The Marshall-Olkin Half Logistic-G Family of Distributions With Applications

Attempts have been made to define new classes of distributions that provide more flexibility for modeling data that is skewed in nature. In this work, we propose a new family of distributions namely the Marshall-Olkin Half Logistic-G (MOHL-G) based on the generator pioneered by [Marshall and Olkin , 1997]. This new family of distributions allows for a flexible fit to real data from several fields, such as engineering, hydrology, and survival analysis. The structural properties of these distributions are studied and its model parameters are obtained through the maximum likelihood method. We finally demonstrate the effectiveness of these models via simulation experiments.


Introduction
More work has recently been done by various authors on the development of new families of distributions through the extension of other existing continuous distributions. These new families have found wider applicability in modeling data in different areas such as engineering, economics, biological studies, environmental sciences, just to mention these few. The main objectives of generalizing this new family of distributions are as follows: achieve skewness for symmetrical models; develop special models exhibiting various shapes of hazard rate function; develop heavy-tailed distributions useful in many real data sets; attain more flexible kurtosis compared to that of the baseline distribution; generate skewed, symmetric, J-shaped or reversed-J shaped distributions with better fits than other generalized distributions having the same underlying model. [Marshall and Olkin , 1997], proposed a new distribution with cumulative distribution function (cdf) and probability density function (pdf) given by and respectively, where δ denotes the tilt parameter and G(x; ξ) is the baseline cdf. The distribution is more flexible compared to the exponential, Weibull and gamma distributions.
The Marshall-Olkin distribution has been generalized from several distributions to construct many family of distributions achieving some of the above characteristics. The examples of these include; Marshall-Olkin-G (MO-G) family by [Marshall and Olkin , 1997], beta Marshall-Olkin-G (BMO-G) by [Alizadeh et al. , 2015a], Kumaraswamy Marshall-Olkin-G (KwMO-G) by [Alizadeh et al. , 2015b], among others. The proposed distribution gives a new flexible parametric model useful in modeling various forms of data in reliability studies, survival analysis, statistical mechanics, quality control, economics, bio medical studies, etc. [Lepetu et al. , 2017], introduced a new class of distributions called the Marshall-Olkin Log-logistic Extended Weibull (MOLLEW) family of distributions. Their work employs the Marshall-Olkin transformation to the Log-logistic Weibull distribution to obtain more new flexible models suitable for reliability data.
Marshall-Olkin transformation was applied to various well-known distributions: Weibull by [Ghitany et al. , 2005], [Zhang and Xie , 2007]. Lately, the general results have been addressed by . [Santos-Neo et al. , 2014], introduces a new class of models called the Marshall-Olkin extended Weibull family of distributions which defines at least twenty-one special models. [Chakraborty and Handique , 2017], presented the generalized Marshall-Olkin Kumaraswamy-G distribution. [Lazhar , 2017] developed and studied the properties of the Marshall-Olkin extended generalized Gompertz distribution. [Kumar , 2016] discussed the ratio and inverse moments of Marshall-Olkin extened Burr Type III distribution from lower generalized order statistics.
On the similar note, the Half-Logistic (HL) distribution has not received much attention from researchers in terms of generalization.  proposed a new flexible family of distributions called the Odd exponentiated half logistic-G (OEHL-G) family of distributions using the HL distribution as the generator and studied its mathematical properties. Their results utilized the flexibility of the baseline distribution in modeling various forms of data.
Other generally known families are the beta-G by [Eugene et al. , 2002], odd log-logistic-G by [Gleaton and Lynch , 2004], the transmuted-G by [Shaw and Buckley , 2009], the gamma-G by [Zografos and Balakrishnan , 2009], the Kumaraswamy-G by [Cordeiro and de Castro , 2011], the logistic-G by [Torabi and Montazeri , 2014], exponentiated generalized-G by , the McDonald-G by [Alexander et al. , 2012], T-X family by [Alzaatreh et al. , 2013], the exponentiated half-logistic generated family by , the beta odd log-logistic generalized by [Cordeiro et al. , 2015], generalized transmuted-G by [Nofal et al. , 2017], generalized odd log-logistic-G by . Various structural properties of the extended distributions may be easily explored using mixture forms of exponentiated-G (exp-G) distributions. It is in the same vein that we develop the new family of distributions called Marshall-Olkin Half Logistic-G (MO-HL-G) family of distributions.
The layout of this paper is as follows; Section 2, presents the new generalized family of distributions with some of its structural properties. The Maximum likelihood estimates are derived under Section 3. Some of the special cases of this new generalized distributions are presented in Section 4. Section 5, presents results from Monte Carlo simulation study. In Section 6. we present applications of the proposed model to real data sets and finally give concluding remarks under Section 7.

The Model
We develop the MO-HL-G distribution using the generalization proposed by [Marshall and Olkin , 1997], and taking the baseline distribution to be the HL-G distribution by [Cordeiro et al. , 2016]. The cdf and pdf of the MO-HL-G family of distributions is given by and respectively, for δ > 0,δ = 1 − δ and ξ is a vector of parameters from the baseline distribution function G(.).

Quantile Function
The quantile function for the MO-HL-G family of distributions is derived by solving the non-linear equation Therefore, the quantiles of the MO-HL-G family of distributions may be obtained by solving the equation via iterative methods in R or Matlab software.

Expansion of Density
This sub-section presents statistical properties of the MO-HL-G family of distributions as derived from the general results for the Marshall and Olkin's family of distributions by ]. Considering we can write equation (7) as where f HL−G (x; ξ) and F HL−G (x; ξ) are as given in equations (5) and (4), respectively. We apply the series expansion which is valid for |z| < 1 and k > 0. If δ ∈ (0, 1), we can obtain where w j,k = w j,k (δ) = δ( j + 1)(1 − δ) j (−1) j−k j k . For δ > 1, we have where For δ ∈ (0, 1), equation (7) becomes Using the series expansion we can write It follows that for δ ∈ (0, 1), the MO-HL-G family of distributions can be expressed as an infinite linear combination of the exponentiated-G (Exp-G) distribution with power parameter (l + j − k) and linear component Furthermore, for δ > 1 equation (7) can be written as Applying the series expansion we can write Therefore, for δ > 1, the MO-HL-G family of distributions can be expressed as a linear combination of the Exp-G distribution with power parameter (l + j) and linear component

Distribution of Order Statistics
Suppose that X 1 , X 2 , ..., X n are independent and identically distributed (i.i.d) random variables distributed according to (7). The pdf of the i th order statistic X i:n , is given by If δ ∈ (0, 1), we have where where For δ ∈ (0, 1), using equation (19) and substituting f (x) by equation (5) and F(x) by equation (4), we get By applying the expansions we can write where g p+ j+l−k+i (x; ξ) = (p+ j+l−k+i)g(x; ξ)[G(x; ξ)] p+ j+l−k+i−1 is an Exp-G distribution with power parameter (p+ j+l−k+i) and Furthermore, for δ > 1, we get By applying the binomial expansions we can write where

Entropy
An Entropy gives a measure of variation of uncertainty for a random variable X with pdf g(x). There are two common measures of entropy, namely Shannon entropy by [Shannon , 1951] and Rényi entropy by [Rényi , 1961]. Rényi entropy is defined by where ν > 0 and ν 1. Using expansion (11), for δ ∈ (0, 1) Thus, Rényi entropy for δ ∈ (0, 1) and δ > 1 are given by and where Now, for δ ∈ (0, 1) and using equation (27), we have By considering the following expansions we can write where and By considering the following expansions we can write where and

Moments and Generating Functions
Let X ∼ MO − HL − G(δ, ξ), then the r th moment can be obtained from equations (14) and (16). For δ ∈ (0, 1), where w * j,m,l is as defined in equation (15) and E(W r l+ j−k ) denotes the r th moment of W l+ j−k which follows an Exp-G distribution with power parameter (l + j − k). For δ > 1 where v * j,m,l is as defined in equation (17) and E(W r l+ j ) denotes the r th moment of W l+ j which follows an Exp-G distribution with power parameter (l + j). The incomplete moments can be obtained as follows: For δ ∈ (0, 1) where I l+ j−k (t) = t 0 x r g l+ j−k (x; ξ)dx and w * j,m,l is as defined in equation (15). Also, For δ > 1 where I l+ j (t) = t 0 x r g l+ j (x; ξ)dx and v * j,m,l is as defined in equation (17). The moment generating function (mgf) of X is as follows: For δ ∈ (0, 1) where E(e tW l+ j−k ) is the mgf of the Exp-G distribution with power parameter (l + j − k) and w * j,m,l is as defined in equation (15). For δ > 1 where E(e tW l+ j ) is the mgf of the Exp-G distribution with power parameter (l + j) and v * j,m,l is as defined in equation (17). Furthermore, we can get the characteristic function and is given by where φ l+ j−k (t) is the characteristic function of Exp-G distribution with power parameter (l + j − k) and w * j,m,l is as defined in equation (15). For δ > 1 where φ l+ j (t) is the characteristic function of Exp-G distribution with power parameter (l + j) and v * j,m,l is as defined in equation (17).

Maximum Likelihood Estimation
If X i ∼ MO − HL − W(δ; ξ) with the parameter vector ∆ = (δ; ξ) T . The total log-likelihood = (∆) from a random sample of size n is given by .
The elements of a score vector, U = ∂ ∂δ , ∂ ∂ξ k are given by: respectively. These partial derivatives does not have closed form and can be solved using numerical methods such as Newton-Raphson procedure..

Some Special Cases
In this Section, we present some special cases of the MO-HL-G family of distributions. We considered cases when the baseline distributions are log-logistic, Weibull and normal distributions.

Figure 2. Plots of the pdf and hrf for the MO-HL-W distribution
The pdfs shown in Figure 2 of the MO-HL-W distribution can take various shapes that include reverse-J, uni-modal, left or right skewed shapes. Furthermore, the MO-HL-W distribution exhibit increasing, decreasing, reverse-J, bathtub and upside bathtub shapes for the hazard rate function.

Simulation Study
A simulation study is conducted to evaluate consistency of the maximum likelihood estimators. We simulated for N=1000 times with sample size n= 50, 100, 200, 400, 800 and 1000. Simulation results are shown in Table 1. From the Monte Carlo simulation results, we conclude that our model produces consistent results when estimating parameters for the model.

Applications
In this Section, three real data examples are used to illustrate the usefulness of the proposed new model compared to other competing known non-nested models. We use 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 * ) and Andersen-Darling (A * ) as described by [Chen and Balakrishnan , 1995], Kolmogorov-Sminorv (K-S) and its p-value to assess the best fitting model. The model with the smaller values of these statistics, is deemed to be the better model.
We used the nlm function in R software to estimate the model parameters and the package AdequacyModel in R software for goodness-of-fit test. Model parameters estimates (standard errors in parenthesis) and the goodness-of-fit-statistics for the three data sets are shown in Tables 2, 3 and 4. Plots of the fitted densities, the histogram of the data and probability plots (Chambers, Cleveland, Kleiner and Tukey (1983)) are also presented to show how well our model fits the observed data sets compared to the selected non-nested models. The plots are shown in Figures 4 (a), 4 (b), 5 (a), 5 (b), 6 (a) and 6 (b).

Strengths of 1.5 cm Glass Fibres Data
The first data set is on strengths of 1.5 cm glass fibres. The data set was also analyzed by  and [Smith and Naylor , 1987] Table 2 and the plots shown in Figures 4 (a) and 4 (b), we can conclude that the new proposed model performs better than the selected non-nested models on strength of 1.5 cm glass fibres.

Concluding Remarks
A new model was developed, which is referred to as the Marshall-Olkin-Half Logistic-G family of distributions. The new distribution is a linear combination of the Exp-G distribution. The new proposed distribution can be applied to heavily skewed data and also data that have non-monotonic hazard rate shapes. From real data examples presented in Tables 2, 3 and 4, we conclude that the MO-HL-W distribution performs better than several competing non-nested models.