A novel Muth generalized family of distributions: Properties and applications to quality control

1.   Introduction

In real-life circumstances, there is always an element of uncertainty which always makes the applied researchers have jitters regarding the selection of an appropriate model. Thus, in order to be on the safe side, applied practitioners always prefer classical distributions such as the Exponential, Weibull or Gamma distribution. To add to the misery, theoreticians generally propose generalizations and modifications of such classical models in order to resolve discrepancies among them. There is an abundance of generalizations of such orthodox distributions as it the discretion of researchers to select the model which they want to explore both theoretically and in applied form.

Despite their importance in the literature, there are a number of distributions that have yet to be fully investigated. The functional complexity of the models may be the most plausible rationale, with the improvement of computational capabilities and numerical optimization techniques such as ${\sf MATLAB}$, ${\sf Python}$ and the ${\sf R}$ language, this claim is easily refuted. In our perspective, the statistical literature should include these overlooked models that are seldom employed. For distributions that are not regularly discussed in the literature, the authors in ^[1] provided a comprehensive list. This motivated us to investigate such models or suggest long-overlooked generalizations based on such models. One such model is the Muth distribution, with the name pioneered by the authors in ^[1]. For a continuous univariate distribution, a random variable $X$ is said to follow a Muth distribution such that $X\sim Muth(a)$ with the following distribution function:

where parameter $a\in (0, 1)$. According to the authors in ^[2], it was Tessi$\acute{e}$r (1934) who initially studied this distribution in the context of an animal ageing mechanism. However, Muth (1977) pointed out instances where it appears as a good model to study the stochastic nature of the variable under consideration as compared to established models. In ^[3], the authors studied the scaled version of the Muth distribution and established its superiority over the existing distribution by using meteorology data. A few other works related to the Muth distribution have been esented in ^{[2,3,4,5,6,7]}. The reader is referred to ^[2], in which an excellent review of the Muth distribution in chronological order has been conducted by the authors.

In this article, we propose a generalization of the Muth distribution. Regarding the generalization of conventional models, the Transformed-transformer T-X approach, introducted in ^[8], is an integral part for the construction of generalized families of distributions. The distribution function (cdf) of the T–X family is defined by

The pdf corresponding to (1) is

where the pdf of any baseline distribution is $g(x; \xi)$.

To the best of our knowledge, very few generalizations of the Muth distribution have been proposed in the literature. These include the Muth-G family by Almarashi and Elgarhy ^[9] using the T-X methodology, the Transmuted Muth-G class of distributions by ^[10] using quadratic rank transformation and the New Truncated Muth generated (NTM-G) family of distributions ^[31] in the context of a unit distribution. This further intrigued us to formulate a generalization of the Muth distribution via an odd random variable, denoted as odd Muth-G (OMG for short), in Eq (2.1) and study its mathematical properties. Similar generalizations based on odd ratios are odd Weibull-G in ^[11], odd generalized-exponential-G in ^[12], alternate odd generalized exponential-G in ^[13], odd gamma-G in ^[14], odd Lindley-G in ^[15], odd Burr-G in ^[16], odd power-Cauchy-G in ^[17], odd half-Cauchy in ^[18], odd additive Weibull-G ^[19], odd power-Lindley-G ^[20], odd Xgamma-G ^[21], etc. For a comprehensive review on generalized families, the reader is referred to ^[22,23].

For a more apt background of the T-X approach, readers are referred to ^[8]. Further motivations to propose the OMG class include the following: The inverse distribution function, median, moment generating function and characteristic function of the Muth distribution are not mathematically tractable, though these properties exist for the OMG family; when the shape parameter $a\rightarrow0$, the Muth distribution converges to the exponential family. Thus, there exists a relation between the OMG and exponential families such as exponentiated-G (EG) by ^[24] and exponentiated generalized-G (EGG) by ^[25]; the OMG class improves the flexibility of the tail properties of the baseline distribution in terms of improving the goodness of fit statistical criterion and the ability to fit symmetric as well as asymmetric real life phenomena; the Muth distribution is applied for the first time in the context of quality control, which is an integral part of reliability analysis.

The manuscript is structured as follows. In Section 2, the odd Muth-G family and its reliability properties are defined. The general properties of the proposed family are depicted in Section 3. Parameters estimation of the proposed family is illustrated in Section 4. A special model called odd Muth-Weibull (OMW) is presented in Section 5 with some essential properties. Section 6 is based on simulation analysis, while Section 7 showes the mathematical and numerical illustration of the group acceptance sampling plan (GASP). The application to real-life data is presented in Section 8. Section 9 ends the manuscript with some concluding remarks.

2.   Layout of the proposed family

In this section, the odd Muth-G (OMG) family of distributions and its reliability properties are defined.

The following are the expressions of cdf, pdf, reliability function (rf), hazard rate function (hrf) and cumulative hazard rate function (chrf) of the OMG family, respectively:

and

3.   Proposed family and its properties

Here, we derive some basic properties of the OMG family.

3.1.   The expression of quantile function

The following expression shows the quantile function (qf) of the OMG:

The above expression contains the Lambert-W function of the negative branch.

3.2.   Densities expansion

In this section, we present a useful expansion for Eq (2.1) by using exponential series expansions, as

and

By using the above exponential series on Eq (2.1), it reduces to

Using reciprocal power series expansion (see ^[26], p. 239) on Eq (3.2), we are given the following result:

where $L_0 = 1/b_0$ when $k = 0$, and

After incorporating results in Eq (3.2), the expression for linear representation will become

where

and $b_k = (-1)^k \binom{j}{k}$.

The expression for the density function after taking the derivative of Eq (3.3) will become

where $h_{j, \, k} = (j+k) g(x)\, G(x; \xi)^{j+k-1}$ is a linear combination of the exp-G family, and one can obtain the various properties by taking into account Eq (3.4).

3.3.   Moments

By using Eq (3.4), the $r$th ordinary or raw moment of the OMG family is given by

By using Eq $(3.5)$, one can get the actual moments and cumulants for $X$ as

respectively, where $\kappa_{1} = \mu_{1}^{\prime}$. By using the relationship between actual moments and ordinary moments, one can get the measures of skewness and kurtosis. The $r$th incomplete moment of OMG can be expressed as

where $I_{j, \, k}^{r}(t) = \int_{0}^{t}x^{r} h(ij, \, k)\, d_x$, and the incomplete moments are vital in order to compute the well-known namely Bonferroni and Lorenz curves.

3.4.   The expression of probability weighted moment

The expression of $(r, q)$th probability weighted moment (PWM) can be founded as

Inserting Eqs (2.1) and (2.2) in Eq (3.7), and using the generalized binomial series expansion, $\rho_{r, q}$ can be expressed as

Applying the power series expansion defined in Section 4 on Eq (3.8), it will become

Applying a power series expansion on quantity A, it will reduce to

The expression for $\rho_{r, q}$ after incorporating the result of quantity A, can be expressed as

Using generalized binomial series expansion on the above equation,

After incorporating the result of the above equation, the expression for $\rho_{r, q}$ can be expressed as

Integrating $(3.10)$, we can obtain the expression of PWMs, where

3.5.   Entropy

The entropy measure is important to underline the uncertainty variation of a $rv$; let $X$ be a $rv$ having pdf $f(x)$. The Rényi entropy can be found by the following expression:

where $\delta > 0$, $\delta \ne 1$, and $I(\delta) = \int_\Re\, f^{\delta}(x)\, dx$.

Inserting Eq (2.2) in $f^{\delta}(x)$, gives

Applying a power series yieldes

After incorporating the result in Eq. (3.11), the expression for Rényi entropy will reduce to

where $V_{k, p} = \sum_{i, j = 0}^{\infty} \frac{(-1)^{i+j+\delta} \Gamma(2 \delta+k+p)}{i!\, j!\, p! \Gamma(2 \delta+k)} \binom{\delta}{j} a^{\delta+k-j-1} (i+j+1)^k$.

4.   Estimation

Here, we demonstrate the estimation of parameters by taking into account the maximum likelihood approach. The log-likelihood (LL) function $\ell(\bf{\Omega })$ for the vector of parameters ${\Omega} = \big(a, \, \xi\big)^\top$ can be expressed as

The first partial derivatives of Eq (4.1) with respect to $a$ and $\xi$ are

where $g_i^\xi\, = \, \frac{\partial}{\partial \xi} g(x_i; \xi)$ and $G_i^\xi\, = \, \frac{\partial}{\partial \xi}\, G(x_i; \xi)$ are derivatives of column vectors of the same dimension of $\xi$.

5.   The OMW distribution

Here we consider Weibull as a baseline model with cdf and pdf, respectively, given as $G(x; \xi) = 1-{\rm e}^{-\alpha x^{\beta}}$ and $g(x; \xi) = \beta\, \alpha\, x^{\beta-1}{\rm e}^{-\alpha x^{\beta}}$, where $\alpha > 0$ is a scale, and $\beta > 0$ is a shape parameter. Then, the cdf, pdf, rf, hrf and chrf of the proposed OMW model, respectively, are given by

and

The graphical illustrations of pdf and hrf based on some selected parametric values of OMW are depicted in Figure 1 and reveal that the OMW model has flexibility in both pdf and hrf.

6.   Properties of OMW model

First, we will derive a linear expression of the OMW density to get the useful mathematical properties of this new model.

Following Eq (3.4), the OMG density will become

where $v_p = (-1)^p \binom{j+k-1}{p}\sum_{j = 1}^{\infty} \sum_{k = 0}^{j}\varpi_{j, k} (j+k)\,$, and $\pi(x; \alpha(p+1), \beta)$ is the Weibull density.

Several properties of the OMW model can be yielded by using Eq (6.2) because it is a linear combination of Weibull densities.

The qf of the OMW distribution is given as

The expression of $r$th moments is given by

The graphical illustrations of skewness and kurtosis are depicted in Figure 2 for the OMW distribution.

The expression for the $r$th incomplete moment can be written as

where $\gamma(s, x) = \int_0^{\infty} x^{s-1}\, \exp(-x) dx$.

The expression for the PWMs can be written as

where $t_s = (-1)^s \binom{j+p}{s}\sum_{j, p = 0}^{\infty}V_{j, p} (j+p+1)\,$.

The expression for Rényi entropy can be written as

where $\omega_n = (-1)^n \binom{k+p}{n} \sum_{k, p = 0}^{\infty} V_{k, p}$.

6.1.   Estimation

Let there be a sample of size $n$ from the OMW model given in Eq (5.2). The LL function $\ell = \ell(\theta)$ for the vector of parameters $\theta = (\, \alpha, \, \beta, \, a)^\top$ is

Equation (6.8) can be easily maximized using the computational software ${\sf R}$ or ${\sf Mathematica}$. The components of the score vector $U(\theta)$ are

One can yield MLEs by setting these equations equal to zero and solving simultaneously.

7.   Simulation study

This section is mainly based on simulation analysis, in order to understand the behavior of MLEs of the OMW distribution at varying sample sizes. We perform simulation analysis by considering $N$ = 1000 and $n$ = 50,100,200,400,500. Three sets of different parameter values are used to perform the simulation study: (1): $\alpha = 0.6$, $\beta = 0.09$ and $a = 0.7$; (2): $\alpha = 0.4$, $\beta = 0.2$ and $a = 0.5$; (3): $\alpha = 0.04$, $\beta = 0.02$ and $a = 0.05$. The simulation analysis biases, mean square errors (MSEs), coverage probability (CP) and average width (AW) show in Tables 1–3 that as sample size increases both biases and MSEs are reduced.

8.   Group acceptance sampling plan (GASP)

This section is based on the illustration of GASP under the assumption that the lifetime distribution of an item follows an OMW model with known parameters $\alpha$ and $\beta$ having cdf in Eq (8.2). In a GASP, say, $n$ is the randomly selected sample size and distributed to $g$ groups, and $r$ for a preassigned time $r$ items in a group are tested. If more than $c$ failures occur in any group during the experiment time, the performed experiment is truncated. The reader is referred to Aslam et al.^[29] and Khan and Alqarni ^[30] for a simple illustration of GASP and an application to real data. Designing the GASP reduced both the time and cost. Several lifetime traditional and extended models are used ^{[10,29,33,34,35]} in designing the GASP by taking into account the quality parameter as mean or median; usually, for skewed distributions median is preferable ^[29].

The GASP is simply the extension of the ordinary sampling plans i.e., the GASP reduces to the ordinary sampling plan by replacing $r = 1$, and thus $n = g$ ^[32].

GASP is based on the following process. First, select $g$ (the number of groups) and allocate predefined $r$ (group size) items to each group so that the sample of size of the lot will be $n = r\times g$. Second, select $c$ and $t_0$ (the experiment time), respectively. Third, do experiment simultaneously for g groups and record the number of failures for each group. Finally, a conclusion is drawn either accepting or rejecting the lot. The lot is accepted if there no more than $c$ failures occur in each and every group and otherwise the lot is rejected. The probability of accepting the lot is represented by the following expression:

where the probability that an item in a group fails before $t_0$ is denoted by $p$ and yielded by inserting (6.3) in (8.2). Let the lifetime of an item or product follow an OMW with known parameters $\alpha$ and $\beta$, with cdf given by for $t > 0$ for convenient we used $G\left(t\right) = 1-\exp\left[-\left(t/\alpha\right)^{\beta}\right]$

the qf of OMW model using (3.1) is given by and if $p$ = 0.5 yielded the median of the lifetime distribution of the product is not smaller than the specified median value.

and by taking $\eta$ as

Eq (8.3) is obtained by writing $\alpha = m/\eta$ and $t = a_1m_0$. The ratio of a of product mean lifetime, the specified life time $m/m_0$ can be used to express the quality level of the product. Replacing the $\alpha = m/\eta$ and $t = m_{0}a_{1}$ in Eq (8.2) yields the probability of failure, given by

From Eq (8.2), for taking the values of $a$ and $\beta$, $p$ can be determined when $a_1$ and $r_2$ are specified, where $r_2 = m/m_0$. Here, we define the two failure probabilities, say, $p_1$ and $p_2$ corresponding to the consumer risk and producer risk, respectively. For given specific values of the parameters $a$, $\beta$, $r_2$, $a_1$, $\beta^*$ and $\gamma$, we need to determine the values of $c$ and $g$ that simultaneously satisfy the following two equations:

and

where the mean ratio at consumer's risk and at producer's risk, respectively denoted by $r_1$ and $r_2$ and the probability of failure to be used in the above expression as follows

and

Tables 4 and 5 are based on arbitrary values of parameters to underline the effect of design parameters. When $r = 5$, from Table (4), with $\beta^* = 0.1$, $a_1$ = 0.5, $r_2$ = 4, there should be 185 items needed for testing (37*5 = 185). On the other hand, under the same condition when r = 10, there should be 60 units tested. So, here, we should prefer when r = 10, which significantly reduces the number of units that need to be tested.

9.   Empirical investigation

The practical implementation of the proposed model is carried out in this section by considering two real-life data sets. The first and second data sets are taken from ^[27,28], respectively.

Data 1: Aircraft Windshield Data:

The first real-life data set deals with the failure times of Aircraft Windshields. The data set is as follows: 0.0400, 1.8660, 2.3850, 3.4430, 0.3010, 1.8760, 2.4810, 3.4670, 0.3090, 1.8990, 2.6100, 3.4780, 0.5570, 1.9110, 2.6250, 3.5780, 0.9430, 1.9120, 2.6320, 3.5950, 1.0700, 1.9140, 2.6460, 3.6990, 1.1240, 1.9810, 2.6610, 3.7790, 1.2480, 2.0100, 2.6880, 3.9240, 1.2810, 2.0380, 2.820, 3.0000, 4.0350, 1.2810, 2.0850, 2.8900, 4.1210, 1.3030, 2.0890, 2.9020, 4.1670, 1.4320, 2.0970, 2.9340, 4.2400, 1.4800, 2.1350, 2.9620, 4.2550, 1.5050, 2.1540, 2.9640, 4.2780, 1.5060, 2.1900, 3.0000, 4.3050, 1.5680, 2.1940, 3.1030, 4.3760, 1.6150, 2.2230, 3.1140, 4.4490, 1.6190, 2.2240, 3.1170, 4.4850, 1.6520, 2.2290, 3.1660, 4.5700, 1.6520, 2.3000, 3.3440, 4.6020, 1.7570, 2.3240, 3.3760, 4.6630.

Data 2: Fiber Strength Data:

The second real-life data set consists of 46 data points representing the strength of 15 cm glass fiber. The data set is as follows: 0.37, 0.40, 0.70, 0.75, 0.80, 0.81, 0.83, 0.86, 0.92, 0.92, 0.94, 0.95, 0.98, 1.03, 1.06, 1.06, 1.08, 1.09, 1.10, 1.10, 1.13, 1.14, 1.15, 1.17, 1.20, 1.20, 1.21, 1.22, 1.25, 1.28, 1.28, 1.29, 1.29, 1.30, 1.35, 1.35, 1.37, 1.37, 1.38, 1.40, 1.40, 1.42, 1.43, 1.51, 1.53, 1.61. The six well-known models exponentiated Weibull (EW), gamma Weibull (GaW) ^[36], Kumaraswamy Weibull (KwW), exponentiated generalized Weibull (EGW), beta Weibull (BW) and Weibull (W) are applied to these data sets.

The analysis of both data sets revealed that the proposed OMW model outperforms the comparative models, as per the least information criterion and higher P-values. The estimated parameters along with standard errors are depicted in Tables 6 and 8, whereas the accuracy measures are given in Tables 7 and 9. The graphical illustrations from Figures 3 and 4 are showing good agreement between the actual and fitted results.

The probability density functions of the comparative models are as follows:

When $r = 5$, from Table 10, with $\beta^* = 0.1$, $a_1$ = 0.5, $r_2$ = 4, there should be 860 items needed for testing (172*5 = 860). On the other hand, under the same condition, when r = 10 there should be 240 units tested. So, here, we should prefer when r = 10, which significantly reduces the number of units that need to be tested.

From Tables 11 and 12, when the true median life increases, the number of groups decreases, and operating characteristics values increases. For data 1 when $\beta^* = 0.05$, $a_1 = 1$, r = 10, $\alpha = 0.3201$ and $\beta = 0.7953$ and for data 2 when $\beta^* = 0.05$, $a_1 = 1$, r = 5, $\alpha = 0.5360$ and $\beta = 1.5825$ are the proposed GASP, when a lifetime of an item follows a OMW model.

10.   Conclusions

We introduced the new odd Muth-G family of distributions with essential properties. A special model called the odd Muth-Weibull is presented with some useful properties. Further, a design of a group acceptance sampling plan is proposed under the OMW model by considering median life as a quality parameter. Real data application revealed that the proposed model yielded better fits compared to some commonly well known models.

Conflict of interest

The authors declare no conflict of interest.