The Type II Exponentiated Half Logistic-Gompertz-G Power Series Class of Distributions: Properties and Applications

We propose and study a new generalized class of distributions called the Type II Exponentiated Half Logistic-Gompertz-G Power Series (TIIEHL-Gom-GPS) distribution. Some structural properties including expansion of density, ordinary and conditional moments, generating function, order statistics and entropy are derived. We present some special cases of the proposed distribution. The maximum likelihood method is used for estimating the model parameters. The usefulness and importance of the new class of distributions are illustrated by means of two applications to real data sets.


Introduction
New families of distributions are widely studied and employed to describe real lifetime data sets.These new families have found wider applicability in various areas of biology, environmental sciences, economics, physics and hydrology.The type II transformations are very important and have led to various new families of distributions.Some well-known type II transformations include type II exponentiated half logistic generated family of distributions by Al-Mofleh et al. [3], type II half logistic Kumaraswamy distribution by ZeinEldin et al. [24], type II power Topp-Leone generated family of distributions by Bantan et al. [4] and type II half logistic family of distributions by Soliman et al. [22].
Al-Mofleh et al. [3], developed the type II exponentiated half logistic family of distributions with the cumulative distribution function (cdf) given by where Ḡ(x; Ψ ) = 1 − G(x; Ψ ) and G(x; Ψ ) is the baseline cdf with parameter vector Ψ and β, α > 0, are the shape parameters.In this work the parameter β is taken to be equal to 1.
The cdf and probability density function (pdf) of the Gompertz-G (Gom-G) family of distributions by Alizadeh et al. [1] are given by respectively, for λ, γ > 0 and Ψ is the parameter vector from the baseline distribution.We set the parameter λ = 1 from equation ( 2) and (3) to avoid the problem of over parameterisation.
The pdf of the Gompertz distribution is either right or left-skewed.Work by Sanku et al. [21] studied several properties and different methods of estimation of the unknown parameters of Gompertz distribution.The Gompertz distribution has a monotonically increasing hazard rate function.Some of its recent generalization include the Topp-Leone-Gompertz-G family of distributions by Oluyede et al. [19] and the Marshall-Olkin-Gompertz-G family of distributions by Chipepa and Oluyede [11].
The study of lifetimes holds significance in numerous scientific and technological areas.Various distributions have been proposed in existing literature to model lifetime data using compounding method.Examples include the type II Exponentiated Half-Logistic-Topp-Leone-G power series class of distributions by Moakofi et al. [17], a new generalized family of lifetime distributions by Goldoust et al. [13], a new two-sided class of lifetime distributions by Kharazmi et al. [15], and the Odd Log-Logistic Transmuted-G family of distributions by Alizadeh et al. [2].
We are motivated by the applicability of the Gompertz-G family of distributions in many fields.Furthermore, the usefulness of power series distributions and the versatility of generalized power series distributions inspired the development of the TIIEHL-Gom-GPS class of distributions.The proposed distribution exhibit both monotonic and non-monotonic hazard rate functions, which is a crucial improvement to the Gompertz distribution.Also, from data modeling examples presented, the new model is a strong alternative model to reliability data.We hope the newly developed model will get attention from various researchers.
This paper is organized as follows: The structural properties of the new TIIEHL-Gom-GPS class of distributions are presented in Section 2. Section 3 contains some special cases of the TIIEHL-Gom-GPS class of distributions.
Monte Carlo simulation study is presented in Section 4. Applications of the proposed model to real data are given in Section 5, followed by concluding remarks.

The Model and Properties
Let N be a discrete random variable following a power series distribution assumed to be truncated at zero, whose probability mass function (pmf) is given by where 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.
We combine the type II exponentiated half logistic-Gompertz-G (TIIEHL-Gom-G) distribution and the power series distribution to obtain a new class of distributions, namely, TIIEHL-Gom-GPS distribution.The cdf of the TIIEHL-Gom-G family of distributions is obtained by inserting the cdf in equation ( 2 Thus, the cdf of the life length of the whole system, X, that is, the cdf of the TIIEHL-Gom-GPS distribution which is the marginal distribution of X = Y (1) , is given by The corresponding pdf and hazard rate function (hrf) are given by and respectively, where α, γ, θ > 0 and Ψ is the parameter vector from the baseline distribution.

Quantile Function
The quantile function of the TIIEHL-Gom-GPS class of distributions is obtained by inverting F (x; α, γ, θ, Ψ ) = u, 0 ≤ u ≤ 1.This is equivalent to solving the equation 0 ≤ u ≤ 1, which can be expressed as The solution of the non-linear equation (9) gives the quantiles of the TIIEHL-Gom-GPS class of distributions.Quantiles for selected parameter values for the type II exponentiated half logistic-Gompertz-Weibull Poisson (TIIEHL-Gom-WP) distribution are shown in Table 2.

Expansion of Density
In this sub-section, we present the series expansion of the TIIEHL-Gom-GPS class of distributions.The pdf in equation ( 6) can be written as where g s+1 (x; Ψ ) = (s + 1) (G(x; Ψ )) s g(x; Ψ ) is the exponentiated-G (Exp-G) distribution with power parameter (s + 1) and Thus, the pdf of the TIIEHL-Gom-GPS class of distributions can be expressed as an infinite linear combination of Exp-G distributions.For derivations visit the appendix.
The coefficients of variation (CV), skewness (CS) and kurtosis (CK) can be readily obtained.The variance (σ 2 ), standard deviation (SD=σ), CV, CS and CK are given by , respectively.Note that the r th cumulant of the random variable X can be readily obtained from the recursive relationship: r , so that the CS and CK are given by τ and . A table of moments, SD, CV, CS, and CK for selected parameter values of the special case of the Type II Exponentiated Half Logistic-Gompertz-Weibull Poisson (TIIEHL-Gom-WP) distribution are given in Table 3.

Conditional Moments
The r th conditional moment of the TIIEHL-Gom-GPS class of distributions is given by where for α, γ, θ > 0, and parameter vector Ψ.In this section, we present the distribution of the k th order statistic and Rényi entropy for the TIIEHL-Gom-GPS class of distributions.

Order Statistics
Order statistics are very useful in probability and statistics.Let X 1 , X 2 , ..., X n be a random sample from TIIEHL-Gom-GPS class of distributions and suppose X 1:n < X 2:n , ... < X n:n denote the corresponding order statistics.The pdf of the k th order statistic is given by where g l+1 (x; Ψ ) = (l + 1)g(x; Ψ )G l (x; Ψ ) is an Exp-G pdf with power parameter (l + 1) and the linear component v * l+1 is given by The t th moment of the distribution of the k th order statistic from TIIEHL-Gom-GPS class of distributions can be readily obtained from equation (13).Visit the appendix for derivations of the pdf of the k th order statistic.

Rényi Entropy
An entropy is a measure of uncertainty or variation of a random variable.Rényi entropy [20] is a generalization of Shannon entropy [23].Rényi entropy for the TIIEHL-Gom-GPS distribution after some simplifications can be written as where ] ν dx is Rényi entropy for an Exp-G distribution with power parameter (k/ν + 1) and Consequently, Rényi entropy of the TIIEHL-Gom-GPS class of distributions can be readily derived from Rényi entropy of the Exp-G.See the appendix for the derivations.

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THE TYPE II EXPONENTIATED HALF LOGISTIC-GOMPERTZ-G POWER SERIES DISTRIBUTION

Parameter Estimation
Let X i ∼ T IIEHL − Gom − GP S(α, γ, θ, Ψ ) and ∆ = (α, γ, θ, Ψ ) T be the parameter vector.The log-likelihood ℓ = ℓ(∆) based on a random sample of size n is given by The maximum likelihood estimates of the parameters, denoted by ∆ is obtained by solving the nonlinear equation ∂Ψ k ) T = 0, using a numerical method such as Newton-Raphson procedure.The multivariate normal distribution N q+3 (0, J( ∆) −1 ), where the mean vector 0 = (0, 0, 0, 0) T and J( ∆) −1 is the observed Fisher information matrix evaluated at ∆, can be used to construct confidence intervals and confidence regions for the individual model parameters.

Special Sub-Models of the TIIEHL-Gom-GPS Distribution
In this section, we look at some special cases of the TIIEHL-Gom-GPS class of distributions.These special cases are the type II exponentiated half logistic-Gom-Weibull Poisson (TIIEHL-Gom-WP), type II exponentiated half logistic-Gom-Weibull Geometric (TIIEHL-Gom-WG), type II exponentiated half logistic-Gom-Log logistic Poisson (TIIEHL-Gom-LLoGP) and type II exponentiated half logistic-Gom-Log logistic Geometric (TIIEHL-Gom-LLoGG) distributions.The cdf and pdf of the

Type II Exponentiated Half Logistic-Gompertz-Weibull Geometric (TIIEHL-Gom-WG) Distribution
The cdf and pdf of the TIIEHL-Gom-WG distribution are given by for α, γ, b > 0 and 0 < θ < 1.  Figure 3 shows the pdf and hrf plots for selected parameter values for the TIIEHL-Gom-WG distribution.The distribution is almost symmetric, right or left-skewed.The hazard rate function exhibits bathtub, upside bathtub followed by bathtub, increasing, decreasing and J shapes. Figure 4 shows 3D plots of skewness and kurtosis of TIIEHL-Gom-WG distribution for some fixed values of parameters.As we fix θ and b we can see that the skewness and kurtosis are increasing.

Exponentiated Half Logistic-Gompertz-Log Logistic Geometric (TIIEHL-Gom-LLoGG) Distribution
The cdf and pdf of the TIIEHL-Gom-LLoGG distribution are given by for α, γ, c > 0 and 0 < θ < 1.The TIIEHL-Gom-LLoGG distribution pdf exhibits extreme tails, almost symmetric and reverse-J shapes.The shapes of the hrf can be monotonic and non-monotonic as shown in Figure 7. Figure 8 shows 3D plots of skewness and kurtosis of TIIEHL-Gom-LLoGG distribution for some fixed values of parameters.

Simulation Study
In this section, we examine the performance of the TIIEHL-Gom-WP distribution by conducting various simulations for different sample sizes (n= 25, 50, 100, 200 and 400) via the R package.We simulate 1000 samples for the true parameter values given in Tables 4 and 5.The average bias (ABIAS) and root mean square error (RMSE) are given by: , respectively.From the results, we can observe 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.Clearly, the model produces consistent estimates.

Applications
In this section, we present applications and empirically establish the flexibility of the TIIEHL-Gom-WP distribution by means of two real data sets.We compared the TIIEHL-Gom-WP distribution to various non-nested models.We use R software to estimate model parameters and standard errors.We assessed model performance using the following goodness-of-fit statistics: -2loglikelihood (-2 log L), Akaike Information Criterion (AIC), Consistent Akaike Information Criterion (AICC), Bayesian Information Criterion (BIC), Cramér-von Mises (W * ), Andersen-Darling (A * ) (see Chen and Balakrishnan [7], for details), and Kolmogorov-Smirnov (K-S) statistic (and its p-value).Tables 6, and 7 shows model parameters estimates (standard errors in parentheses) and several goodness-of-fit statistics.We also provide fitted densities, fitted cdf, Kaplan-Meier (K-M) survival curves, Total Time on Test (TTT) plots and probability plots (as described by Chambers et al. [6]) to demonstrate how well our model fits the selected data sets.

Glass Fibre Data
The first data set is on strengths of 1.5 cm glass fibres.The data set was also analyzed by Makubate et al. [16] From the results shown in Table 6, we can conclude that the TIIEHL-Gom-WP model performs better on glass fibre data compared to the several competing non-nested models included in this paper.Based on Figures 9(a) and 9(b), we observe that TIIEHL-Gom-WP offers more flexibility on fitting the glass fibre data set.From Figure 10, the closeness of the fitted Kaplan-Meier survival curve and fitted cdf using TIIEHL-Gom-WP to the empirical Kaplan-Meier and empirical cdf is clear.The TTT plot indicates an increasing hazard rate function which means that the TIIEHL-Gom-WP distribution can be used to model this data.

Figure 1 .Figure 2 .
Figure 1.Plots of the pdf and hrf for the TIIEHL-Gom-WP distribution

Figure 3 .Figure 4 .
Figure 3. Plots of the pdf and hrf for the TIIEHL-Gom-WG distribution

Figure 5 .Figure 6 .
Figure 5. Plots of the pdf and hrf for the TIIEHL-Gom-LLoGP distribution

Figure 9 .Figure 10 .
Figure 9. Fitted densities and probability plots for glass fibre data

Table 1 .
Special Cases of the TIIEHL-Gom-GPS Distribution

Table 2 .
Table of Quantiles for Selected Parameters of the TIIEHL-Gom-WP Distribution

Table 4 .
Monte Carlo Simulation Results for TIIEHL-Gom-WP Distribution: Mean, RMSE and Average Bias

Table 5 .
Monte Carlo Simulation Results for TIIEHL-Gom-WP Distribution: Mean, RMSE and Average Bias

Table 7
presents model parameters estimates (standard errors in parentheses) and several goodness-of-fit statistics of the TIIEHL-Gom-WP distribution and the non-nested distributions.The K-S value is smallest and p-value corresponding