The extended Burr-R class: properties, applications and modified test for censored data

1.   Introduction

Marshall Olkin in 1997 gave a general method to extend any existing distribution by adding one additional parameter called tiled parameter for the seek of goodness of fit. Later on many researchers adopted this method and proposed many new extended distributions, available in literature some of them are the Marshall Olkin Pareto distribution ^[7], Marshall Olkin beta distribution ^[14], Marshall Olkin extended Weibull distribution ^[10], Marshall Olkin exponentiated Burr XII distribution ^[11], Marshall Olkin additive Weibull distribution ^[2], Marshall Olkin alpha power family ^[22], Marshall Olkin Burr III family ^[1], and Marshall Olkin power generalized Weibull distribution ^[3], among many more.

In this study, we introduced a new family of distributions called Marshall Olkin Burr-R (MOB-R) family by inserting the generalized Burr-G (Arslan et al. ^[20]) in the Marshall Olkin family (Marshall and Olkin ^[16]), we obtain the following cdf and pdf of the MOB-R class as

and

Its hazard rate function (hrf) has the form

The random variable ($rv$) $X$ having the density (1.2)is denoted by $X\sim$MOB-R$(\eta, \theta, \rho)$. The quantile function (qf), $Q_x(u)$, of the MOB-R family reduces to

where $x = Q(u)$ follows the MOBG family with $U\sim uniform(0, 1)$.

Motivation

The MOB-R family can be justified physically as follows. Cosider $N$ independent components which are related by a series system and suppose that $N$ is a $rv$ with a probability mass function $P(N = n) = \delta(1-\delta)^n$, $0 < \theta < 1$ and $n = 0, 1, ...$. Let $X_1, X_2, ..., X_n$ represents the lifetimes of the components, which are assumed to be independently and identically distributed $rv$'s with cdf $F(x; \theta, \rho, \xi)$. The $rv$ $Y = \min(X_1, ..., X_N)$ refers to time of the first failure and its cdf takes the form

where $R(x)$ is any baseline cdf, i.e $0 < X < \infty$ such as Lomax, Frechet, log-logistic and Weibull distributions.

The rest of the article is followed as. In Section 2, a useful linear mixture representation. The general properties of the MOB-R class are provided in Section 3. In Section 4, estimation of the MOB-R parameters is carried out by maximum likelihood. In Section 5, we study the MOB-Lomax model as a special model of the proposed family. In Section 6, a modified chi-square test statistic for right censored data is applied to validate the MOB-L model. In Section 7, we provide simulation results to check the performance of the maximum likelihood and to validate the test statistic. Real data applications are analyzed to show the flexibility of the MOB-Lomax model in Section 8. Further, a real censored data is analyzed to validate the considered test statistic. Some concluding remarks are reported in Section 9.

2.   Useful expansion

This section provides an infinite linear mixture for the cdf and pdf of the MOB-R class in (1.1) and (1.2). By using following two series expansions

The log-power expansion has the form

where $\varrho_{j, \rho} = \frac{1}{\rho}\sum\limits_{m = 1}^\rho(jm-\rho+m)\theta_m \, \, \varrho_{j, \rho-m}$, $\, \, p_{j, 0} = 1$ and $\theta_\rho = \frac{(-1)^\rho}{\rho+1}.$

The equation (1.1) reduce to

where $a_m = \eta\, \sum\limits^{\infty}_{i = 0}\bar{\eta}^i\, \sum\limits_{j = 0}^{\infty} \binom{\rho(i+1)+j-1}{j}, \theta\, \, j\, \, \binom{m-\theta j}{m} \sum\limits_{l = 0}^{m}\frac{(-1)^{l+cj+m+j}}{cj-l} \binom{m}{l}\, \, \varrho_{l, m}$

and $a_m$ are weights and $\sum a_m = 1$. Equation (2.1) can be written as

where $b_0 = 1-a_0$, $b_m = -a_m$ and $H_m(x) = R(x)^m$.

Similarly

where $h_{m+1}(x) = (m+1)\, \, r(x)\, \, R(x)^m$. Equations (2.3) and (2.2) refer to the infinite linear representation for the cdf and pdf of the MOB-R class in terms of their baseline distributions, and they are helpful to derive the MOB-R properties.

3.   General properties

In this section, we derive some properties of MOB-R family, such as moments, generating function (mgf), stochastic ordering, reliability parameter, and order statistics.

3.1.   Moments

The $r$th moments of MOB-R family is derived using the following expression

Using mixture representation given in (2.3), we have

where $\Delta^r = \int\limits_{-\infty}^{\infty} x^r\, h_{m+1}(x)\, dx$. Similarly, incomplete moments of MOB-R family can be calculated using the following formula

Applications of Equation (3.2) are related to mean deviation, Zenga index, income quantiles such as Lorenz and Bonferroni curves, mean waiting time, and mean residual life.

The mgf of the MOB-G family can be obtained by using the following expression

Using mixture representation given in (2.3), we have

Note that these integrals only depends only for the arbitrary base line distribution.

3.2.   Stress-strength analysis

In the field of reliability, stress-strength model has an important role which defines the life time of a component which has a random strength, say $X_1$, which is subject to an accidental stress, say $X_2$. The component will still work when $X_1 > X_2$. It has many application in engineering. Let $X_1$ and $X_2$ be two $rv$'s follow the MOB-R family i.e. $X_1\sim$MOB-R($\theta_1, \rho_1, \eta_1$) and $X_2\sim$MOB-R($\theta_2, \rho_2, \eta_2$) with a common shape and scale parameters.

Using Equations (2.2) and (2.3), we can write

where $h_{p+1}(x)$ and $H_m(x)$ are already defined in the previous section.

3.3.   Stochastic ordering

The stochastic ordering is commonly used in showing the ordering mechanism in life time distribution. A $rv$ $X$ is stochastically greater than the $rv$ $Y$ if $F_X(x)\leq F_Y(x)$ for all $x$'s. Further, there are some important stochastic orderings namely, stochastic order, hazard rate order, mean residual order, likelihood ratio order, and reversed hazard rate order which are related to each other according to the following chain of stochastic orders

Further details about different stochastic orderings and their definitions can be explored in Shaked and Shanthikumar ^[23]. Let $X_1\sim MOBG(\theta, \rho, \eta_1)$ and $X_2\sim MOBG(\theta, \rho, \eta_2)$. Then, using likelihood ratio ordering defined by $\left[\frac{f(x)}{g(x)}\right]$, we can write

where $\xi = \left(1+(H_R(x))^\theta\right)^{-\rho}$ and $\xi^\prime = \theta\, \rho\, h_R(x)H_R(x)^{\theta-1}\, \left(1+(H_R(x))^\theta\right)^{-\rho-1}.$ Therefore,

From the above expression we see that $\frac{d}{dx}\frac{f(x)}{g(x)} < 0$, if $\eta_1 < \eta_2$, so the likelihood ratio ordering exists among the variables i.e. $X\leq_{lr}Y$ and the remaining stochastic orderings follow simply from the above chain.

3.4.   Order statistics

Let $X_1, ..., X_n$ be a random sample for MOB-R family. The $i$th order statistic, $X_{i:n}$, has the following pdf

Using Equations (2.2), (2.3) and the power series expansion in ^[13] (Pages 17–18), we have

where $v_{j, r, m} = \frac{\rho\, (m+1)\, n!(-1)^j\, b_{r+1}\, e_{j+i-1:r}}{(i-1)!(n-i-j)!j!}(r+m+1),$ $h_{r+m}(x) = (r+m+1)g(x)G^{r+m}(x)$ is already defined in previous section.

4.   Estimation

In this section, the estimation of the MOB-R parameters are obtained using the maximum likelihood (ML) method. Let $X_1, X_2, ..., X_n$ be a random sample from the MOB-R class. The log-likelihood function for $\Theta = (\eta, \theta, \rho, \xi)^T$ takes the form

where $S_{i} = \left\{-\log [1-R(x_{i})] \right\}^{\theta}$. The score vector elements take the forms

Equating the above equations by zero and solving them simultaneously yields the ML estimates.

5.   The MOB-Lomax model

If $X\sim$Lomax$\, (a, b)$, then its cdf takes the form $R(x) = 1-\left(1+\frac{x}{a}\right)^{-b}$. Using Equations (1.2) and (1.1), we obtain the cdf and pdf of the MOB-Lomax (MOB-L) distribution as follows

and

The hrf of the MOB-L distribution reduces to

In Figure 1, the plots for pdf and hrf are presented for the MOB-L distribution for several values of parameters. As seen in Figure 1, MOB-L distribution is very flexible with skewed shapes.

The mixture representations for the cdf and pdf of the MOB-L distribution follow from Equations (2.2) and (2.3) as

and

The qf of the MOB-L distribution follows from Equation (1.4) as

where $A = 1-\exp \left[-\left\{\left(\frac{u-1}{1-\bar{\eta}}\right)^{-\frac{1}{\rho}}-1\right\}^\frac{1}{\theta}\right].$

The $r^{th}$ moment of the MOB-L model can be calculated using (3.1) as

Its $r^{th}$ incomplete moment takes the form

where $v_{j, m} = \sum\limits_{m = 0}^{\infty}\, \, b_{m+1} \, \, (m+1) \, \, \binom{m+1}{j}(-1)^j.$

Setting $r = 1$ in (5.1), the first incomplete moment reduces to

The mgf of MOB-L distribution follows from Equation (3.3) as

where $v_{i, j, m} = \sum\limits_{m, j = 0}^{\infty}\, \, b_{m+1}\, \, (m+1)\, \, \binom{m+1}{j} \binom{\beta(j+1)+i}{i} (-1)^{i+j}.$

The pdf of $i^{th}$ order statistic for the MOB-L model is

where

If we have two MOB-L distributions, such as MOB-L$(\eta_1, \theta_1, \rho_1, a, b_1)$ and MOB-L$(\eta_2, \theta_2, \rho_2, a, b_2)$, with a common parameter $a$ then from Equation (3.5), we obtain the reliability function as

5.1.   Estimation of the MOB-L Parameters

The log-likelihood function for the MOB-L model takes the form

where $B_{i} = \left[b\log(d_{i})\right]^\theta$ and $d_{i} = \left(1+\frac{x_{i}}{a}\right)$.

The components of score vector are

The log-likelihood function can be maximized directly using the R-package (AdequecyModel). In AdequecyModel package, there are some maximization algorithms such as NR (Newton-Raphson), BFGS (Broyden-Fletcher-Goldfarb-Shanno), BHHH (BerndtHall-Hall-Hausman), NM (Nelder-Mead), SANN (Simulated-Annealing) and limited memory quasi-Newton code for Bound-constrained optimization (L-BFGS-B). However, the MLEs here are computed using the BFGS algorithm.

6.   Modified Chi-Square test for right censored data

In this section, we provide a modified chi-square test statistic for right censored data from the MOB-L distribution based on the modified chi-square type test which is proposed by Bagdonavicius et al. ^[8] and Bagdonavicius and Nikulin ^[9], for parametric models with right censored data. Using the maximum likelihood estimators (MLEs) for non-grouped data, this test statistic is also based on the differences between the numbers of observed failures and the numbers of expected failures in the chosen grouped intervals. Here, random grouping intervals are considered as data functions. Voinov et al. ^[24] developed the description of construction of this chi-square type test. The test statistic can be defined as follows.

Suppose that $X_{1}, X_{2}, ....., X_{n}$ is a random sample with right censoring from a parametric model, and a finite time $\tau.$

The test statistic takes the form

where $U_{j}$ and $e_{j}$ are the observed and expected numbers of failure in grouping intervals, and $Q$ has the form

where $h(x_{i}, \hat{\xi})$ is the hrf $\xi = (b, \theta, \rho, \eta, \bar{\eta})$ and $\widehat{\xi }$ is the MLE of $\xi$ on initial non-grouped data.

The limits $a_{j}$ of $r$ random grouping intervals $I_{j} = [a_{j-1}, a_{j}[$ are chosen such as the expected failure times to fall into these intervals which are the same for each $j = 1, .., r-1,$ $\hat{a}_{r} = \max \left(X_{\left(l\right) }, \tau \right).$ The estimated $\hat{a}_{j}$ is

where $H(x)$ is the cdf of the considered distribution. This test statistic $Y_{n}^{2}$ follows a chi-square distribution.

The expected failure times $e_{j}$ to fall into these intervals are $e_{j} = \frac{E_{r}}{r}$ for any $j$, with $E_{r} = \sum_{i = 1}^{n}H\left(x_{i}, \hat{\gamma}\right)$.

The limit intervals $a_{j}$ are considered as random variables such that the expected numbers of failures in each interval $I_{j}$ are the same, so the expected numbers of failures $e_{j}$ can be calculated by the following formula

6.1.   Quadratic form $Q$

To calculate the quadratic form $Q$ of the statistic $Y_{n}^{2}$, and as its distribution does not depend on the parameters, so we can use the estimated matrices $\hat{W}$ $\hat{C}$ and the estimated information matrix $\hat{I}.$ The elements of $\hat{C}$ are defined by

where $\varpi _{i} = b\log (1+\frac{x}{a})$ and

The elements of $\hat{C}$ take the forms

and

As the above components of the statistic have explicit forms, then we can obtain the test statistic for the MOB-L$(\hat{\xi})$ distribution with unknown parameters and right censored data. This statistic, follows a chi-square distribution with $r$ degrees of freedom, takes the form

7.   Simulations

In this section, we provide two simulation studies to assess the performance of the MLEs and to validate the test statistic $Y_{n}^{2}(\xi)$.

7.1.   Simulation for the MLEs

Now, we will study the performance of the maximum likelihood in estimating the MOB-L parameters using simulations which are conducted for sample sizes $n = 50,150,300$, and for different parameter combinations (I: $a = 3, b = 4.5, \theta = 0.5, \rho = 4, \eta = 2$), (II: $a = 0.2, b = 0.8, \theta = 1.5, \rho = 8, \eta = 7$) and (III: $a = 0.2, b = 0.8, \theta = 0.5, \rho = 5, \eta = 5$). To obtain the average values of estimates (AEs), mean square errors (MSEs) and absolute biases (ABs) of the parameters, we generated $N = 3000$ samples from the MOB-L model using the R program.

The MSEs and ABs were determined by the following equations:

where $\pmb \vartheta = (a, b, \theta, \rho, \eta)^{'}$.

The simulation results are shown in Table 1. The small values of ABs and MSEs prove that the maximum likelihood performs very well in estimating the MOB-L parameters.

7.2.   Simulation for test statistic $Y^{2}$

For testing the null hypothesis $H_{0}$ that right censored data are from MOB-L model, we calculate the test statistic $Y_{n}^{2}(\xi)$, defined above, using 10,000 simulated samples from the hypothesized distribution with different sizes $n = 15, 25, 50,130,350,500, 1000$ using the package "bb solve algorithm" in the R software. Then, the empirical levels of significance are calculated, for $Y{{}^2} > \chi _{\varepsilon }^{2}(r)$, with theoretical levels of significance $\varepsilon = 0.10, \, 0.05, \, 0.01$, and $r = 5$. The simulated levels of significance for $Y_{n}^{2}(\xi)$ of the MOB-L model are reported in Table 1.

The null hypothesis $H_{0}$ for which simulated samples are fitted by MOB-L distribution, is widely validated for different significance levels. Therefore, the proposed test statistic can be used to fit the data from the MOB-L distribution.

8.   Data analysis

In this section, we provide two real data analysis to prove the importance and flexibility of the MOB-L distribution and another censored data to validate the MOB-L model using the modified test statistic.

8.1.   Applications

In this section, we illustrate the performance and flexibility of the MOB-L distribution, as a sub-model of the MOB-R class, using two real-life data sets. The first data on 63 of strengths of 1.5 cm glass fibres which are used by ^[6] and ^[19]. The data are: 0.55, 0.93, 1.25, 1.36, 1.49, 1.52, 1.58, 1.66, 1.69, 1.76, 1.84, 2.24, 0.81, 1.13, 1.29, 1.77, 1.84, 1.27, 1.39, 1.49, 1.53, 1.59, 1.61, 1.66, 1.68, 1.76, 1.48, 1.5, 1.55, 1.61, 1.62, 1.81, 2, 1.82, 2.01, 0.77, 1.61, 0.74, 1.04, 1.62, 1.66, 1.7, 1.64, 1.68, 1.73, 1.11, 1.28, 1.42, 1.5, 1.54, 1.3, 1.48, 1.51, 1.55, 1.61, 1.6, 0.84, 1.24, 1.63, 1.67, 1.7, 1.78, 1.89. The second data on 128 bladder cancer patients on their remission times (in months) which are reported in ^[15] and are analyzed by ^[4] and ^[5]. The data are: 0.08, 2.09, 3.48, 4.87, 6.94, 8.66, 6.97, 9.02, 13.29, 13.11, 23.63, 0.20, 2.23, 3.52, 4.98, 0.40, 2.26, 13.80, 25.74, 0.50, 2.46, 3.57, 5.06, 7.09, 9.22, 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, 3.64, 5.09, 7.26, 9.47, 14.24, 14.83, 34.26, 0.90, 2.69, 4.18, 5.34, 7.59, 10.66, 15.96, 36.66, 10.75, 16.62, 43.01, 1.19, 2.75, 1.05, 2.69, 4.23, 5.41, 7.62, 4.26, 5.41, 7.63, 4.33, 5.49, 7.66, 11.25, 17.14, 17.12, 46.12, 1.26, 2.83, 79.05, 11.64, 17.36, 1.40, 3.02, 1.35, 2.87, 5.62, 7.87, 4.34, 5.71, 7.93, 11.79, 8.26, 11.98, 19.13, 1.76, 18.10, 1.46, 4.40, 5.85, 3.25, 4.50, 6.25, 6.54, 8.53, 12.03, 20.28, 2.02, 8.37, 12.02, 2.02, 3.31, 4.51, 3.36, 6.76, 2.07, 3.36, 12.07, 21.73, 6.93, 8.65, 12.63, 22.69.

The MOB-L model is compared with some existing models namely, Kumaraswamy-Lomax (Kw-L) ^[17], generalized-exponentiated exponential-Weibull (GE-EW), beta-Lomax (B-L) ^[17], generalized exponentiated-exponential (GEE), exponentiated-Weibull (EW) ^[21] and Lomax (L) distributions. The fitted competing models are assessed using the Anderson-Darling (AnD) and Cramer von Mises (CvM) measures.

The MLEs and the discrimination measures for all competing models are listed in Tables 3 and 4 for both data sets, respectively. It is clear from Tables 3 and 4, that the MOB-L model provides better fit for both data sets as compared with other competing models.

The hrf plots of the MOB-L model for the data sets are depicted in Figure 2. The TTT plots for the two data sets are shown in Figure 3. The TTT plot for glass fibres data is concave that refers to increasing failure rate, whereas the TTT plot for remission times data is concave then convex which refers to a unimodal hazard rate. As shown, Figures 2 and 3 are consistent, where the hrf of the MOB-L model is increasing for glass fibers data and unimodal for remission times data, hence we conclude that the MOB-L is a suitable distribution for fitting the two data sets. Further, the estimated pdf, cdf, sf and pp plots of the MOB-L model are displayed in Figures 4 and 5, for the two data sets.

8.2.   Testing a real censored data using the $Y_{n}^{2}$ statistic

For now, we analyze the lymphoma data which represent times from diagnosis to death (in months) for 31 individuals with advanced non Hodgkin's lymphoma clinical symptoms, using the MOB-L model. This data have been analyzed by Matthews et al. ^[18] and Gijbels and Gurler ^[12]. Among these 31 observations 11 of the times are censored, because the patients were alive at the last time of follow-up. The data are: 2.5, 4.1, 4.6, 6.4, 6.7, 7.4, 7.6, 7.7, 7.8, 8.8, 13.3, 13.4, 18.3, 19.7, 21.9, 24.7, 27.5, 29.7, 30.1*, 32.9, 33.5, 35.4*, 37.7*, 40.9*, 42.6*, 45.4*, 48.5*, 48.9*, 60.4*, 64.4*, 66.4*. The * denotes a censored observation.

The test statistic provided in Section 7 is used to verify if these data can be modeled by the MOB-L distribution. To this end, we first calculate the MLEs of the MOB-L parameters

Data are grouped into $r = 5$ intervals $I_{j}$. The numerical result are listed in Table 5.

Then, we obtain the value of the test statistic $Y_{n}^{2}$ as

For significance level $\varepsilon = 0.05$, the critical value $\chi _{5}^{2} = 11.0705$ is superior than the value of $Y_{n}^{2} = 7.\, 258,$ so we can conclude that the proposed MOB-L model fit these data very well.

9.   Conclusions

In this paper, we present a new family called, Marshall-Olkin Burr-R family. Some general properties of this family are studied. The estimation of its parameters is carried out by the maximum likelihood approach. One special sub-model namely, Marshall-Olkin Burr-Lomax (MOB-L) distribution is discussed in detail. Two applications are used to check the performance of the MOB-L model. A modified chi-square test statistic for censored data is used to verify the validity of the MOB-Lomax distribution. This test statistic shows that the MOB-Lomax model can be used as a good candidate for analyzing real censored data. The work in the present paper can be extended in some ways. For example, bivariate Marshall-Olkin Burr family can be studied. Further, the parameters of the MOB-L distribution can be estimated using classical and Bayesian estimation methods and compare between them to determine the best estimation method.

Acknowledgments

This project is supported by Researchers Supporting Project number (RSP-2020/156) King Saud University, Riyadh, Saudi Arabia. The first author, therefore, gratefully acknowledges the KSU for technical and financial support. The authors would like to thank the Editor and reviewer for their constructive comments that improved the final version of the paper.

Conflict of interest

There is no conflict of interest declared by the authors.