Marshall – Olkin Power Generalized Weibull Distribution with Applications in Engineering and Medicine

This paper proposes a new flexible four-parameter model called Marshall – Olkin power generalized Weibull (MOPGW) distribution which provides symmetrical, reversed-J shaped, left-skewed and right-skewed densities, and bathtub, unimodal, increas-ing,constant,decreasing,Jshaped,andreversed-Jshapedhazardrates.SomeoftheMOPGWstructuralpropertiesarediscussed.ThemaximumlikelihoodisutilizedtoestimatetheMOPGWunknownparameters.Simulationresultsareprovidedtoassesstheperformanceofthemaximumlikelihoodmethod.Finally,weillustratetheimportanceoftheMOPGWmodel,comparedwithsomerivalmodels,viatworealdataapplicationsfromtheengineeringandmedicinefields.


INTRODUCTION
The Weibull distribution has been used in modeling lifetime data with monotonic failure rates. It does not provide adequate fits to real data with unimodal or bathtub-shaped hazard rates, often encountered in engineering, medicine and reliability fields. Hence, several authors have constructed different generalizations and extended forms of the Weibull distribution to increase its flexibilty. For example, exponentiated Weibull [1], Marshall Olkin extended Weibull [2], beta Weibull [3], Kumaraswamy Weibull [4], transmuted complementary Weibull geometric [5], Weibull-Weibull [6], odd log-logistic exponentiated Weibull [7], alpha logarithmic transformed Weibull [8], and odd Lomax Weibull [9] distributions. Another Weibull extension called the power generalized Weibull (PGW) model pioneered by [10] and they used it in accelerated failure time models. The hazard rate function (HRF) of PGW model can be monotone, unimodal, and bathtub shaped.
The PGW model can be specified by the cumulative distribution function (CDF) where represents a scale parameter, and represent shape parameters.
The corresponding probability density function (PDF) and HRF take the forms, Some special cases of the PGW distribution are the Weibull distribution with parameters and for = 1; the exponential distribution with parameter for = = 1; the Rayleigh distribution with parameter for = 1 and = 2; and the Nadarajah-Haghighi (NH) model [11] with parameters = 1 and for = 1. Further applications of the PGW model can be explored in [12,13] and [14].
This paper is devoted to propose and study a new flexible extension of the PGW model called the MOPGW distribution, which has some desirable motivations as follows: • The MOPGW contains a number of well-known lifetime sub-models called, Marshall Olkin Weibull [15], Marshall Olkin-NH [16], NH [11], PGW [10], Weibull [17], exponential, and Rayleigh distributions, among others, see Table 1.
• The PDF of the MOPGW distribution can be reversed-J shaped, right-skewed, concave down, left-skewed, or symmetric, see Figure 1. Further, its HRF takes some important forms such as, constant, monotone (decreasing or increasing), bathtub, upside down bathtub, and reversed-J shaped. Hence, the MOPGW distribution can be considered as a superior lifetime distribution to other models, which exhibit only monotonic and constant hazard rates, see Figure 2.
• The MOPGW distribution is considered as a suitable model for modeling skewed data that cannot be properly modeled by other extensions of the Weibull distribution. Further, it can be utilized to model real data in many applied areas, such as engineering, survival analysis, medicine, and industrial reliability. Table 1 Special cases of the MOPGW distribution. • The kurtosis of the MOPGW distribution is more flexible as compared to the baseline PGW model, whereas its skewness varies within the interval (1.43529, 5.62470), whereas the PGW skewness can only range in the interval (-0.68927, 4.25756). Further, the MOPGW distribution can be leptokurtic (kurtosis > 3) or platykurtic (kurtosis < 3), see Table 2.

Distribution Authors
• Two real-life data applications from the engineering and medicine sciences, prove that the MOPGW distribution outperforms nine other well-known competing lifetime distributions, motivating its usage in applied fields.  The MOPGW distribution is constructed by incorporating the PGW as a baseline model in the Marshall Olkin-G (MO-G) family proposed by [15]. This family has been widely used to provide more flexible extensions of the well-known classical distributions in the statistical literature. See, for example, [2,18,19], and the references therein.
The CDF of the MO-G family has the form The corresponding PDF and HRF of the MO-G class are The MO-G class reduces to the baseline distribution for = 1. Further details about the MO-G class can be explored in [15] and [20].
The rest of the article is outlined as follows: The MOPGW distribution, its special cases, and PDF and HRF plots are provided in Section 2. Several structural properties of the MOPGW distribution are derived in Section 3. In Section 4, the estimation of the MOPGW parameters is demonstrated via the maximum likelihood and its performance is evaluated by simulation results. In Section 5, we illustrate the importance of the MOPGW model by using two real data applications. Finally, the paper is concluded in Section 6.

THE MOPGW DISTRIBUTION
In this section, we introduce the four-parameter MOPGW distribution. By inserting (1) in Equation (2), the CDF of the MOGPW distribution follows as The corresponding PDF of the MOPGW distribution takes the form Henceforth, the random variable with PDF (4) is denoted by X ∼MOPGW ( ), where = ( , , , ).
Figures 1 and 2 display some plots of the probability density and hazard functions of the MOPGW distribution for some different values of its parameters. Figures 1 and 2 reveal that the MOGPW density exhibits reversed-J shaped, left-skewed, symmetric, or right-skewed shapes, whereas its HRF can exhibit constant, monotone (increasing or decreasing), or nonmonotone (unimodal or bathtub) hazard rate shapes.
Further, the MOPGW distribution contains some important special sub-models which are displayed in Table 1.

STATISTICAL PROPERTIES OF THE MOPGW DISTRIBUTION
Some properties of the MOPGW distribution including quantile function (QF), ordinary and conditional moments, generating function (MGF), mean deviation, Lorenz and Bonferroni curves, and residual life and reversed residual life moments are derived.

Quantile Function
The QF Q(u), of the MOPGW distribution follows, by inverting its CDF, as Equation (5) can be easily used to generate the MOPGW random variates, and the median of X follows from it with u = 0.5.

Moments
Here, we derive the rth moment of the MOPGW distribution.
Using the generalized binomial expression for |d| < 1 and k > 0 (a real non-integer) Using Equation (7) in (6), we obtain Setting and after some algebra, we have Consider the power series with real number power w where is a binomial coefficient. The above power series converges for w ≥ 0 an integer, or | | x a | | < 1 (see [22]). Applying (9) in Equation where the complementary incomplete gamma is defined by Γ(s, t) = ∫ ∞ t x s−1 e −x dx, for all real numbers except negative integers. Table 2 shows some numerical values for the mean, , variance, 2 , skewness, 1 , and kurtosis, 2 , of the MOPGW distribution which can be computed, using the R software, for several values of , , and , with = 1. Table 2 shows that the MOPGW skewness can range in the interval (−1.43529, 5.62470), whereas the PGW skewness can range only in (−0.68927, 4.25756) for = 1 and the shape parameters have values from 0.5 to 5. The kurtosis spread for the MOPGW model is much larger ranging in the interval (2.76358, 55.9161), whereas the kurtosis spread for the PGW model can only vary from 2.54958 to 33.2167 for the same parameter values. Further, the MOPGW model can be right skewed and left skewed. Table 2 illustrates that the MOPGW model is leptokurtic (kurtosis > 3) or platykurtic (kurtosis < 3). Hence, the MOPGW distribution can be utilized in modeling skewed data.

Generating Function
The MGF is useful for several reasons, one of which is its application in analyzing the sums of random variables.

Proof.
The MGF can be defined as Since the series expansion of e tx is given by e tx = ∑ ∞ k=0 Then, substituting from Equations (10) into (11), we get which completes the proof.

Conditional Moments
The rth lower incomplete moment of X can be defined (for any real After some algebra, we get where the lower incomplete gamma function is denoted by Λ(s, t) = ∫ t 0 x s−1 e −x dx. The first incomplete moment of X, v 1 (t), is computed using Equation (12) by setting s = 1, and it takes the form .
Similarly, the rth upper incomplete moment of X can be defined (for any real s > 0) as where the upper incomplete gamma function is denoted by Γ(s, t) = ∫ ∞ t x s−1 e −x dx.

Mean Deviation, Lorenz, and Bonferroni Curves
This section is devoted to derive the mean deviation about the mean, denoted by 1 (x), and the mean deviation about the median, denoted by 2 (x), for the MOPGW model.
The mean deviations about the mean of the MOPGW distribution is The mean deviations about the median of the MOPGW distribution has the form Using Equation (10), we get the mean of X, ′ 1 = E(X), the median of X is M = Q(0.5), F( ′ 1 ) follows simply from Equation (3) and v 1 ( ′ 1 ) denotes the first incomplete moment.
Further, the Lorenz and Bonferroni curves have useful applications in income inequality measures, demography, reliability, insurance, and medicine.
Lorenz curve has the form The Bonferroni curve takes the form

Moments of Residual and Reversed Residual Lives
The rth moment of the residual life is Using the binomial series to the term (x − t) r and the PDF of the MOPGW in (4), the r (t) reduces to where upper incomplete gamma is Γ(s, t) = ∫ ∞ t x s−1 e −x dx. The mean residual life (MRL) of the MOPGW distribution follows from the last equation, with r = 1, as The variance residual life of the MOPGW distribution is obtained directly using 2 (t) and (t).
Further, the rth moment of reversed residual life is Using the binomial series to the term (t − x) r and the MOPGW PDF in (4), the m r (t) follows as where the lower incomplete gamma has the form Λ(s, t) = ∫ The variance and coefficient of variation of reversed residual life for the MOPGW distribution follow simply using m(t) and m 2 (t).

ESTIMATION AND SIMULATION
In this section, we discuss the estimation of the MOPGW parameters using the maximum likelihood. Let x 1 , x 2 , ..., x n be a random sample from the MOPGW distribution with parameters vector = ( , , , ) T .
The log likelihood function, log ℓ( ), has the form The maximum likelihood estimates (MLEs) of , , , and are obtained by maximizing Equation (13) with respect to these parameters. Further, the MLEs of the MOPGW parameters can be obtained by solving the following four nonlinear equations, which represent the score vector elements given by  Tables 3 and 4.   The values of AVEs and MSEs reveal that • All parameters estimates show consistency property, that is, the MSEs decrease when the sample size increases.
• For fixed , , and , the MSEs of̂increase, in most cases when increases.

APPLICATIONS
Two real data applications are provided in this section to study empirically the importance and flexibility of the MOPGW distribution. The first set of data represents the gauge lengths of 20 mm and it consists of 74 observations. This data is reported in [23], and it is studied by [24,25] and [26]. The second set of data consists of n = 128 observations of remission times (months) of bladder cancer patients studied by [27]. The MOPGW provides better fits to the cancer data than the odd Lindley Burr XII [7], generalized odd Lindley Burr XII [29], quasi xgamma-geometric [30], and Weibull Marshall Olkin Lindley [31] distributions.
The W * (Cramér Von Mises), A * (Anderson Darling), KS (Kolmogorov Smirnov), and its PV (p-value) measures, are considered to compare the fits of the MOPGW distribution with other aforementioned models.
Tables 5 and 6 provide the MLEs and associated standard errors (SEs) (between parentheses) of the parameters of all fitted models and the values of W * , A * , KS, and PV, for both data sets, respectively. They show that the MOPGW distribution gives the best fit to the given data sets and it can be considered a very competitive distribution to aforementioned extensions of the Weibull distribution. Figure 3 displays the histogram plots for the gauge lengths and cancer data, and the fitted PDF, CDF, SF, and PP plots of the MOPGW distribution.
The TTT plot of gauge lengths data and the HRF plot of the MOPGW distribution are displayed in Figure 4, whereas the TTT plot of cancer data and MOPGW HRF plot are shown in Figure 5. It is clear that the MOPGW HRF is increasing for gauge lengths data, whereas it is unimodal (upside down bathtub) for cancer data. Furthermore, the scaled TTT plot for gauge lengths data is concave which indicates an increasing HRF, whereas it is concave then convex for cancer data which indicates a unimodal HRF. Hence, the MOPGW distribution is a suitable for modeling both data sets.