Beta Linear Failure Rate Geometric Distribution with Applications

: This paper introduces the beta linear failure rate geometric (BLFRG) distribution, which contains a number of distributions including the exponentiated linear failure rate geometric, linear failure rate geometric, linear failure rate, exponential geometric, Rayleigh geometric, Rayleigh and exponential distributions as special cases. The model further generalizes the linear failure rate distribution. A comprehensive investigation of the model properties including moments, conditional moments, deviations, Lorenz and Bonferroni curves and entropy are presented. Estimates of model parameters are given. Real data examples are presented to illustrate the usefulness and applicability of the distribution.


Introduction
Let ( ; ) be the cumulative distribution function (cdf) of an absolutely continuous random variable , where ϕ ∈ Ω is the parameter vector. A general class generated from the logit of a beta random variable was introduced by Eugene where a > 0 and b > 0 are two additional parameters whose role is to introduce skewness and to vary tail weight, G(x; ϕ) is an arbitrary parent/baseline cdf, B y (a, b) = ∫ −1 (1 − 0 ) −1 is the incomplete beta function with B(a, b) = B 1 ( , ) and I y ( , ) = ( , ) ( , ) is the incomplete beta function ratio. One major benefit of this class of distributions is its ability of fitting skewed data that cannot be properly fitted by existing distributions.
which is called the exponentiated G distribution (or the Lehmann type-I distribution). See for example, the exponentiated Weibull (Pal et al. (2006), Mudholkar et al. (1995)) and exponentiated exponential (Gupta and Kundu (1999)) distributions. Indeed, if Z is a beta distributed random variable with parameters a and , b then the cdf of ) ( = 1 Z G X  agrees with the cdf given in equation (1). As usual, a random variable X with the cdf (1) is said to have a beta G  (BG) distribution and will be denoted by ~ ( , ; ). Some special cases of BG distributions are given by Bidram et al. (2013). (  x G . Several well-known distributions that belong to the resilience parameter family include the exponentiated Weibull (EW) distribution (see Mudholkar et al. (1995), generalized (or exponentiated) exponential distribution proposed by Gupta and Kundu (1999), and exponentiated type distributions introduced by Nadarajah and Kotz (2006). The generalized exponential-geometric (GEG) distribution of Silva et al. (2010) also belongs to the resilience parameter family.
For general a and b , we can express the cdf given in equation (1)   (  x G distribution in equation (1) which, in principle, follow from the properties of the hypergeometric function that are well established in the literature; see, for example, section 9.1 of Gradshteyn and Ryzhik (2000). The probability density function (pdf) and hazard (failure) rate functions of a BG distribution corresponding to the cdf in equation (1) is the survival function of a BG distribution corresponding to the cdf in equation (1).
The beta Generalized (Beta-G) distribution (beta modified Weibull (BMW)) distribution introduced by Silva et al. (2010) is a rich class of generalized distributions. This class has captured considerable attention over the last few years. Sepanski and Kong (2007) applied the Beta-G distribution to model the size distribution of income. This distribution has been studied in the literature for various forms of the baseline distribution .
G The beta-G distributions that have been explored include the beta normal (BN) (Eugene et al. (2002)), beta Fre c het (BFr) distribution (Nadarajah and Kotz (2004)), beta exponential (BE) distribution (Nadarajah and Kotz (2006) In this article, we attempt to generalized the linear failure rate geometric (LFRG) distribution of Mahmoudi and Jafari (2014) by taking ) ; (  x G in equation (1) to be the cdf of a LFRG distribution, when ( ) = 1− , 0 < < 1. Mahmoudi and Jafari (2014) compounded the linear failure rate distribution with a geometric distribution to obtained new lifetime distribution with five possible shapes for the hazard rate function, that is, increasing, decreasing, upside -down bathtub (unimodal), bathtub and increasing -decreasing -increasing shaped, which are common in reliability and biological studies. Recent generalizations of the linear failure rate distributions include the gamma linear failure rate distribution ), and the Poisson generalized linear failure rate model (Cordeiro et al. (2015)).
The reminder of the paper is organized as follows. In Section 2, we define the beta linear failure rate geometric distribution, expansion for the cumulative and density functions, hazard and reverse hazard functions and some special cases are presented. Moments, moment generating function and conditional moments are discussed in Section 3. In Section 4, we obtain the mean deviations about the mean and the median, Bonferroni and Lorenz curves. Section 5 contains the distribution of the order statistics and uncertainty measures including Rényi and s-entropies. Maximum likelihood estimation is performed in Section 6. Applications are given in section 7, followed by concluding remarks.

Beta Linear Failure Rate Geometric Distribution
Consider the linear failure rate geometric (LFRG) distribution of Mahmoudi and Jafari (2014) with the cdf 0, > , 1 The corresponding pdf of this new distribution is given by where the parameters 0 > a and 0 > b are shape parameters, which characterize the skewness, kurtosis, and unimodality of the distribution.
Graphs of the pdf of BLFRG distribution are given in the Figure 1. The plots show that the BLFR pdf can be decreasing or right skewed among several other possible shapes as seen in the figure. The distribution has positive asymmetry. The reliability (survival) function (RF) of the BLFRG distribution, denoted by The hazard and reverse hazard functions of the BLFRG distribution are given by ) ( respectively. The density and hazard functions can exhibit different behavior depending on the values of the parameters when chosen to be positive, as shown in these plots. However, it is hard to analyze the shape of both the density and hazard function due to their complicated forms. Plots of the hazard rate function for different combinations of the parameter values are given in Figure 2. The plot shows various shapes including monotonically increasing, and bathtub shapes for five combinations of the values of the parameters. This flexibility makes the BLFRG hazard rate function suitable for both monotonic and non-monotonic empirical hazard behaviors that are likely to be encountered in real life situations.

Special Cases of the BLFRG Distribution
The BLFRG distribution is a very flexible model that approaches different distributions when its parameters are changed. The BLFRG distribution contains several special-models including the following distributions.

Quantile Function
In this subsection, we present the quantile function of the BLFRG distribution. The quantile of the BLFRG distribution is obtained by solving the nonlinear equation and U is a uniform variate on the unit interval [0,1]. It follows that the BLFRG variate X are the roots of the equation , denotes the inverse of the incomplete beta function ratio. Quantiles of the BLFRG distribution for selected values of the model parameters are given in Table 1. An R algorithm for the computation of the quantiles of the BLFRG distribution is given in the appendix.

Expansion for the Cumulative and Density Functions
In this subsection, we present some series representations of the cdf and pdf of the BLFRG distribution. The mathematical relation given below will be useful in subsequent sections. Here and henceforth, we let X be a random variable having the ) , , , , ( We obtain some alternative expressions for the cdf The following series representations will be useful in studying the properties of the BLFRG distribution. If b is a positive real non-integer and 1, Applying the series representation, one can re-write the BLFRG cdf as follows: Consequently, the BLFRG cdf can be expressed as a mixture of generalized linear failure rate geometric (GLFRG) distributions with parameters    , , and . j a  Also, when 0 > b is an integer the index j in the series representation stops at 1  b . Using the series representation below: the BLFRG pdf can be expressed as is the pdf of the BLFR distribution that was introduced by Mahmoudi and Jafari (2014). Another form of equation (0.15) is as follows: , and using Taylor expansion of the function In this section, we present the moments and conditional moments of the BLFRG distribution. Moments are necessary and important in any statistical analysis, especially in applications. It can be used to study the most important features and characteristics of a distribution (e.g., tendency, dispersion, skewness and kurtosis). A table of the first six moments and related statistics for selected values of the model parameters is also presented.

Moments
If X has the BLFRG distribution, then the th r moment of X is given by The coefficients of variation, skewness and kurtosis of the BLFRG distribution can be readily obtained according to the following relation SD and CK = 4 −4 1 3 +6 1 2 2 −3 1 4

Conditional Moments
For lifetime models, it is also of interest to find the conditional moments and the mean residual lifetime function. The th n conditional moments for BLFRG distribution is given by is the upper incomplete gamma function. The mean residual lifetime function is given by

Mean Deviations, Bonferroni and Lorenz Curves
In this section we obtain mean deviation about the mean and the mean deviation about the median as well as Bonferroni and Lorenz curves for the BLFRG distribution.

Mean Deviations
The amount of scatter in a population is evidently 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. They are defined by , respectively, so that the mean deviation about the mean is and the mean deviation about the median is

Bonferroni and Lorenz Curves
In this subsection, we present Bonferroni and Lorenz Curves. Bonferroni and Lorenz curves (Bonferroni (1930)) 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 are given by is the lower incomplete gamma function.

Order Statistics and Measures of Uncertainty
In this section, the distribution of order statistics and measures of uncertainty for the BLFRG distribution are presented. 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.

Distribution of Order Statistics
Suppose that n X X , , 1  is a random sample of size n from a continuous pdf, ) (x f . Let The corresponding cdf of

Rényi Entropy
Rényi entropy tends to Shannon entropy as

 v
Note that by using the series representations in equations (12) and (15), we have

s -Entropy
The s -entropy for BLFRG distribution is defined by

Estimation and Inference
In this section, we present the maximum likelihood estimates (MLEs) of the parameters of the BLFRG distribution from complete samples only. Let The associated score function is given by The log-likelihood can be maximized either directly or by solving the nonlinear likelihood equations obtained by differentiating the log likelihood function. The components of the score vector are given by  . These equations cannot be solved analytically, and statistical software can be used to solve them numerically via iterative methods. We can use iterative techniques such as a Newton-Raphson type algorithm to obtain the estimate φ.
The convergence of the estimation procedures often depends on the choice of starting or initial values of the parameters, so one has to be cautious or care must be taken when obtaining the numerical approximations of the expected information matrix. For methods such as the BFGS, approximation of the Hessian matrix is used for the computations of each iteration and this approximation may not be reliable when convergence of the methods occurs too fast, thereby leading to an unreliable approximate Hessian matrix. Other methods of estimation such as generalized method of moments to obtain initial values followed by Newton or quasi-Newton methods to obtain better and reliable parameter estimates may be used. In this paper, we maximize the likelihood function using NLmixed in SAS and nlm in R. The function was 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.
For interval estimation and hypothesis tests on the model parameters, we require the information matrix. The Fisher information matrix is given by quantile of the standard normal distribution.
We can use the likelihood ratio (LR) test to compare the fit of the BLFRG distribution with its sub-models for a given data set. For example, to test

Simulation
In this section, we examine the performance of the BLFRG distribution by conducting various simulations for different sample sizes. We simulate 1000 samples for the true parameters values  Table 3, 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.

Applications
In this section, we present examples to illustrate the flexibility of the BLFRG distribution and its sub-models for data modeling. We also compare the five parameters BLFRG distribution to the beta-Weibull-geometric (BWG) distribution (Bidram et al. (2013)). The cdf and pdf of BWG distribution are given by

Time to failure of kevlar 49/epoxy strands tested at various stress level
The data consists of a real life example is taken from Cooray and Ananda (2008), where 101 data points 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 data set with the exact times of failure is recorded. These failure times in hours, are originally given in Andrews and Herzberg (1985) and Barlow et al. (1984). Initial value for BLFRG model in the R code are Estimates of the parameters of BLFRG distribution and its related sub-models (standard error in parentheses), AIC, AICC, BIC, * W , * A and SS for stress-rupture life of kevlar 49/epoxy strands data are given in Table   4. respectively. Plots of the fitted densities and the histogram, observed probability vs predicted probability, and empirical survival function for the data are given in Figure 3. . We can conclude that there is a significant difference between BLFRG and BR distributions.
Considering the values of the statistics AIC, BIC and the values of SS given in Table 4, we observe that the BLFRG distribution gives a better fit for the data. The values of the goodnessof-fit statistics * W and * A shows that the BLFRG distribution is a better fit that its submodels except for the staistics * W for the BR and R distributions. When the BLFRG distribution is compared to the non-nested five parameter BWG distribution, it is seen that the BLFRG distribution is a competitive distribution.

Fatigue fracture of kevlar 373/epoxy
The data set is on the life of fatigue fracture of Kevlar 373/epoxy that are subject to constant pressure at the 90% stress level until all had failed. The complete data with the exact times of failure was also studied by Andrews and Herzberg (1985), and Barlow, Toland and Freeman (1984). Initial value for BLFRG model in the R code are  Table 5. respectively. Plots of the fitted densities and the histogram, observed probability vs predicted probability, and empirical survival function for the data are given in Figure 4.  Table 5, we observe that the BLFRG distribution gives a better fit for the data. When the BLFRG distribution is compared to the non-nested five parameter BWG distribution, it is seen that the BLFRG distribution is a better fit.

Concluding Remarks
A new and generalized linear failure rate distribution called the beta linear failure rate geometric (BLFRG) distribution is proposed and studied. The BLFRG distribution has several well known distributions including the LFRG, LFR, EG, RG, Rayleigh and exponential distributions as special cases. The density of this new class of distributions can be expressed as a linear combination of BLFR density functions. The BLFRG distribution possesses hazard function with flexible behavior. We also obtain closed form expressions for the moments, mean and median deviations, distribution of order statistics and entropy. Maximum likelihood estimation technique is used to estimate the model parameters. Finally, the BLFRG distribution is fitted to real data sets to illustrate its applicability and usefulness.