A New Class of Generalized Weibull-G Family of Distributions : Theory , Properties and Applications

We propose a generalized class of distributions called the Webull-G Power Series (WGPS) family of distributions and its sub-model Weibull-G logarithmic (WGL) distributions. Structural properties of the WGPS family of distributions and its sub-model WGL distribution including hazard function, moments, conditional moments, order statistics, Rényi entropy and maximum likelihood estimates are derived. A simulation study to examine the bias, mean square error of the maximum likelihood estimators for each parameter is presented. Finally, real data examples are presented to illustrate the applicability and usefulness of the proposed model.


Introduction
There has been tremendous interest in the generalization or modification of the well-known classical distributions in order to provide better and flexible models for different real life applications.In several areas such as reliability, finance and insurance, there is need for generalized form of these distributions.This has led to several new methods for generating generalized new families of distributions.
Weibull distribution is not ideal for modeling non-monotone hazard rates, since in real life applications, empirical hazard rate curves often exhibit non-monotonic shapes such as a bathtub, upside-down bathtub (uni-modal) and others, thus the need for generating new families of distributions that can provide more flexibility in general situations including lifetime modeling.
We propose a new class of generalized distributions called the Weibull-G power series (WGPS) family of distributions and its sub-model Weibull-G logarithmic family of distributions.We compound the Weibull-G family and power series distributions, and introduce a new class of distributions and its sub-model called the Weibull-G logarithmic (WGL) family of distributions.This content of this paper is organized as follows.In section 2, the generalized distribution, it's corresponding probability density function (pdf) and sub-models are given.Some structural properties including the hazard function, quantile function, and various sub-models, moments, moment generating function, conditional moments, the distribution of the order statistics, L-moments, Rényi entropy and estimates of model parameters are developed.In section 3, the special case of the Weibull-G Logarithmic distribution is presented.Monte Carlo simulation study is conducted to examine the bias and mean square error of the maximum likelihood estimators for each parameter in section 4. Applications of the proposed model to real data are given in section 5, followed by summary remarks.

The Model, Sub-models and Properties
The generalized family of distributions called the Weibull-G power series (WGPS) family of distributions and some of its properties including expansion of the density, hazard function, quantile function and sub-models, moments, conditional moments and maximum likelihood estimation of model parameters are derived.Bourguignon, Silva & Cordeiro (2014) proposed the Weibull-G (WG) family of distributions with the cumulative distribution function (cdf) and probability density function (pdf) given by and respectively, for α > 0, β > 0, and x ∈ R, where G(x; ξ) and g(x; ξ) are the cdf and pdf of any baseline distribution that depend on a parameter vector ξ and Ḡ(x; ξ) = 1 − G(x; ξ).
We develop a generalization or an extension of the WG family of distributions by compounding it with the power series class of distributions.Let N be a zero truncated discrete random variable having a power series distribution, whose probability mass function (pmf) is given by where C(θ) = ∑ ∞ n=1 a n θ n is finite, θ > 0 and {a n } n≥1 a sequence of positive real numbers.The power series family of distributions includes binomial, Poisson, geometric and logarithmic distributions (Johnson, Kotz & Balakrishnan (1994)).
Given N, let X 1 , X 2 , ..., X N be identically and independently distributed (iid) random variable following the Weibull-G distributions.Let The Weibull-G Power Series (WGPS) distribution denoted by WGPS(α, β, θ, ξ) is defined by the marginal distribution of X (n) , and is given by The pdf is given by The hazard function is given by

Quantile Function
Let X be a random variable with cdf as in (5).The quantile function

Expansion of Density
The series expansion of the pdf of the WGPS class of distributions is presented in this sub-section.We apply the series expansion where for θ > 0, α( j+1) > 0, β > 0 and ξ > 0, is the Weibull-G (WG) family of distributions and ω j,n (θ) = {(−1) j n!a n θ n }/{C(θ)(n− j − 1)!Γ( j)} are the weights.It follows that the WGPS class of distributions can be written as a linear combination of WG class of distributions.Consequently, the mathematical and statistical properties of the WGPS class of distributions can be readily obtained from those of the WG class of distributions.

Moments
Moments are very crucial in any statistical analysis.In this subsection, we present the r th moment and moment generating function (MGF) of the WGPS class of distributions.The r th moment is given by where Y ∼ WG(α( j + 1), β, ξ).Note that Thus, the moments of any WG distribution can be expressed as an infinite weighted sum of probability weighted moments (PWMs) of the parent distribution.The integral ∫ ∞ −∞ y r g(y; ξ)G(y; ξ) β(k+1)+m−1 dy can be based on the parent quantile where the integral is now calculated over the interval (0, 1).The MGF of the WGPS class of distributions is given by where E(X r ) is given by the equation ( 9).Note also that the r th moment of the WGPS class of distributions can be directly obtained as follows:

Conditional Moments
The conditional moments of the WGPS class of distributions is presented in this subsection.They can be used to calculate the mean residual function and other useful quantities in reliability and survival analysis.The r th conditional moment for the WGPS class of distributions is given by The WGPS vitality and mean residual life functions are given by E(Y|Y > t) and E(Y|Y > t) − t, respectively.

Order Statistics, L-Moments and Rényi Entropy
This section contain the distribution of the order statistic, L-moments and Rényi entropy of the WGPS class of distributions.

Order Statistics
The pdf of the i th order statistic from the WGPS pdf f WGL (x) is given by where f m+i (x) is the pdf of the exponentiated WGPS distribution with parameters θ, α, β, ξ and (m + i), B(., .) is the beta function and the weights w i,m are given by The t th moment of the i th order statistics from the WGPS distribution can be obtained by using the result given by Barakat & Abdelkhader (2004) as follows: Now, applying the result on power series raised to a positive integer from (Gradshetyn & Ryzhik (2000)), with a s = a s+1 θ s , that is, where ] β dx.

L−Moments
The L−moments (Hoskings (1990)) are expectations of some linear combinations of order statistics and they exist whenever the mean of the distribution exits, even when some higher moments may not exist.They are given by The L−moments of the WGPS distribution are obtained from equation ( 14).The first four L−moments are given by λ

Rényi Entropy
Rényi entropy is an extension of Shannon entropy and is given by Rényi entropy tends to Shannon entropy as Rényi entropy for the WGPS class of distributions is given by for v 1, and v > 0.

The Sub-model of WGPS Class of Distributions and Properties
In this section, we derive some properties of the Weibull-G logarithmic (WGL) family of distributions.
Let the random variable X denote the lifetime of a system defined by X = max(X 1 , X 2 , ..., X N ), where we assume that the distribution of each X i is identically and independently distributed following the Weibull-G family of distributions with cdf and pdf given in equations ( 1) and (2).Let the random variable N follow the logarithmic distribution with pmf given as: , n = 1, 2, ..., and 0 < p < 1. ( 18) Then the marginal cdf and pdf of X are given by and for α > 0, 0 < p < 1, β > 0, respectively.

Quantile Function
The WGL quantile function can be obtained by inverting where Note that the quantile function of the WGL distribution is obtained by solving the nonlinear equation Thus, the quantile function of the WGL distribution reduces to Consequently, random numbers can be generated based on equation ( 21), when the function G(x) is specified.Consider a special case where G(x) is Log-logistic (LLo) distribution.

Expansion of Density
Expansion of the density of the WGL distribution is presented in this sub-section.We apply the series expansion (1−y to rewrite the WGL density function as follows: where } are the weights.It follows that the WGL class of distributions can be written as a linear combination of WG class of distributions.

Hazard Function
The hazard function of the WGL distribution is given by for p, α > 0, β > 0, and the parameter vector ξ.Similarly, the reverse hazard function is given by: for p, α > 0, β > 0, and the parameter vector ξ.Plots of the hazard function can be readily obtained for any specified baseline cdf G(x; ξ).

Sub-models of WGL Distribution
Some special models of the WGL distributions are presented in this subsection.Given the baseline cdf G(x; ξ), and pdf g(x; ξ), we present the cdf, pdf and plots of the hazard functions of the Weibull exponential logarithmic, Weibull-Weibull logarithmic, Weibull log-logistic logarithmic, Weibull uniform logarithmic and Weibull Burr XII logarithmic distributions.
Plots of the WWL density and hazard function for the selected parameter values are given in Figure ( 2a) and (2b), respectively.The density and hazard functions reveal different shapes for various values of the parameters, as shown in these plots.The graph of the hazard function exhibit various shapes including: increasing, decreasing, bathtub and J-shape for selected values of the parameters.The cdf of the log-logistic distribution for x > 0, c > 0, is G(x) = 1 − (1 + x c ) −1 and pdf is given as g We define the Weibull log-logistic logarithmic (WLLoL) cdf and the corresponding pdf by respectively, for α > 0, β > 0, c > 0 and 0 < p < 1.
Plots of the WLLoL density and hazard function for selected parameter values are given in Figure ( 3a) and (3b), respectively.The graph of the hazard function exhibit increasing and decreasing shapes for selected values of the parameters, while the density function reveals a decreasing, left skewed and right skewed shapes for various selected values of the parameters, which may also be suitable for analysing extreme values.The cdf of the Uniform distribution is G(x) = x θ and pdf given as g(x) = 1 θ for 0 < x < θ.Then we define the Weibull uniform logarithmic (WUL) cdf by respectively, for α > 0, β > 0, 0 < p < 1 and 0 < x < θ.
Plots of the WBXIIL density and hazard function for the selected parameter values are given in

Moments
In this section, we present the moments of the WGL distribution.The first five non-central moments, standard deviation (SD), coefficients of variation (CV), skewness (CS), and kurtosis (CK) are also given.The r th moment of the WGL distribution is given by where Y ∼ WG(α( j + 1), β, ξ).To obtain the moment generating function (MGF) of the WGL distribution, recall the Taylor's series expansion of the function e tx = ∑ ∞ j=0 (tx) j j! , so that, we have where E(X r ) is given by equation ( 30).
Table 3 lists the first five moments together with the standard deviation (SD or σ), coefficient of variation (CV), coefficient of skewness (CS) and coefficient of kurtosis (CK) of the WLLoL distribution for selected values of the parameters.These values can be determined numerically using R and MATLAB.The variance σ 2 , standard deviation σ, CV, CS and CK are given by , respectively.

Conditional Moments
Conditional moments are very useful in probability and statistics, particularly in reliability and lifetime studies.The mean residual life function and vitality function can be readily obtained from the conditional moments.The r th conditional moments for the WGL distribution is given by x r g(x; ξ)G(x; ξ) β(k+1)+m−1 dx. (31)

Order Statistics and Rényi Entropy
This section contain the distribution of the order statistics and Rényi entropy for the WGL distribution.

Order Statistics
The pdf of the i th order statistic from the WGL pdf f WGL (x) is given by where f m+i (x) is the pdf of the exponentiated WGL distribution with parameters α, β, p and (m + i), B(., .) is the beta function and the weights w i,m are given by

Rényi Entropy
Rényi entropy of the WGL distribution is given by Note that [ f WGL (x; α, β, p, ξ)] v = f v WGL (x) can be written as Consequently, Rényi entropy for the WGL distribution reduces to

Maximum Likelihood Estimation
Let X ∼ WGL(p, α, β) and ∆ = (p, α, β, ξ) T be the parameter vector.The log-likelihood ℓ = ℓ(∆) based on a random sample of size n is given by where H(x i ; ξ) = G(x i ; ξ)/ Ḡ(x i , ξ).Elements of the score vector U = ( ∂ℓ ∂p , ∂ℓ ∂α , ∂ℓ ∂β , ∂ℓ ∂ξ k ) are given by: and respectively.The equations obtained by setting the above partial derivatives to zero are not in closed form and the values of the parameters p, α, β, ξ k must be found via iterative methods.The maximum likelihood estimates of the parameters, denoted by ∆ is obtained by solving the non-linear equation ( ∂ℓ ∂p , ∂ℓ ∂α , ∂ℓ ∂β , ∂ℓ ∂ξ k ) T = 0, using a numerical method such as Newton-Raphson procedure.The Fisher information matrix is given by I(∆) = [I θ i ,θ j ] tXt = E(− ∂ 2 ℓ ∂θ i ∂θ j ), i, j = 1, 2, ..., t, can be numerically obtained by NLMIXED in SAS or R software.The total Fisher information matrix nI(∆) can be approximated by where t is the number of model parameters.For a given set of observations, the matrix given in equation ( 33) is obtained after the convergence of the Newton-Raphson procedure via NLMIXED in SAS or R software.

Simulation Study
In this section, we examine the performance of the WLLoL distribution by conducting various simulations for different sample sizes (n=25, 50, 100, 200, 400, 800) via the R package.We simulate 1000 samples for the true parameters values given in Table 4. Table 4 lists the mean MLEs of the four model parameters along with the respective biases and root mean squared errors (RMSEs).The bias and RMSE are given by: respectively, where θ stand for the parameters c, α, β and p. From the tabulated results, we note that as the sample size n increases, the mean estimates of the parameters tend to be closer to the true parameter values, since RMSEs decay toward zero.
than the non-nested NMW and GDa distributions based on all the statistics presented in Table 5.The probability plots of the compared distributions based on the fracture toughness of alumina data set is shown in Figure 6.The value of SS from the probability plots is smallest for the WLLoL distribution.The LR test statistic of the hypothesis H 0 : WW against H a : WWL and H 0 :W against H a : WWL are 6.44 (p-value = 0.011) and 6.8 (p-value = 0.079).We can conclude that there is significant difference between the fit of the WWL and WW distributions.However, there is a significance difference between the fits of the WWL and W distributions at the 10% significance level.The values of the goodness-of-fit statistics W * , A * and KS in Table 6 point to the WWL distribution as the better fit.The probability plots of the compared distributions based on the fracture toughness of alumina data set is shown in Figure 7.The value of SS from the probability plots in Table 6 is smallest for the WWL distribution.
The LR test statistic of the hypothesis H 0 : WBXII against H a : WBXIIL and H 0 :BXII against H a : WBXIIL are 8.33 (p-value = 0.0039) and 321.32 (p-value < 0.0001).We can conclude that there is significant difference between the fit of WBXIIL and WBXII distributions and also between the fits of WBXIIL and BXII distributions.The WWL and WBXIIL distributions also seemed to be better than the non-nested BEW and GD distributions based on the values of the statistics in Table 6 and Table 7, respectively.Also, the probability plots of the fitted models based on the fracture toughness of alumina data set is shown in Figure 7.The value of SS from the probability plots in Table 7 is smallest for the WBXIIL distribution.
The plots of the WEL density and hazard function for selected parameter values are given in Figure(1a) and (1b), respectively.The density and hazard functions reveal different shapes for various values of the parameters, as shown in these plots.The graph of the hazard function exhibit increasing, decreasing and upside down bathtub shapes for selected values of the parameters.This flexibility makes the WEL hazard function suitable for non-monotonic empirical hazard behavior.
Figure 1.Plots of WEL Density and Hazard Function Figure 2. Plots of WWL Density and Hazard Function Figure 3. Plots of WLLoL Density and Hazard Function The plots of the WUL density and hazard function for selected parameter values are given in Figure (4a) and (4b) respectively.The density and hazard functions revealed different shapes for various values of the parameters, as shown in these plots.The graph of the hazard function exhibit increasing, decreasing and bathtub shapes for the selected values of the parameters while the density function shows decreasing, left and right skewed shapes for selected values of the parameters.(a) Plot of WUL density function (b) Plot of WUL hazard function

Figure 4 .
Figure 4. Plots of WUL Density and Hazard Function Figure 5. Plots of WBXIIL Density and Hazard Function

Figure 6 .
Figure 6.Fitted densities and probability plots fracture toughness of Alumina data

Figure 8 .
Figure 8. Fitted densities and probability plots for fatigue time of 101 6061-T6 aluminum coupons data

Table 1 .
Table 1 lists the quantile for selected values of the parameters of the WLLoL distribution.Table of Quantiles for Selected Parameters Values of the WLLoL Distribution

Table 2 .
Distributions and corresponding G(x; ξ)/G(x; ξ) functions Table of Moments for Selected Parameters Values of the WLLoL Distribution

Table 5 .
Parameter estimates and goodness-of-fit statistics for various fitted models for the alumina data set

Table 6 .
Parameter estimates and goodness-of-fit statistics for various fitted models for the alumina data set

Table 7 .
Parameter estimates and goodness-of-fit statistics for various fitted models for the alumina data set

Table 8 .
Parameter estimates and goodness-of-fit statistics for various fitted models for the aluminum coupons data set