31, 4.51, 6.54, 8.53, 12.03, 20.28, 2.02, 3.36, 6.76, 12.07, 21.73, 2.07, 3.36, 6.93, 8.65, 12.63, 22.69.

    Tables 711 provide the values of $ {W}^{\mathrm{*}} $, $ {A}^{\mathrm{*}} $, KS, and the KS $ p $-value of the fitted models for the five datasets, respectively. Further, Tables 711 display the MLEs and standard errors (SEs) (appear in parentheses) of the parameters of the GKME, GE, GDUSE, MOE, APE, GIE, E, and KME models. The values in these tables indicate that the GKME distribution has the lowest values of $ {W}^{\mathrm{*}} $, $ {A}^{\mathrm{*}} $, and KS statistics and largest $ p $-value, among all fitted models. The fitted functions of the GKME model including the fitted PDF, CDF, SF, and probability-probability (PP) plots for all datasets are displayed in Figures 68, respectively.

    Table 7.  The findings of the fitted distributions for failures times data.
    Distribution Estimates (SEs) $ {W}^{*} $ $ {A}^{*} $ KS $ p $-value
    GKME $ \alpha $ 2.4499 (0.7020) 0.0170 0.1233 0.0626 0.9996
    $ \lambda $ 0.9103 (0.2017)
    GE $ \alpha $ 1.9724 (0.5241) 0.0209 0.1425 0.0681 0.9984
    $ \lambda $ 1.0317 (0.2051)
    GDUSE $ \alpha $ 1.6627 (0.4901) 0.0301 0.1952 0.0823 0.9818
    $ \lambda $ 1.1436 (0.2178)
    MOE $ \alpha $ 3.2566 (1.8757) 0.0530 0.3341 0.0837 0.9784
    $ \lambda $ 1.1985 (0.3041)
    APE $ \alpha $ 13.999 (16.1660) 0.0404 0.2590 0.0851 0.9746
    $ \lambda $ 1.1822 (0.2417)
    GIE $ \alpha $ 3. 0.990 (0.2161) 0.1228 0.8122 0.1460 0.5026
    $ \lambda $ 1.6293 (0.4407)
    E $ \lambda $ 0.6914 (0.1222) 0.0246 0.1645 0.1787 0.2583
    KME $ \lambda $ 0.4985 (0.0998) 0.0182 0.1276 0.1996 0.1562

     | Show Table
    DownLoad: CSV
    Table 8.  The findings of the fitted distributions for waiting times data.
    Distribution Estimates (SEs) $ {W}^{*} $ $ {A}^{*} $ KS $ p $-value
    GKME $ \alpha $ 2.7513 (0.4562) 0.0170 0.1322 0.0385 0.9984
    $ \lambda $ 0.1417 (0.0174)
    GE $ \alpha $ 2.1837 (0.3343) 0.0208 0.1431 0.0402 0.9970
    $ \lambda $ 0.1592 (0.0175)
    GDUSE $ \alpha $ 1.8592 (0.3156) 0.0384 0.2423 0.0492 0.9691
    $ \lambda $ 0.1764 (0.0185)
    MOE $ \alpha $ 4.1167 (1.3542) 0.1073 0.6563 0.0597 0.8687
    $ \lambda $ 0.1924 (0.0260)
    APE $ \alpha $ 21.1797 (14.1717) 0.0667 0.4179 0.0528 0.9430
    $ \lambda $ 0.1831 (0.0197)
    GIE $ \alpha $ 7.8532 (0.9383) 0.2716 1.8253 0.1075 0.1979
    $ \lambda $ 1.8662 (0.2924)
    E $ \lambda $ 0.1013 (0.0101) 0.0271 0.1794 0.1730 0.0050
    KME $ \lambda $ 0.0725(0.0082) 0.0176 0.1280 0.1920 0.0013

     | Show Table
    DownLoad: CSV
    Table 9.  The findings of the fitted distributions for flood levels data.
    Distribution Estimates (SEs) $ {W}^{*} $ $ {A}^{*} $ KS $ p $-value
    GKME $ \alpha $ 95.5974(71.3536) 0.04648 0.2851 0.1217 0.9284
    $ \lambda $ 10.9424 (2.0872)
    GE $ \alpha $ 57.5708(42.0625) 0.0465 0.2861 0.1218 0.9281
    $ \lambda $ 11.0157(2.0564)
    GDUSE $ \alpha $ 61.2250 (49.1245) 0.0499 0.3129 0.1311 0.8821
    $ \lambda $ 12.0039 (2.2007)
    MOE $ \alpha $ 480.3879(600.3460) 0.0928 0.5943 0.1433 0.8061
    $ \lambda $ 15.1397(2.9917)
    APE $ \alpha $ 14860830 (23726.57) 0.0566 0.3562 0.1544 0.7270
    $ \lambda $ 7.9302 (0.6336)
    GIE $ \alpha $ 1.6448 (0.2888) 0.0691 0.4325 0.1584 0.6974
    $ \lambda $ 37.7007 (23.1976)
    E $ \alpha $ 2.3632 (0.5284) 0.0741 0.4620 0.4654 0.0003
    KME $ \lambda $ 1.5806 (0.4018) 0.0683 0.4266 0.4585 0.0004

     | Show Table
    DownLoad: CSV
    Table 10.  The findings of the fitted distributions for vehicle fatalities data.
    Distribution Estimates (SEs) $ {W}^{*} $ $ {A}^{*} $ KS $ p $-value
    GKME $ \alpha $ 1.9136 (0.4726) 0.0355 0.2572 0.08720 0.9281
    $ \lambda $ 0.0589 (0.0124)
    GE $ \alpha $ 1.5726 (0.3616) 0.0465 0.3228 0.0929 0.8893
    $ \lambda $ 0.0674 (0.0128)
    GDUSE $ \alpha $ 1.3016 (0.3328) 0.0640 0.4288 0.1079 0.7543
    $ \lambda $ 0.0746 (0.0135)
    MOE $ \alpha $ 2.0222 (1.0401) 0.0779 0.5099 0.0992 0.8374
    $ \lambda $ 0.0718 (0.0183)
    APE $ \alpha $ 4.9155 (4.9562) 0.0739 0.4902 0.1033 0.7998
    $ \lambda $ 0.0731 (0.01637)
    GIE $ \alpha $ 9.4533 (1.9872) 0.1905 1.1852 0.1826 0.1486
    $ \lambda $ 1.2388 (0.2864)
    E $ \alpha $ 0.0512 (0.0082) 0.0505 0.3453 0.1383 0.4443
    KME $ \lambda $ 0.0375 (0.0068) 0.0379 0.2721 0.1636 0.2477

     | Show Table
    DownLoad: CSV
    Table 11.  The findings of the fitted distributions for bladder cancer data.
    Distribution Estimates (SEs) $ {W}^{*} $ $ {A}^{*} $ KS $ p $-value
    GKME $ \alpha $ 1.4334(0.1863) 0.0660 0.4133 0.0565 0.8086
    $ \lambda $ 0.1034(0.0128)
    GE $ \alpha $ 1.2179(0.1488) 0.1122 0.6741 0.0725 0.5113
    $ \lambda $ 0.1212(0.0136)
    GDUSE $ \alpha $ 0.9879(0.1346) 0.2038 1.2876 0.7136 0.000
    $ \lambda $ 0.1343(0.0144)
    MOE $ \alpha $ 1.0558(0.3216) 0.1254 0.7510 0.0811 0.3685
    $ \lambda $ 0.1099(0.0199)
    APE $ \alpha $ 1.1744(0.8437) 0.1283 0.7672 0.0793 0.3963
    $ \lambda $ 0.1113(0.0226)
    GIE $ \alpha $ 1.9945 (0.2705) 1.1615 6.8690 0.2067 0.0000
    $ \lambda $ 0.7463 (0.0883)
    E $ \lambda $ 0.1068(0.0094) 0.1193 0.7160 0.0846 0.3184
    KME $ \lambda $ 0.0797 (0.0079) 0.0707 0.4384 0.1092 0.0943

     | Show Table
    DownLoad: CSV
    Figure 6.  The fitted GKME PDF, CDF, SF, and PP plots for failures times data (left panel) and waiting times data (right panel).
    Figure 7.  The fitted GKME PDF, CDF, SF, and PP plots for flood levels data (left panel) and vehicle fatalities data (right panel).
    Figure 8.  The fitted GKME PDF, CDF, SF, and PP plots for bladder cancer data.

    We proposed a new class of continuous distributions called the GKM-G family. The GKM-G family generalizes the KM family and provides greater flexibility. Some special models of the GKM-G family are provided. Some of its basic properties are studied. Eight methods are used for estimating the parameters of the GKME distribution. The performance of the estimators is assessed by simulation studies for small and large samples. Our study shows that all considered estimation approaches are consistent. Five real-life data applications from medicine, environment, and reliability fields are analyzed to illustrate the flexibility of the GKME distribution. These applications indicate that the GKME distribution provides a better fit as compared to other existing exponential distributions.

    For a possible direction of future works, the research in this article can be extended in some ways. For example, construction of autoregressive processes based on the special sub-models of the GKM-G family, constructing regression models by exploiting the flexibility of the GKME distribution, and a discrete version of the GKME distribution may be established. Furthermore, considering the works of Alsadat et al. [43] and Tolba et al. [44], the parameters of the GKME distribution can be explored using the Bayesian approach under complete and censored samples. Additionally, considering the work of Chinedu et al. [45], the GKME distribution and other special models of the family may have some applications to the single acceptance sampling plan under different scenarios of failure rates. Also, the detailed study of theoretical statistical properties that distinguishes the GKME distribution and makes it distinct from some corresponding distributions such as the skewness and kurtosis can be addressed according to the works of Barakat [46] and Barakat and Khaled [47].

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia for funding this research work through project number 445-9-854.

    The authors also would like to thank the Editor and the reviewers for their constructive comments and suggestions, which greatly improved the paper.

    The authors declare no conflict of interest.

    [1] Huttenhower C, Gevers D, Knight R, et al. (2012) Structure, function and diversity of the healthy human microbiome. Nature 486: 207–214. doi: 10.1038/nature11234
    [2] Backhed F (2011) Programming of host metabolism by the gut microbiota. Ann Nutr Metab 58: 44–52. doi: 10.1159/000328042
    [3] Cani P, Everard A (2015) Talking microbes: when gut bacteria interact with diet and host organs. Mol Nutr Food Res 60: 58–66.
    [4] Cotillard A, Kennedy S, Kong L, et al. (2013) Dietary intervention impact on gut microbial gene richness. Nature 500: 585–588. doi: 10.1038/nature12480
    [5] Wu G, Compher C, Chen E, et al. (2014) Comparative metabolomics in vegans and omnivores reveal constraints on diet-dependent gut microbiota metabolite production. Gut 65: 63–72.
    [6] Lim M, Rho M, Song Y, et al. (2014) Stability of gut enterotypes in korean monozygotic twins and their association with biomarkers and diet. Sci Rep 4: 7348.
    [7] Org E, Parks B, Joo J, et al. (2015) Genetic and environmental control of host-gut microbiota interactions. Genome Res 25: 1558–1569. doi: 10.1101/gr.194118.115
    [8] Tilg H, Cani P, Mayer E, et al. (2016) Gut microbiome and liver diseases. Gut 65: 2035–2044. doi: 10.1136/gutjnl-2016-312729
    [9] Tabibian J, Varghese C, LaRusso N, et al. (2015) The enteric microbiome in hepatobiliary health and disease. Liver Int 36: 480–487.
    [10] Lazaridis K, LaRusso N (2016) Primary sclerosing cholangitis. New Engl J Med 375: 2500–2502. doi: 10.1056/NEJMc1613273
    [11] Trivedi P, Adams D (2016) Gut-liver immunity. J Hepatol 64: 1187–1189. doi: 10.1016/j.jhep.2015.12.002
    [12] Kummen M, Holm K, Anmarkrud J, et al. (2016) The gut microbial profile in patients with primary sclerosing cholangitis is distinct from patients with ulcerative colitis without biliary disease and healthy controls. Gut 66: 611–619.
    [13] Bajer L, Kverka M, Kostovcik M, et al. (2017) Distinct gut microbiota profiles in patients with primary sclerosing cholangitis and ulcerative colitis. World J Gastroentero 23: 4548. doi: 10.3748/wjg.v23.i25.4548
    [14] Sabino J, Vieira-Silva S, Machiels K, et al. (2016) Primary sclerosing cholangitis is characterised by intestinal dysbiosis independent from IBD. Gut 65: 1681–1689. doi: 10.1136/gutjnl-2015-311004
    [15] Tabibian J, O-Hara S, Trussoni C, et al. (2015) Absence of the intestinal microbiota exacerbates hepatobiliary disease in a murine model of primary sclerosing cholangitis. Hepatology 63: 185–196.
    [16] Tabibian J, O'Hara S, Splinter P, et al. (2014) Cholangiocyte senescence by way of n-ras activation is a characteristic of primary sclerosing choalngitis. Hepatology 59: 2263–2275. doi: 10.1002/hep.26993
    [17] Tabibian J, O'Hara S, Lindor K (2014) Primary sclerosing cholangitis and the microbiota: current knowledge and perspectives on etiopathogenesis and emerging therapies. Scand J Gastroentero 49: 901–908.
    [18] Eaton J, Juran B, Atkinson E, et al. (2015) A comprehensive assessment of environmental exposures among 1000 north american patients with primary sclerosing cholangitis, with and without inflammatory bowel disease. Aliment Pharm Therap 41: 980–990. doi: 10.1111/apt.13154
    [19] Lichtman S, Keku J, Schwab J, et al. (1991) Hepatic injury associated with small bowel bacterial overgrowth in rats is prevented by metronidazole and tetracycline. Gastroenterology 100: 513–519. doi: 10.1016/0016-5085(91)90224-9
    [20] Tabibian J, Talwalkar J, Lindor K (2013) Role of the microbiota and antibiotics in primary sclerosing cholangitis. Biomed Res Int 2013: 1–7.
    [21] Tabibian J, Weeding E, Jorgensen R, et al. (2013) Randomised clinical trial: vancomycin or metronidazole in patients with primary sclerosing cholangitis-a pilot study. Aliment Pharm Therap 37: 604–612. doi: 10.1111/apt.12232
    [22] Mistilis S, Skyring A, Goulston S (1965) Effect of long-term tetracycline therapy, steroid therapy and colectomy in pericholangitis associated with ulcerative colitis. Australas Ann Med 14: 286–294. doi: 10.1111/imj.1965.14.4.286
    [23] Färkkilä M, Karvonen A, Nurmi H, et al. (2004) Metronidazole and ursodeoxycholic acid for primary sclerosing cholangitis: a randomized placebo-controlled trial. Hepatology 40: 1379–1386. doi: 10.1002/hep.20457
    [24] Silveira M, Torok N, Gossard A, et al. (2009) Minocycline in the treatment of patients with primary sclerosing cholangitis: results of a pilot study. Am J Gastroenterol 104: 83–88. doi: 10.1038/ajg.2008.14
    [25] Rankin J, Sydney M, Boden R, et al. (1959) The liver in ulcerative colitis; treatment of pericholangitis with tetracycline. Lancet 274: 1110–1112. doi: 10.1016/S0140-6736(59)90098-4
    [26] Mathew K (1983) Metronidazole in primary cholangitis. J Indian Med Assoc 80: 31–33.
    [27] Kozaiwa K, Tajiri H, Sawada A, et al. (1998) Case report: three paediatric cases of primary sclerosing cholangitis treated with ursodeoxycholic acid and sulphasalazine. J Gastroen Hepatol 13: 825–829. doi: 10.1111/j.1440-1746.1998.tb00740.x
    [28] Cox KL, Cox KM (1998) Oral vancomycin: treatment of primary sclerosing cholangitis in children with inflammatory bowel disease. J Pediatr Gastr Nutr 27: 580–583. doi: 10.1097/00005176-199811000-00015
    [29] Broccoletti T, Ciccimarra E, Spaziano M, et al. (2002) Refractory primary sclerosing cholangitis becoming responsive after sulphasalazine treatment of an underlying silent colitis. Ital J Pediatr 28: 515–517.
    [30] Tada S, Ebinuma H, Saito H, et al. (2006) Therapeutic benefit of sulfasalazine for patients with primary sclerosing cholangitis. J Gastroenterol 41: 388–389. doi: 10.1007/s00535-005-1758-x
    [31] Boner A, Peroni D, Bodini A, et al. (2007) Azithromycin may reduce cholestasis in primary sclerosing cholangitis: a case report and serendipitous observation. Int J Immunopath Ph 20: 847–849. doi: 10.1177/039463200702000423
    [32] Davies Y, Cox K, Abdullah B, et al. (2008) Long-term treatment of primary sclerosing cholangitis in children with oral vancomycin: an immunomodulating antibiotic. J Pediatr Gastr Nutr 47: 61–67. doi: 10.1097/MPG.0b013e31816fee95
    [33] Pohl J, Ring A, Stremmel W, et al. (2006) The role of dominant stenoses in bacterial infections of bile ducts in primary sclerosing cholangitis. Eur J Gastroen Hepat 18: 69–74. doi: 10.1097/00042737-200601000-00012
    [34] Hiramatsu K, Harada K, Tsuneyama K, et al. (2000) Amplification and sequence analysis of partial bacterial 16S ribosomal RNA gene in gallbladder bile from patients with primary biliary cirrhosis. J Hepatol 33: 9–18. doi: 10.1016/S0168-8278(00)80153-1
    [35] Olsson R, Bjornsson E, Backman L, et al. (1998) Bile duct bacterial isolates in primary sclerosing cholangitis: a study of explanted livers. J Hepatol 28: 426–432. doi: 10.1016/S0168-8278(98)80316-4
    [36] Sasatomi K, Noguchi K, Sakisaka S (1998) Abnormal accumulation of endotoxin in biliary epithelial cells in primary biliary cirrhosis and primary sclerosing cholangitis. J Hepatol 29: 409–416. doi: 10.1016/S0168-8278(98)80058-5
    [37] Vleggaar F, Monkelbaan J, Van Erpecum K (2008) Probiotics in primary sclerosing cholangitis: a randomized placebo-controlled crossover pilot study. Eur J Gastroen Hepat 20: 688–692. doi: 10.1097/MEG.0b013e3282f5197e
    [38] Allegretti J, Kassam Z, Carrellas M, et al. (2017) Fecal microbiota transplantation improves microbiome diversity and liver enzyme profile in primary sclerosing cholangitis. World Congress of Gastroenterology at ACG2017 Meeting Abstracts.
    [39] David L, Maurice C, Carmody R, et al (2013) Diet rapidly and reproducibly alters the human gut microbiome. Nature 505: 559–563.
    [40] Andersen I, Tengesdal G, Lie B, et al. (2014) Effects of coffee consumption, smoking, and hormones on risk for primary sclerosing cholangitis. Clin Gastroenterol H 12: 1019–1028. doi: 10.1016/j.cgh.2013.09.024
    [41] Lammert C, Juran B, Schlicht E, et al. (2014) Reduced coffee consumption among individuals with primary sclerosing cholangitis but not primary biliary cirrhosis. Clin Gastroenterol H 12: 1562–1568. doi: 10.1016/j.cgh.2013.12.036
    [42] Torres J, Palmela C, Bao X, et al. (2016) The gut microbiota in primary sclerosing cholangitis and inflammatory bowel disease: correlations with disease and diet. Microbiology 150: S583–S584.
    [43] Boonstra K, Beuers U, Ponsioen C (2012) Epidemiology of primary sclerosing cholangitis and primary biliary cirrhosis: a systematic review. J Hepatol 56: 1181–1188. doi: 10.1016/j.jhep.2011.10.025
    [44] Dyson J, Hirschfield G, Adams D, et al. (2015) Novel therapeutic targets in primary biliary cirrhosis. Nat Rev Gastro Hepat 12: 147–158. doi: 10.1038/nrgastro.2015.12
    [45] Mattner J (2016) Impact of microbes on the pathogenesis of primary biliary cirrhosis (PBC) and primary sclerosing cholangitis (PSC). Int J Mol Sci 17: 1864. doi: 10.3390/ijms17111864
    [46] Kaplan M, Gershwin E (2005) Primary biliary cirrhosis. New Engl J Med 353: 1261–1273. doi: 10.1056/NEJMra043898
    [47] Selmi C, Bowlus C, Gershwin E, et al. (2011) Primary biliary cirrhosis. Lancet 377: 1600–1609. doi: 10.1016/S0140-6736(10)61965-4
    [48] Chassaing B, Etienne-Mesmin L, Bonnet R, et al. (2013) Bile salts induce long polar fimbriae expression favouring crohn's disease-associated adherent-invasive escherichia coli interaction with peyer's patches. Environ Microbiol 15: 355–371. doi: 10.1111/j.1462-2920.2012.02824.x
    [49] Kakiyama G, Pandak W, Gillevet P, et al. (2013) Modulation of the fecal bile acid profile by gut microbiota in cirrhosis. J Hepatol 58: 949–955. doi: 10.1016/j.jhep.2013.01.003
    [50] Lv L, Fang D, Shi D, et al. (2016) Alterations and correlations of the gut microbiome, metabolism and immunity in patients with primary biliary cirrhosis. Environ Microbiol 18: 2272–2286. doi: 10.1111/1462-2920.13401
    [51] Sasatomi K, Noguchi K, Sakisaka S, et al. (1998) Abnormal accumulation of endotoxin in biliary epithelial cells in primary biliary cirrhosis and primary sclerosing cholangitis. J Hepatol 29: 409–416. doi: 10.1016/S0168-8278(98)80058-5
    [52] Maldonado R, Sa-Correia I, Valvano M (2016) Lipopolysaccharide modification in gram-negative bacteria during chronic infection. FEMS Microbiol Rev 40: 480–493. doi: 10.1093/femsre/fuw007
    [53] Wang A, Migita K, Ito M, et al. (2005) Hepatic expression of toll-like receptor 4 in primary biliary cirrhosis. J Autoimmun 25: 85–91. doi: 10.1016/j.jaut.2005.05.003
    [54] Benias P, Gopal K, Bodenheimer H, et al. (2012) Hepatic expression of toll-like receptors 3, 4, and 9 in primary biliary cirrhosis and chronic hepatitis c. Clin Res Hepatol Gas 36: 448–454. doi: 10.1016/j.clinre.2012.07.001
    [55] Seki E, Brenner D (2008) Toll-like receptors and adaptor molecules in liver disease: update. Hepatology 48: 322–335. doi: 10.1002/hep.22306
    [56] Wahlström A, Sayin S, Marschall H, et al. (2016) Intestinal crosstalk between bile acids and microbiota and its impact on host metabolism. Cell Metab 24: 41–50. doi: 10.1016/j.cmet.2016.05.005
    [57] Visschers R, Luyer M, Schaap F, et al. (2013) The gut-liver axis. Curr Opin Clin Nutr 16: 576–581. doi: 10.1097/MCO.0b013e32836410a4
    [58] Lindor K, Gershwin E, Poupon R, et al. (2009) Primary biliary cirrhosis. Hepatology 50: 291–308. doi: 10.1002/hep.22906
    [59] Beuers U, Boberg K, Chapman R (2009) EASL clinical practice guidelines: management of cholestatic liver diseases. J Hepatol 51: 237–267. doi: 10.1016/j.jhep.2009.04.009
    [60] Lee J, Arai H, Nakamura Y, et al. (2013) Contribution of the 7β-hydroxysteroid dehydrogenase from ruminococcus gnavus N53 to ursodeoxycholic acid formation in the human colon. J Lipid Res 54: 3062–3069. doi: 10.1194/jlr.M039834
    [61] Tang R, Wei Y, Che W, et al. (2017) Gut microbial profile is altered in primary biliary cholangitis and partially restored after UDCA therapy. Gut 67: 534–541.
    [62] Hopf U, Moller B, Stemerowicz R, et al. (1989) Relation between Escherichia coli R(rough)-forms in gut, lipid A in liver, and primary biliary cirrhosis. Lancet 334: 1419–1422. doi: 10.1016/S0140-6736(89)92034-5
    [63] Bogdanos D, Baum H, Grasso A, et al. (2004) Microbial mimics are major targets of crossreactivity with human pyruvate dehydrogenase in primary biliary cirrhosis. J Hepatol 40: 31–39.
    [64] Olafsson S, Gudjonsson H, Selmi C, et al. (2004) Antimitochondrial antibodies and reactivity to N. aromaticivorans proteins in icelandic patients with primary biliary cirrhosis and their relatives. Am J Gastroenterol 99: 2143–2146.
    [65] Mattner J, Savage P, Leung P, et al. (2008) Liver autoimmunity triggered by microbial activation of natural killer T cells. Cell Host Microbe 3: 304–315. doi: 10.1016/j.chom.2008.03.009
    [66] Smyk D, Bogdanos D, Kriese S, et al. (2012) Urinary tract infection as a risk factor for autoimmune liver disease: from bench to bedside. Clin Res Hepatol Gas 36: 110–121. doi: 10.1016/j.clinre.2011.07.013
    [67] Rashid T, Ebringer A (2012) Autoimmunity in rheumatic diseases is induced by microbial infections via crossreactivity or molecular mimicry. Autoimmune Dis: 1–9.
    [68] Stemerowicz R, Hopf U, Moller B, et al. (1988) Are antimitochondrial antibodies in primary biliary cirrhosis induced by R(rough)-mutants of enterobacteriaceae? Lancet 332: 1166–1170. doi: 10.1016/S0140-6736(88)90235-8
    [69] Gossard A, Lindor K (2012) Autoimmune hepatitis: a review. J Gastroenterol 47: 498–503. doi: 10.1007/s00535-012-0586-z
    [70] Boberg K, Aadland E, Jahnse J, et al. (1998) Incidence and prevalence of brimary biliary cirrhosis, primary sclerosing cholangitis, and autoimmune hepatitis in a norwegian population. Scand J Gastroentero 33: 99–103. doi: 10.1080/00365529850166284
    [71] Lee Y, Teo E, Ng T, et al. (2001) Autoimmune hepatitis in singapore: a rare syndrome affecting middle-aged women. J Gastroen Hepatol 16: 1384–1389. doi: 10.1046/j.1440-1746.2001.02646.x
    [72] Hurlburt K, McMahon B, Deubner H, et al. (2002) Prevalence of autoimmune liver disease in Alaska natives. Am J Gastroenterol 97: 2402–2407. doi: 10.1111/j.1572-0241.2002.06019.x
    [73] Primo J, Merino C, Fernandez J, et al. (2004) Incidence and prevalence of autoimmune hepatitis in the area of the Hospital de Sagunto (spain). Gastroent Hepat-Barc 27: 239–243. doi: 10.1016/S0210-5705(03)70452-X
    [74] Ngu J, Bechly K, Chapman B, et al. (2010) Population-based epidemiology study of autoimmune hepatitis: a disease of older women? J Gastroen Hepatol 25: 1681–1686. doi: 10.1111/j.1440-1746.2010.06384.x
    [75] Delgado J, Vodonos A, Malnick S, et al. (2013) Autoimmune hepatitis in southern Israel: a multicenter study of 15 years. J Digest Dis 14: 611–618.
    [76] Czaja A (2016) Factoring the intestinal microbiome into the pathogenesis of autoimmune hepatitis. World J Gastroentero 22: 9257–9278. doi: 10.3748/wjg.v22.i42.9257
    [77] Czaja A (1998) Frequency and nature of the variant syndromes of autoimmune liver disease. Hepatology 28: 360–365. doi: 10.1002/hep.510280210
    [78] Czaja A (2013) The overlap syndromes of autoimmune hepatitis. Digest Dis Sci 58: 326–343.
    [79] Lin R, Zhou L, Zhang J, et al. (2015) Abnormal intestinal permeability and microbiota in patients with autoimmune hepatitis. Int J Clin Exp Patho 8: 5153–5160.
    [80] Cai W, Ran Y, Li Y, et al. (2017) Intestinal microbiome and permeability in patients with autoimmune hepatitis. Best Pract Res Cl Ga 31: 669–673. doi: 10.1016/j.bpg.2017.09.013
    [81] Yuksel M, Wang Y, Tai N, et al. (2015) A novel humanized mouse model for autoimmune hepatitis and the association of gut microbiota with liver inflammation. Hepatology 62: 1536–1550. doi: 10.1002/hep.27998
    [82] Reumaux D, Duthilleul P, Roos D (2004) Pathogenesis of diseases associated with antineutrophil cytoplasm autoantibodies. Hum Immunol 65: 1–12. doi: 10.1016/j.humimm.2003.09.013
    [83] Terjung B, Sohne J, Lechtenberg B, et al. (2010) p-ANCAs in autoimmune liver disorders recognise human beta-tubulin isotype 5 and cross-react with microbial protein FtsZ. Gut 59: 808–816. doi: 10.1136/gut.2008.157818
  • Reader Comments
  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(5169) PDF downloads(848) Cited by(4)

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog