The Burr-Weibull Power Series Class of Distributions

A new generalized class of distributions called the Burr-Weibull Power Series (BWPS) class of distributions is developed and explored. This class of distributions generalizes the Burr power series and Weibull power series classes of distributions, respectively. A special model of the BWPS class of distributions, the new Burr-Weibull Poisson (BWP) distribution is considered and some of its mathematical properties are obtained. The BWP distribution contains several new and well known sub-models, including Burr-Weibull, Burrexponential Poisson, Burr-exponential, Burr-Rayleigh Poisson, Burr-Rayleigh, Burr-Poisson, Burr, Lomax-exponential Poisson, Lomax-Weibull, Lomax-exponential, Lomax-Rayleigh, Lomax-Poisson, Lomax, Weibull, Rayleigh and exponential distributions. Maximum likelihood estimation technique is used to estimate the model parameters followed by a Monte Carlo simulation study. Finally an application of the BWP model to a real data set is presented to illustrate the usefulness of the proposed class of distributions.


Introduction
The Burr XII (Burr) distribution is a very useful model that was first discussed by Burr (1942) as a two-parameter family.An additional scale parameter was introduced by Tadikamalla (1980).Examples of data modeled by the Burr distribution include household income, crop prices, insurance risk, travel time, flood levels, and failure data.Other applications include simulation, quantal response, approximation of distributions, and development of non-normal control charts.A number of standard theoretical distributions are limiting forms of the Burr distributions.Rodriguez (1977) presented a comprehensive guide to the Burr distribution and showed that this distribution covers various well-known and useful distributions, including the normal, log-normal, gamma, logistic and extreme-value type-I distributions.Morais and Bareto-Souza (2011) presented results on a compound class of Weibull and power series distributions.The Burr power series distribution was given by Silva and Cordeiro (2015).
The primary motivation for developing the class of Burr-Weibull Power Series (BWPS) distributions is the versatility and flexibility derived from compounding continuous distributions The Burr-Weibull Power Series Class of Distributions including the new distribution called Burr-Weibull distribution with power series distributions to obtain a new class of distributions with desirable properties including hazard function that exhibits increasing, decreasing, bathtub and upside down bathtub shapes.Another important reason for the development of the BWPS class of distributions is the modeling of income and lifetime data with a model that takes into consideration not only shape and scale but also skewness, kurtosis and tail variation.Also, motivated by various applications of power series distributions in several areas including reliability, exponential tilting (weighting) in finance and actuarial sciences, as well as economics we construct and develop the statistical properties of this new class of generalized compound distribution called the Burr-Weibull Power Series class of distributions and apply it to real lifetime data in order to demonstrate the usefulness of the proposed distribution.This class of distributions generalizes the Burr power series and Weibull power series class of distributions, and their sub-classes, respectively.
The results in this paper are organized in the following manner.In section 2, the Burr-Weibull Power Series (BWPS) distribution is presented.The cumulative distribution function (cdf) of the specific following cases: Burr-Weibull Poisson (BWP), Burr-Weibull geometric (BWG), Burr-Weibull logarithmic (BWL) and Burr-Weibull binomial (BWB) distributions are given.Maximum likelihood estimates of the BWPS model parameters are given in section 3. The special case of BWP distribution is discussed in section 4. A Monte Carlo simulation study to examine the bias and mean square error of the maximum likelihood estimates for the BWP distribution is also presented in section 4. Section 5 contains an application of the BWP model to a real data set.A short concluding remark is given in section 6.

Burr-Weibull power series distribution
In this section, the Burr-Weibull Power Series (BWPS) class of distributions is presented.In a recent note, Mdlongwa, Oluyede, Amey, and Huang (2017) presented important results on a new distribution called the Burr-modified Weibull (BMW) distribution which includes Burr-Weibull (BW) distribution as a special case of BMW when λ = 0.An important motivation for this family of distributions, particularly for use in survival and reliability studies is as follows.Suppose the failure of a device is due to the presence of an unknown number of initial defects of the same kind say N , which is identifiable only after causing failure and are repaired perfectly.Let Y i , i = 1, ..., N, denote the time to the failure of the device due to the i th defect and assume the Y i 's are independent and identically distributed (iid) BW random variables independent of N which is a truncated power series random variable, then the time to the first failure can be modeled by a distribution in the class of BWPS distributions.
The proposed class of distributions can be used for series systems with identical components, which is often the case in many industrial applications and biological organisms.Now, consider a sequence of N iid random variables, say Y i , i = 1, . . ., N , from the BW distribution.If Y is a random variable following BW distribution with parameters c, k, α, β > 0, its cdf is given by (1) The corresponding BW survival function and pdf are given by S(y) = (1 + y c ) −k exp(−αy β ), and g(y for c, k, α, β > 0, and y ≥ 0, respectively.Now, let N be a discrete random following a power series distribution assumed to be truncated at zero, whose probability mass function (pmf) is given by where C(θ) = ∞ n=1 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 (Johnson, Kotz, and Balakrishnan 1994).
Thus, the cdf of the life length of the whole system, X, say F θ , is given by which can be done using numerical methods.Consequently, random number can be generated based on equation (4).Some special cases of the BWPS class of distributions are presented in Table 1 for c, k, α, β, θ > 0.
Table 1: Special Cases of the BWPS Distribution The r th moment of the BWPS distribution is given by dt is the beta function (see Appendix for the derivation).

Estimation and inference
Let X ∼ BW P S(c, k, α, β, θ) and ∆ = (c, k, α, β, θ) T be the parameter vector.The loglikelihood function = (∆) based on a random sample of size n is given by where the S(x) and g(x) are given in equation ( 2).The elements of the score vector are given in the Appendix.
The equations obtained by setting the elements of the score vector to zero are not in closed form and the values of the parameters c, k, α, β, θ must be found via iterative methods.The maximum likelihood estimates of the parameters, denoted by ∆ is obtained by solving the nonlinear equation ( ∂ ∂c , ∂ ∂k , ∂ ∂α , ∂ ∂β , ∂ ∂θ ) 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, 5, can be numerically obtained by NLMIXED in SAS or mle2 package in R software.The total Fisher information matrix nI(∆) can be approximated by , i, j = 1, ..., 5. (5) Note that the expectations in the Fisher Information Matrix (FIM) can be obtained numerically.
Let ∆ = (ĉ, k, α, β, θ) be the maximum likelihood estimate of ∆ = (c, k, α, β, θ).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 5 (0, J( ∆) −1 ), where the mean vector 0 = (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.

Burr-Weibull Poisson distribution and simulation study
In this section, we present some results the Burr-Weibull Poisson (BWP) distribution including Monte Carlo simulation study to examine the bias and mean square error of the maximum likelihood estimates.Recall the BWP distribution is a special case of the BWPS class of distributions with C(θ) = e θ − 1 and a n = 1 n! .The cdf is given by for c, k, α, β, θ > 0.

Sub-models of the BWP distribution
The BWP distribution contains several new and known sub-models.In this subsection, we present some of the sub-models.
We observe that the estimates approach the true parameter values as the sample size increases, thus implying consistency of the estimates.Also, from the results, we can verify 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.The bias and RMSEs are given by respectively.

Application
In this section, we present an example to illustrate the flexibility of the BWP distribution and its sub-models for data modeling.BWP distribution is fitted to real data set and these fits are compared to the fits using the sub-models: Burr-Weibull (BW), Burr-exponential Poisson (BEP), Burr-exponential (BE), Burr-Rayleigh Poisson (BRP), Burr-Rayleigh (BR), Burr Poisson (BP), Burr (B), Lomax-Weibull Poisson (LWP), Lomax-Rayleigh Poisson (LRP), Lomax Poisson (LP), Weibull Poisson (WP), and Weibull (W) distributions.We also compare the BWP distribution with the gamma log-logistic Weibull (GLLoGW) (Foya, Oluyede, Fagbamigbe, and Makubate 2017) and beta modified Weibull (BetaMW) (Nadarajah, Cordeiro, and Ortega 2011) distributions.The pdf of the BetaMW distribution is given by Also, the pdf of the gamma log-logistic Weibull (GLLoGW) distribution (Foya et al. 2017) is  The likelihood ratio test statistic for testing H 0 : BW against H a : BWP and H 0 : BRP against H a : BWP are 3.65 (p-value = 0.05607) and 6.9 (p-value = 0.00862).We can conclude that there are significant differences in the fit of the BWP and the BW distribution as well as the fit of BWP and BRP distribution.We can also conclude that there are significant differences in the fit of the BWP and the LWP distribution as well as the fit of the BWP and the LP distribution.There are significant differences in the fit of the BWP and the LRP distribution based on the likelihood ratio test.There is no significant difference in the fit of the BWP and .

Figure 2 :
Figure 2: Plots of BWP Hazard Function

Figure 3 :
Figure 3: Fitted Densities and Probability Plots for Time to Failure of kevlar 49/epoxy strands tested at various stress level data

Table 3 :
MLEs of the parameters, SEs in parenthesis and the goodness-of-fit statistics for kevlar 49/epoxy failure time data