Marshall-Olkin Log-Logistic Extended Weibull Distribution : Theory , Properties and Applications

Marshall and Olkin (1997) introduced a general method for obtaining more flexible distributions by adding a new parameter to an existing one, called the Marshall-Olkin family of distributions. We introduce a new class of distributions called the Marshall Olkin Log-Logistic Extended Weibull (MOLLEW) family of distributions. Its mathematical and statistical properties including the quantile function hazard rate functions, moments, conditional moments, moment generating function are presented. Mean deviations, Lorenz and Bonferroni curves, Rényi entropy and the distribution of the order statistics are given. The Maximum likelihood estimation technique is used to estimate the model parameters and a special distribution called the Marshall-Olkin Log Logistic Weibull (MOLLW) distribution is studied, and its mathematical and statistical properties explored. Applications and usefulness of the proposed distribution is illustrated by real datasets.


Introduction
Marshall and Olkin (1997) derived an important method of including an extra shape parameter to a given baseline model thus defining an extended distribution.The Marshall-Olkin transformation provides a wide range of behaviors with respect to the baseline distribution (Santos-Neo et al. 2014).Adding parameters to a well-established distribution is a time-honored device for obtaining more flexible new families of distributions (Cordeiro and Lemonte 2011).Several new models have been proposed that are some way related to the Weibull distribution which is a very popular distribution for modelling data in reliability, engineering and biological studies.Extended forms of the Weibull distribution and applications in the literature include Xie : Marshall-Olkin, Log-Logistic Weibull distribution, Maximum imum likelihood estimates of the model parameters are given in section 7. A Monte Carlo simulation study to examine the bias and mean square error of the maximum likelihood estimates are presented in section 8. Section 9 contains applications of the new model to real data sets.A short conclusion remark is given in section 10.

Marshall-Olkin Log-Logistic Extended Weibull Distribution
In this section, the model, hazard function and quantile function of the Marshall-Olkin Log-Logistic Extended Weibull (MOLLEW) distribution are presented.First, we present the class of extended Weibull distributions, Log-Logistic Weibull distribution and the Marshall-Olkin Log-Logistic Extended Weibull distribution.Gurvich et al. (1997) pioneered the class of extended Weibull (EW) distributions.The cumulative distribution function (cdf) is given by where D is a subset of R, and H(x; ξ) is a non-negative function that depends on the vector of parameters ξ.The corresponding probability density function (pdf) is given by g(x; α, ξ) = α exp[−αH(x; ξ)]h(x; ξ), where h(x; ξ) is the derivative of H(x; ξ).The choice of the function H(x; ξ) leads to different models including for example, exponential distribution with H(x; ξ) = x, Rayleigh distribution is obtained from H(x; ξ) = x 2 and Pareto distribution from setting H(x; ξ) = log(x/k).Now consider a general case called log-logistic extended Weibull family of distributions.The distribution is obtained via the use of competing risk model and is given by combining both the log-logistic and extended Weibull family of distributions as given below (Oluyede et al. 2016).The corresponding pdf is given by f (x; c, α, ξ) = e −αH(x;ξ) (1 + x c ) −1 αh(x; ξ) for c, α, ξ > 0 and x ≥ 0. The cumulative distribution function (cdf) of the distribution is given by  Marshall and Olkin (1997).The survival function is given as where δ = 1 − δ and for α, δ > 0. The associated hazard rate reduces to h(x; α, δ, ξ) = δh(x; ξ) 1 − δ exp[−αH(x; ξ)] , x ∈ D, α > 0, δ > 0. (2.6) The corresponding pdf is given by where δ = 1 − δ .The general case of Marshall-Olkin Log-Logistic Extended Weibull (MOLLEW) family of distributions has survival function that is given by The pdf is given by (2.9) for c, α, ξ, δ > 0 and x ≥ 0. The parameter α control the scale of the distribution and c and ξ controls the shape of the distribution and δ is the tilt parameter.The hazard and reverse functions of the MOLLEW distribution are given by and for c, α, ξ, δ > 0 and x ≥ 0, respectively.

Quantile Function
The MOLLEW quantile function can be obtained by inverting (2.12) The quantile function of the MOLLEW distribution is obtained by solving the non-linear equation using numerical methods.Consequently, random number can be generated based on equation (2.13).

Expansion for the Density Function
Using the generalized binomial expression the MOLLEW pdf can be rewritten as follows: where w(j, δ) = δδ j and f BW (x, c, j + 1, α(j + 1), ξ) is the Burr XII Extended Weibull pdf with parameters for c, α(j + 1), ξ, δ > 0 and x ≥ 0, respectively.Thus MOLLEW pdf can be written as a linear combination of Burr XII Extended Weibull density functions.The mathematical and statistical properties of the MOLLEW density function follows directly from those of the Burr XII Extended Weibull density function.

Moments, Moment Generating Function and Conditional Moments
In this section, moments, moment generating function and conditional moments are given for the MOLLEW distribution.The r th moment of the MOLLEW distribution is given by Applying the power series expansion we have (3.1)

Conditional Moments
The r th conditional moment for MOLLEW distribution is given by Note that once H(x; ξ) is specified, the moment and conditional moments can be readily obtained.

Bonferroni and Lorenz Curves
In this subsection, we present Bonferroni and Lorenz curves.Bonferroni and Lorenz curves have applications not only in economics for the study income and poverty, but also in other fields such as reliability, demography, insurance and medicine.Bonferroni and Lorenz curves for the MOLLEW distribution are given by and respectively.The special cases for specified H(x; ξ) can be readily computed.

Rényi Entropy
The concept of entropy plays a vital role in information theory.The entropy of a random variable is defined in terms of its probability distribution and can be shown to be a good measure of randomness or uncertainty.In this section, we present Rényi entropy for the MOLLEW distribution.Rényi entropy is an extension of Shannon entropy.Rényi entropy is defined to be Rényi entropy tends to Shannon entropy as v → 1.Note that [g(x; c, α, ξ, θ)] v = g v (x) can be written as Thus, Consequently, Rényi entropy is given by for v = 1, and v > 0.

Maximum Likelihood Estimation
Let X ∼ M OLLEW (c, α, ξ, δ) and ∆ = (c, α, ξ, δ) T be the parameter vector.The log-likelihood function = (∆) based on a random sample of size n is given by The elements of the score vector , and The equations are obtained by setting the above partial derivatives to zero are not in closed form and the values of the parameters c, α, ξ and δ must be found via iterative methods.The maximum likelihood estimates of the parameters, denoted by ∆ is obtained by solving the nonlinear equation , ∂ ∂δ ) T = 0, using a numerical method such as Newton-Raphson procedure.The Fisher information matrix is given by ∂θ i ∂θ j ), i, j = 1, 2, 3, 4, can be numerically obtained by MATLAB or NLMIXED in SAS or R software.The total Fisher information matrix nI(∆) can be approximated by For a given set of observations, the matrix given in equation (5.1) is obtained after the convergence of the Newton-Raphson procedure via NLMIXED in SAS or R software.
Now consider the Log-Logistic-Weibull distribution.This distribution is obtained via the use of competing risk model and is given by combining both the Log-Logistic and Weibull distribution.In this case, we take H(x; ξ) = x β in the MOLLEW distribution.The MOLLW cdf is given by for c, α, β > 0. The corresponding pdf is given by for c, α, β > 0 and x ≥ 0. Recall the extended case, Marshall-Olkin Log Logistic Extended Weibull (MOLLEW) distribution has survival function given by with corresponding pdf given by (6.4)In this section, we study a special case of the family, namely Marshall-Olkin Log-Logistic Weibull (MOLLW) distribution, by taking H(x; ξ) = x β .The MOLLW survival function is given by The corresponding pdf is given by for c, α, β, δ > 0 and x ≥ 0. Note that the parameter α control the scale of the distribution, c and β controls the shape of the distribution and δ is the tilt parameter.The plots of the MOLLW pdf is given in Figure 6.1.The plots shows that the pdf can be L-shaped, decreasing, unimodal depending on selected parameter values.

Quantile Function
The quantile function of the MOLLW distribution is obtained by solving the non-linear equation The quantile function of the MOLLW distribution is obtained by solving the non-linear equation using numerical methods.Consequently, random number can be generated based on equation (6.7).Table 6.1 lists the quantile for selected values of the parameters of the MOLLW distribution.

Hazard and Reverse Hazard Functions
The hazard and reverse functions of the MOLLW distribution are given by and for c, α, β, δ > 0 and x ≥ 0, respectively.Plots of the MOLLW hazard function are given in Figure 6.2.The plots shows that the hazard function of the MOLLW can either be decreasing, bathtub followed by upside down bathtub or increasing-decreasing depending on selected parameter values.

Some Sub-models
There are several new as well as well known distributions that can be obtained from the MOLLW distribution.The sub-models include the following distributions: • When δ = 1, we obtain the Log-Logistic Weibull (LLW) distribution.
• If c = 1, then the Marshall-Olkin Log-Logistic Weibull distribution reduces to the 3 parameter distribution with survival function (6.10) • If c = 1 and β = 1, then the Marshall-Olkin Log-Logistic Weibull distribution reduces to the 2 parameter distribution with survival function (6.11) • If c = 1 and β = 2, then the Marshall-Olkin Log-Logistic Weibull distribution reduces to the 2 parameter distribution with survival function given by (6.12)

Expansion for the Density Function
Using the generalized binomial expression the MOLLW pdf can be rewritten as follows: where w(j, δ) = δδ j and f BW (x, c, j + 1, α(j + 1), β) is the Burr XII Weibull pdf with parameters for c > 0, j + 1 > 0, α(j + 1) > 0, and β > 0 respectively.Thus MOLLW pdf can be written as a linear combination of Burr XII-Weibull density functions.The mathematical and statistical properties of the MOLLW distribution follows directly from those of the Burr XII Weibull density function.
In this section, moments, moment generating function and conditional moments are given for the MOLLW distribution.The r th moment of the MOLLW distribution is given by Applying the power series expansion we have where B(a, b) = , where E(X n ) is given above.Table 6.2 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 MOLLW distribution for selected values of the parameters, by fixing β = 1.5 and δ = 1.5.Table 6.3 lists the first five moments, SD, CV, CS and CK of the MOLLW distribution for selected values of the parameters, by fixing c = 1.0 and α = 1.5.These values can be determined numerically using R and MATLAB.The SD, CV, CS and CK are given by σ and respectively.Plots of the skewness and kurtosis for selected choices of the parameter β as a function of c as well as for some selected choices of c as a function of β are

Conditional Moments
The r th conditional moment for MOLLW is given by Lornah Lepetu, Broderick O. Oluyede, Boikanyo Makubate, Susan Foya and Precious Mdlongwa 707 where B y (a, b) = y 0 x a−1 (1 − x) b−1 dx is the incomplete beta function.The mean residual lifetime function of the MOLLW distribution is given by E(X|X > t) − t.

Bonferroni and Lorenz Curves
In this subsection, we present Bonferroni and Lorenz Curves.Bonferroni and Lorenz curves have applications not only in economics for the study income and poverty, but also in other fields such as reliability, demography, insurance and medicine.Bonferroni and Lorenz curves for the MOLLW distribution are given by and respectively, where T (q) = ∞ q xg(x)dx, is given by and q = G −1 (p), 0 ≤ p ≤ 1.

Rényi Entropy
The concept of entropy plays a vital role in information theory.The entropy of a random variable is defined in terms of its probability distribution and can be shown to be a good measure of randomness or uncertainty.In this subsection, Rényi entropy of the MOLLW distribution is derived.An entropy is a measure of uncertainty or variation of a random variable.Rényi entropy is an extension of Shannon entropy.Note that [g(x; c, α, β, θ)] v = g v (x) can be written as Consequently, Rényi entropy is given by for v = 1, and v > 0. Let X ∼ M OLLW (c, α, β, δ) and ∆ = (c, α, β, δ) T be the parameter vector.The log-likelihood function = (∆) based on a random sample of size n is given by The elements of the score function are given by Note that the expectations in the Fisher Information Matrix (FIM) can be obtained numerically.Let ∆ = (ĉ, α, β, δ) be the maximum likelihood estimate of ∆ = (c, α, β, δ).Under the usual regularity conditions and that the parameters are in the interior of the parameter space, but not on the boundary, we have: , where I(∆) is the expected Fisher information matrix.The asymptotic behavior is still valid if I(∆) is replaced by the observed information matrix evaluated at ∆, that is J( ∆).The multivariate normal distribution N 4 (0, J( ∆) −1 ), where the mean vector 0 = (0, 0, 0, 0) T , can be used to construct confidence intervals and confidence regions for the individual model parameters and for the survival and hazard rate functions.That is, the approximate 100(1 − η)% two-sided confidence intervals for c, k, α, β and δ are given by: and respectively, where Table 8.1 presents the Average Bias, RMSE, CP and AW values of the parameters c, α, δ and β for different sample sizes.From the results, we can verify that as the sample size n increases, the RMSEs decay toward zero.We also observe that for all the parametric values, the biases decrease as the sample size n increases.Also, the table shows that the coverage probabilities of the confidence intervals are quite close to the nominal level of 95% and that the average confidence widths decrease as the sample size increases.Consequently, the MLE's and their asymptotic results can be used for estimating and constructing confidence intervals even for reasonably small sample sizes. and respectively.The maximum likelihood estimates (MLEs) of the MOLLW parameters c, α, β, δ are computed by maximizing the objective function via the subroutine NLMIXED in SAS as well (bbmle) package in R. The estimated values of the parameters (standard error in parenthesis), -2log-likelihood statistic, Akaike Information Criterion, AIC = 2p−2 ln(L), Consistent Akaike Information Criterion, (AICC = AIC + 2p(p+1) n−p−1 ) and Bayesian Information Criterion, BIC = p ln(n)−2 ln(L), where L = L( ∆) is the value of the likelihood function evaluated at the parameter estimates, n is the number of observations, and p is the number of estimated parameters are presented in Tables 9.2 and 9.4 for the MOLLW distribution and its sub-models MOLLE, MOLLR, MOLL, LLW, LLE and LW distributions and alternatives (non-nested) GD and BW distributions.The Cramer von Mises and Anderson-Darling goodness-of-fit statistics W * and A * , are also presented in these tables.These statistics can be used to verify which distribution fits better to the data.In general, the smaller the values of W * and A * , the better the fit.The AdequacyModel package was used to evaluate the statistics stated above.
We can use the likelihood ratio (LR) test to compare the fit of the MOLLW distribution with its sub-models for a given data set.For example, to test β = 1, the LR statistic is ω = 2[ln(L(ĉ, α, β, δ)) − ln(L(c, α, 1, δ))], where ĉ, α, β and δ are the unrestricted estimates, and c, α and δ are the restricted estimates.The LR test rejects the null hypothesis if ω > χ 2 , where χ 2 denote the upper 100 % point of the χ 2 distribution with 1 degrees of freedom.

Glass Fibers Data
Specifically we consider the following data set which consists of 63 observations of the strengths of 1.5 cm glass fibers, originally obtained by workers at the UK National Physical Laboratory.The data was also studied by (Smith and Naylor 1987).The data observations are given below:  We can conclude that there are significant differences between MOLLW and its sub-models: MOLLE, MOLLR, MOLL, and LLW distributions at the 5% level of significance.The values of the statistics: AIC, AICC and BIC also shows that the MOLLW distribution is a better fit than the non-nested GD and BW distributions for the glass fibers data.There is also clear evidence based on the goodness-of-fit statistics W * and A * that the MOLLW distribution is by far the better fit for the glass fibers data.Plot of the fitted densities, and the histogram of the data are given in Figure 9.1.The asymptotic covariance matrix of the MLE's for the MOLLW model parameters which is the observed Fisher information matrix I −1 n ( ∆) is given by: The LR test statistic of the hypothesis H 0 : LLW against H a : MOLLW, H 0 : LLE against H a : MOLLW, and H 0 : MOLL against H a : MOLLW and are 83.3 (p-value = 0.00001), 421.8 (p-value = 0.00001), 13.9 (p-value= 0.000959).
We can conclude that there are significant differences between MOLLW and the sub-models: LLW, LLE, and MOLL distributions at the 5% level of significance.The values of the statistics: AIC, AICC and BIC also shows that the MOLLW distribution is a better fit than the non-nested GD and BW distributions for the carbon fiber data.Infact, there is also clear evidence based on the Cramer von Mises and Anderson-Darling goodness-of-fit statistics W * and A * that the MOLLW distribution is by far the better fit for the carbon fiber data.Plot of the fitted densities, and the histogram of the data are given in Figure 9.2.A new class of distributions called the MOLLEW distribution and its special case MOLLW are presented.This general and specific class of distributions and some of its structural properties including hazard and reverse hazard functions, quantile function, moments, conditional moments, Bonferroni and Lorenz curves, Rényi entropy, maximum likelihood estimates, asymptotic confidence intervals are presented.Applications of the specific MOLLW model to real data sets are given in order to illustrate the applicability and usefulness of the proposed distribution.MOLLW distribution has a better fit than some of its sub-models and the non-nested GD and BW distributions for the glass fibers data and carbon fibers data.
) for c, α, ξ > 0. Santos-Neto et al. (2014) proposed a new class of models called the Marshall-Olkin extended Weibull (MOEW) family of distributions based on the work by

Figure 6 . 3 :
Figure 6.3: Plots of skewness and kurtosis for selected parameter values

Table 8 .
1: Monte Carlo Simulation Results: Average Bias, RMSE, CP and AW In this section, we present examples to illustrate the flexibility of the MOLLW distribution and its sub-models for data modeling.We fit the density function of the MOLLW, Marshall-Olkin log logistic exponential (MOLLE), Marshall-Olkin log logistic Rayleigh (MOLLR), Marshall-Olkin log logistic and log logistic Weibull (LLW) distributions.We also compare the MOLLW distribution to other models including the gamma-Dagum (GD) (Oluyede et al. 2016) and beta Weibull (Lee et al. 2007) distributions.The GD and BW pdfs are given by

Table 9 .
2: Estimates of Models for Glass fibers Data

Table 9 .
4: Estimates of Models for Carbon fibers Data