The Beta Log-logistic Weibull Distribution : Model , Properties and Application

We propose and develop the properties of a new generalized distribution called the beta log-logistic Weibull (BLLoGW) distribution. This model contain several new distributions such as beta log-logistic Rayleigh, beta log-logistic exponential, exponentiated log-logistic Weibull, exponentiated log-logistic Rayleigh, exponentiated log-logistic exponential, log-logistic Weibull, log-logistic Rayleigh and log-logistic distributions as special cases. Structural properties of this generalized distribution including series expansion of the probability density function and cumulative distribution function, hazard function, reverse hazard function, quantile function, moments, conditional moments, mean deviations, Bonferroni and Lorenz curves, Rényi entropy and distribution of order statistics are presented. The parameters of the distribution are estimated using maximum likelihood estimation technique. A Monte Carlo simulation study is conducted to examine the bias and mean square error of the maximum likelihood estimates. A real dataset is used to illustrate the applicability and usefulness of the new generalized distribution.


Introduction
Generalized distributions are of tremendous practical importance and have recieved considerable attention by many authors in recent years.Eugene, Lee and Famoye (2002) introduced a generalized beta distributions and presented results on the beta normal distribution.The work presented by Jones (2004) is also related to the beta-generated family.Recently, Barreto-Souza, Santos and Cordeiro (2010) introduced the beta generalized exponential distribution where they derived mathematical properties of the distribution.Also, Pescim et al. (2010) studied the beta generalized half-normal distribution.Nadarajah, Cordeiro & Ortega (2012) presented general results on the beta modified Weibull distribution.Famoye, Lee & Olumolade (2005) studied beta Weibull distribution and showed that the lifetime distribution is unimodal and Cordeiro and Nadarajah (2011) extended the beta Weibull distribution by beta exponentiated Weibull distribution.There are several generalizations of the log-logistic distribution including beta log-logistic distribution, presented by Lemonte (2014) and the log-logistic Weibull distribution by Oluyede et al. (2016).
The primary motivation for the development of the beta log-logistic Weibull distribution is the modeling of lifetime data and other data types with a diverse model that takes into consideration not only shape, and scale but also skewness, kurtosis and tail variation.Also, motivated by various applications of log-logistic, Weibull and beta distributions in several areas including reliability, exponential tilting (weighting) in finance and actuarial sciences, as well as economics, where loglogistic distribution plays an important role in income, we construct and develop the statistical properties of this new class of generalized distribution called the beta log-logistic Weibull distribution and apply it to real lifetime data in order to demonstrate the usefulness of the proposed distribution.
In this paper, the results are organized in the following manner.The beta log-logistic Weibull (BLLoGW) distribution, its sub-models, quantile function, hazard and reverse hazard functions are given in section 2. In section 3, moments, moment generating function and conditional moments are presented.Mean deviations, Lorenz and Bonferroni curves are given in section 4. Section 5 contain results on Rényi entropy, density of the order statistics and L-moments.Maximum likelihood estimates of the model parameters are given in section 6.A Monte Carlo simulation study to examine the bias and mean square error of the maximum likelihood estimates are presented in section 7. Section 8 contains an application of the new model to real data set.A short conclusion is given in section 9.

The Model Definition
If G(x) denote the cumulative distribution function of a random variable X, then a generalized class of distribution can be defined by where the parameters a and b are shape parameters, B(a, b) = Γ(a)Γ(b)/Γ(a + b) denote the beta function and Γ(.) is the gamma function.The corresponding probability density function (pdf) is given by where g(x) is the pdf of the baseline cumulative distribution function (cdf) G(x).

Beta Log-Logistic Weibull Distribution
In this subsection, the model is presented.A series expansion of the BLLoGW cdf is given.The new BLLoGW cdf is given by where (Oluyede et al., 2016) with parameters c, α and β > 0. If |t| < 1 and b > 0 is real non-integer, we apply series representation to the BLLoGW cdf to obtain where G ELLoGW (x) is the exponentiated log-logistic Weibull cdf with the exponentiated parameter a + j > 0. The corresponding pdf from equation ( 2) is given by

Expansion of Density Function
In this section, the expansion of the BLLoGW pdf is presented.Applying equation (4) to the expression the BLLoGW pdf can be written as for x > 0, c > 0, a > 0, b > 0, α > 0, β > 0, where parameter α control the scale of the distribution and the parameters c, β, a, and b control the shape, skewness and tail weight of the distribution.
Consequently, the BLLoGW pdf can be written as where

Hazard and Reverse Hazard Functions
The hazard and reverse hazard functions of the BLLoGW distribution are presented in this subsection.The hazard rate and reverse hazard functions of the BLLoGW distribution are given by and respectively.Graphs of the hazard function are presented in Figure 2. The graphs show different shapes including monotonically decreasing and increasing, bathtub followed by upside-down bathtub and upside-down bathtub shapes.The BLLoGW distribution is flexible to accommodate both monotonic and non-monotonic hazard behaviors that are likely to be encountered when dealing with lifetime and reliability data.

Some Sub-models
In this subsection, the nested-models of the BLLoGW distribution are presented.
• When β = 1, we obtain beta log-logistic exponential (BLLoGE) distribution.The cdf of BLLoGE distribution is given by • When β = 2, we obtain beta log-logistic Rayleigh (BLLoGR) distribution.The cdf of BLLoGR distribution is given by • When α → 0 + , we obtain beta log-logistic (BLLoG) distribution.The cdf of BLLoG distribution is given by • When b = 1, the BLLoGW distribution becomes the exponentiated log-logistic Weibull (ELLoGW) distribution.The cdf of ELLoGW distribution is given by • When b = 1 and β = 1, the BLLoGW distribution becomes the exponentiated log-logistic exponential (ELLoGE) distribution and the cdf is given by • When α → 0 + and b = 1, the BLLoGW distribution becomes the exponentiated log-logistic (ELLoG) distribution.The cdf of ELLoG distribution is given by • When a = 1, we obtain Lehmann type-II log-logistic Weibull (LELLoGW) distribution.The cdf of LELLoGW distribution is given by • When a = 1 and β = 1, we obtain Lehmann type-II log-logistic exponential (LELLoGE) distribution.The cdf is given by • When a = 1 and β = 2, we obtain Lehmann type-II log-logistic Rayleigh (LELLoGR) distribution.The cdf is given by • When α → 0 + , and a=1, we obtain Lehmann type-II log-logistic (LELLoG) distribution.The cdf is given by • When a = b = 1, the BLLoGW cdf reduces to the log-logistic Weibull (LLoGW) distribution.The cdf is given by • When a = b = β = 1, the BLLoGW cdf reduces to the log-logistic exponential (LLoGE) cdf and is given by • When a = b = 1 and α → 0 + , the BLLoGW cdf reduces to the log-logistic (LLoG) cdf and is given by

Quantile Function
To obtain the quantile function of the EBW distribution, we can invert the following equation: ).Consequently, we obtain the quantile function of the BLLoGW distribution by solving the equation ( 25) using numerical methods.
log (1 Consequently, random number can be generated based on equation ( 25).The quantile for selected values of the BLLoGW distribution parameters are listed in Table 1.

Moments and Conditional Moments
In this section, we present moments and conditional moments of the BLLoGW distribution.

Moments
Moments are used to understand different characteristics of a distribution such as skewness, kurtosis, central tendency and dispersion.
Theorem: The r th non-central moment of the BLLoGW distribution is given by ) .
Proof: Now, we apply the expansion is the beta function.The first six moments, standard deviation (SD), coefficient of variation (CV), coefficient of skewness (CS) and coefficient of kurtosis (CK) for some selected parameters values of the BLLoGW distribution are given in Table 2.The variance (σ 2 ), (SD (σ)), CV, CS and CK are given by , respectively.

Conditional Moments
The r th conditional moment of the BLLoGW distribution is given by where The mean residual life function E(X − t|X > t) can be readily obtained.

Mean Deviations, Bonferroni and Lorenz Curves
The amount of scatter in a population can be measured to some extent by the totality of deviations from the mean and median.These are known as the mean deviation about the mean and the mean deviation about the median.
Bonferroni and Lorenz curves are widely used tool for analyzing and visualizing income inequality.Lorenz curve, L(p) can be regarded as the proportion of total income volume accumulated by those units with income lower than or equal to the volume x, and Bonferroni curve, B(p) is the scaled conditional mean curve, that is, ratio of group mean income of the population.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.Mean deviations, Lorenz and Bonferroni curves for the BLLoGW distribution are presented in this section.

Mean Deviations
The mean deviation about the mean µ and the mean deviation about the median M are defined by and respectively.The mean deviation about the mean µ for BLLoGW distribution is given by where ) .
The mean deviation about the median M for BLLoGW distribution is given by where ) .

Bonferroni and Lorenz Curves
Lorenz and Bonferroni curves for the BLLoGW distribution are presented in this subsection.Bonferroni curve for the BLLoGW distribution is given by where ) .
Lorenz curves for the BLLoGW distribution is given by where T (q) is given above.

Rényi Entropy and Order Statistics
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.Order statistics play an important role in probability and statistics, particularly in reliability and lifetime data analysis.Rényi entropy and distribution of order statistics from the BLLoGW distribution are presented in this section.

Rényi Entropy
Rényi entropy is defined as Note that, Therefore, Using the binomial expansion, we have ) .
As a result, Rényi entropy is given by for v > 0, v 1.

Order Statistics
Suppose that X 1 , X 2 , ...., X n denote continuous independent BLLoGW random variables with common cdf and pdf.The pdf of the k th order statistic for a random sample of size n from BLLoGW distribution is given by by using the identity ( ∑ ∞ j=0 a j x j ) m = ∑ ∞ j=0 c j,m x j for n positive integer (see Gradshteyn and Ryzshik (2000)), where we have The distribution of the k th order statistics is given by: The t th moment of the distribution of the i th order statistics of BLLoGW distribution can be derived from the result of Barakat and Abdelkader (2004), that is, where Using the identity x k for n positive integer (see Gradshteyn and Ryzshik (2000)), we have where c 0,l = a l 0 and c j,l =( ja 0 ) −1 ∑ j q=1 (lq − j + q)a q c j−q,l .Therefore, ∫ ∞ 0 Note that Using the series expansion above, we have Therefore, the t th moment of the distribution of the i th order statistic from BLLoGW distribution is given by (41)

Estimation and Inference
Let x 1 , x 2 , ..., x n denote a random sample of size n from BLLoGW distribution and φ = (c, a, b, α, β) T the vector of parameters.The log-likelihood function of the parameter vector can be written as The associated score vector is where the elements are given by and respectively.Solving the nonlinear system equations, U n (φ) = 0 yields the maximum likelihood estimates.These equations can be solved numerically via iterative methods such as Newton-Raphson technique using statistical software.
We maximized the likelihood function using NLmixed in SAS as well as the function nlm in R (R Development Core Team ( 2011)).These functions were applied and executed for wide range of initial values.This process often results or lead to more than one maximum, however, in these cases, we take the MLEs corresponding to the largest value of the maxima.In a few cases, no maximum was identified for the selected initial values.In these cases, a new initial value was tried in order to obtain a maximum.
The issues of existence and uniqueness of the MLEs are theoretical interest and has been studied by several authors for different distributions including Seregina(2010), Santos Silva and Tenreyros (2010), Zhou (2009), and Xia, Mi, and Zhou (2009).At this point we are not able to address the theoretical aspects (existence, uniqueness) of the MLE of the parameters of the BLLoGW distribution.

Asymptotic Confidence Interval
The Fisher information of the BLLoGW distribution is a 5 × 5 symmetric matrix given by 1, 2, 3, 4, 5.The total Fisher information matrix nI(φ) can be approximated by , for i, j = 1, 2, 3, 4, 5. Let φ = (â, b, ĉ, α, β) be the maximum likelihood estimates of φ=(a, b, c, α, β).Under the usual regularity conditions and that the parameters are in the interior of the parameter space, but not on the boundary, are met, we have , where I(φ) is the expected Fisher information matrix.The multivariate normal distribution with mean vector 0 and covariance matrix I(φ) can be used to construct confidence intervals for the BLLoGW distribution parameters.A large sample 100(1 − ξ)% two-sided confidence intervals for BLLoGW distribution parameters: a, b, c, α and β are given by: â , where Z ξ/2 is the 1 − ξ 2 quantile of the standard normal distribution.

Likelihood Ratio Test
The likelihood ratio (LR) test can be used to compare the fit of the BLLoGW distribution with its nested-models for a given data set.For example, to test a where ĉ, â, b, α, and β are the unrestricted estimates, and α and β are the restricted estimates.The LR test rejects the null hypothesis if ω > χ 2 d , where χ 2 d denote the upper 100d% point of the χ 2 distribution with 3 degrees of freedom.

Simulation Study
The performance and accuracy of MLEs of the BLLoGW distribution parameters is examined by conducting a simulation study in this section.and RMS E = , respectively.The mean MLEs of the BLLoGW distribution parameters along with their respective root mean squared errors (RMSE) and average bias for different sample sizes are listed in Table 3. From the results in the Table 3, it can be verified that the mean estimates of the parameters converges to the true parameter and the RMSEs decay toward zero as the sample size n increases.We also notice that for all the parametric values, the biases decrease as the sample size n increases.

Application
In this section, we present an example to illustrate the flexibility of the BLLoGW distribution and its sub-models for data fitting.We compared the fit of the BLLoGW distribution with the fits of the nested BLLoG, ELLoG, LLoG distributions and the non-nested beta Weibull log-logistic (BWLLoG) distribution.The pdf of the BWLLoG distribution (Makubate et al. (2018)) is given by for a, b, c, α, β > 0.

Stress-Rupture Life of Kevlar 49/Epoxy Strands Data
This data reported by Cooray and Ananda (2008) consist of observations that represent the stress-rupture life of kevlar 49/epoxy strands which are subjected to constant sustained pressure at the 90% stress level until all have failed, so that the complete dataset with the exact times of failure is recorded and shown in Table ??.Andrews and Herzberg (1985) and Barlow, Toland and Freeman (1984) gives the recorded failure times in hours.Initial values for BLLoGW model in R code are a = 0.8, b = 1.0, c = 0.2, α = 1.0, β = 2.0.The asymptotic covariance matrix of the MLE's for the BLLoGW distribution I −1 n ( φ) is given by:  and the approximate 95% two-sided confidence interval for the parameters a, b, c, α and β are given by: 6.46809 ± 29.9107982, 5.07658 ± 26.1505485, 0.22372 ± 0.5841883, 0.244637 ± 0.7246439, and 0.93981 ± 0.7414126, respectively.
Table 5 gives the maximum likelihood estimates (MLE), standard errors (in parentheses), Akaike Information Criterion  The LR statistics for testing the hypothesis H 0 : BLLoG against H a : BLLoGW, H 0 : ELLoG against H a : BLLoGW and H 0 : LLOG against H a : BLLoGW are 258.0299(p-value < 0.0001), 374.6059 (p-value < 0.0001), 33.5119 (p-value < 0.0001), respectively.Since the p-values are small, we reject the null hypothesis and conclude that the BLLoGW distribution is significantly different from BLLoG, ELLoG and LLoG distributions.The BLLoGW distribution is the better model since it has the smallest value for each of the statistics: AIC, AICC, BIC and HIQC.The values of the goodness-of-fit statistics W * and A * gives clear and convincing evidence that the BLLoGW distribution fits better for kevlar 49/epoxy strands data.Plots of the estimated pdf of the BLLoGW, BLLoG, ELLoG, LLoG and BWLLoG distributions fitted to the dataset are given in Figure 7.The Figure showed that BLLoGW distribution is good lifetime distribution when compared with the three sub-models and the non-nested BWLLoG distribution.Also, the Probability Plots of the compared distributions using the failure times of the kevlar data set is shown in Figure 8.The value of SS from the probability plots is smallest for the BLLoGW distribution.
Figure 1.Plots of the BLLoGW Density Function

Figure 2 .
Figure 2. Plots of the BLLoGW Hazard Function

Figure 3 .
Figure 3. Plot of Skewness and Kurtosis for selected parameter values of the BLLoGW distribution

Figure 5 .
Figure 5. Plot of Skewness and Kurtosis for selected parameter values of the BLLoGW distribution The simulation study is repeated N = 2000 times with each sample size n = 35, 50, 70, 100, 200, 400, 800, 1600 and true parameters valuesI : a = 1, b = 1, c = 1, α = 0.02, β = 2.2 and II : a = 1, b = 1, c = 0.8, α = 0.5, β = 3.In this simulation study, we computed three quantities: the mean estimate, root mean square error and average bias.The mean estimate, average bias (Abias) and root mean square error (RMSE) of the MLE φ for the parameter φ = a, b, c, α, β are given by (AIC), Bayesian Information Criterion (BIC), Consistent Akaike Information Criterion (AICC) and Hannan-Quinn information criterion (HQIC) of BLLoGW, BLLoG, ELLoG, LLoG and BWLLoG distributions.The values of the goodnessof-fit statistics W * and A * are also presented.The smaller the values of these statistics the better the fit.For the probability plot, we plotted F(y ( j) ; ĉ, α, β, â, b) against j − 0.375 n + 0.25 , j = 1, 2, • • • , n, where y ( j) are the ordered values of the observed data.The measures of closeness are given by the sum of squares

Figure 7 .
Figure 7. Fitted PDF for Failure Times of Kevlar Data Set

Table 3 .
Monte Carlo Simulation Results: Mean, Average Bias and RMSE

Table 4 .
Failure times data of kevlar 49/epoxy strands with pressure at 90%