Complementary Kumaraswamy Weibull Power Series Distribution: Some Properties and Application

In this paper, we propose Complementary Kumaraswamy Weibull Power Series (CKWPS) Distributions. The method is obtained by compounding the Kumaraswamy-G distribution and Power Series distribution on a latent complementary distance problem base. The mathematical properties of the proposed class of distribution are studied. The method of Maximum Likelihood Estimation is used for obtaining the estimates of the model parameters. A member of the family is investigated in detail. Finally an application of the proposed class is illustrated using a real data set.


Introduction
In recent times, several compound models have been developed by complementary risk motivation and applied in several areas. Complementary risk problems arise in several areas, such as engineering, public health, economics actuarial science, biomedical studies, demography and industrial reliability.
The event of interest in latent complementary risk scenarios is related to causes which are not completely observable; rather we observe only the maximum lifetime value among all risks. Since it is not possible to observe the lifetime of the event of interest, the event of interest is modeled as a function of the available information, which is taken as the maximum ordered lifetime value among all causes. Example of industrial applications shows that, the failure of a system can be due to several competing causes such as error in design, contamination from dirt and failure of a component, an assembly One of the most important distributions used in modeling lifetime data is the Weibull distribution, whose cumulative distribution function (cdf) and probability density function (pdf) are respectively given by where α is a scale parameter and β is a shape parameter.
[10] proposed a generalized class of distribution called the Kumaraswamy Generated (K-G) distribution which has a cdf given by and the corresponding (pdf) given by where a and b are additional shape parameters.
The cdf of the Kumaraswamy Weibull distribution is given as: and the corresponding pdf of Kumaraswamy Weibull distribution is given as: The Kumaraswamy Weibull Power Series (KWPS) models are obtained by compounding the Kumaraswamy Weibull and Power Series distributions. The compounding procedure follows the same set-up pioneered by [11].
The power series family of discrete univariate distributions is credited to [12] even though the earliest work on this family of distributions is due to [13]. Let N be a discrete random variable having a power series distribution with probability mass function (pmf) given by ( ) ( ) , 1, 2, ...

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Now, let 1 2 , , ..., M X X X be independent and identically distributed (iid) random variable of size m from the Kumaraswamy-G family of distributions as defined in (2.3) and (2.4). Suppose N is discrete and follows the power series distribution in (2.7), then the cdf and pdf of the Complementary Kumaraswamy-G Power Series (CK-GPS) class of distribution are given by with ( ) G x as Weibull distribution in (2.9), the cdf and pdf of the proposed CKWPS family of distributions are given by respectively.

The survival and hazard rate functions
For a continuous distribution function with pdf (2.10) and cdf (2.11), the survival function of the CKWPS is given by and the hazard rate function of the CKWPS is given by This is the cdf of the Kumaraswamy-Weibull distribution. x ab Hence, this completes the proof

Expansion of the CKWPS density function
Using Proposition 3.2, ; , By the series representation are the weights and ( ) ; [ 1], g x m α + β is the Weibull pdf with scale parameter ( 1) m α + and shape parameter . β ( )

Quantile function, moments and order statistics of the CKWPS distribution Proposition The quantile function of the CKWPS distribution is obtained by
where U is a uniform random variable on unit interval (0, 1).
The median of the CKWPS distribution is obtained by setting 0.5 U = in (3.11) and obtain 1 1 Proposition 3.4. The th r moment of a CKWPS distributed random variable X is given by Proof. Since D′ is a non-decreasing function, describes the density of CKWPS as a mixture, then where m Y has m f as its density function, and Substituting (3.15) into (3.14) gives equation (3.13).
The first moments about the origin of the CKWPS distribution (the mean) is obtained by setting The variance of the CKWPS distribution is given by The coefficient of skewness of the CKWPS distribution is given by and the moment generating function of the CKWPS distribution is given as

Order Statistics
Let the set of observations be ordered as 1 2 3 , , , ..., n X X X X whereby 1 X denote the minimum time to failure and n X denote the maximum time to failure. The trials are independent and identically distributed. The pdf of the th k order statistics from the CKWPS distribution is given as Using the identity Using (2.8) and (2.9) in (3.22), we have denotes the beta function, and Substituting (3.25) and (3.26) into (3.23), we have To obtain the pdf of smallest order statistics, substituting The pdf of largest order statistics is obtained by substituting

Sub-models of the CKWPS Family of Distribution
In this section, some sub-models of the CKWPS family of distributions are studied. In particular, the Complementary Kumaraswamy Weibull Poisson (CKWP) distribution is discussed in details.
The sub-models considered are as follows: x

The complementary Kumaraswamy Weibull Poisson (CKWP) distribution
Here, the pdf, reliability, hazard function, quantile and moments for the CKWP distribution are studied. The pdf of the CKWP distribution corresponding to (33) is given as where a, b, β are shape parameters and α is a scale parameter.
The plots of the pdf and cdf for the CKWP distribution for some selected parameter values are presented in Figures 1 and 2 respectively.  x

Complementary Kumaraswamy Weibull Power Series Distribution: …
The plot of the survival function for the CKWP distribution for some selected parameter values is presented in Figures 3. x The plot of the hazard rate function for the CKWP distribution for some selected parameter values is given in Figures 4.
where u is a uniform random variable on unit interval (0, 1). The median ( ) 2 Q of the CKWP distribution is obtained by setting The first moments about the origin (the mean) of the CKWP distribution is obtained by setting The variance of the CKWP distribution is given by The moment generating function of the CKWPS distribution is given by The pdf of the CKWP distribution in (4.19) can be rewritten as where ( ) exp 1. The likelihood of the CKWP distribution function is given by  , , , , .
The maximum likelihood estimate of Θ can be obtained by solving the non-linear system of equation Setting (5.6), (5.7), (5.8), (5.9) and (5.10) to zero and solving for the solution of the non-linear system of equations produce the maximum likelihood estimates of parameters ˆ, , , a b α β and θ . However these solutions can only be obtained numerically with the aid of suitable statistical software like R, SAS etc. Hence, some datasets are considered in the next section to fit the proposed distribution with other related distributions using "maxLik" package in R software.

Applications
This section presents a real life datasets, the descriptive statistics, graphical summary and application. where ƖƖ denotes the log-likelihood value evaluated with the maximum likelihood estimates, k is the number of model parameters and n is the sample size. The model with the lowest values of these statistics would be chosen as the best model to fit the dataset.

Data:
The data set consists of 63 observations of the strengths of 1.5 cm glass fibers, originally obtained by workers at the UK National Physical Laboratory. Unfortunately, the units of measurement are not given in the paper. It has been used by [16], [17], [18], [19], [20], [21], [22], [23]. It is given as 0   The result in Table 2 and the graphical display in Figure 5 reveal that the dataset is negatively skewed, and therefore would be flexible for skewed distributions.  Figure 6 displayed the histogram and estimated densities and cdfs of the fitted models for dataset.   Table 3 clearly shows that the CKWPD has smallest values of -ƖƖ, AIC, BIC, CAIC and HQIC compared to the other four distributions using the real life dataset. This provides evidence to show that the CKWPD fits the real life data better than the other four models. The plot in Figure 6 also reveals that the CKWPD performs better than the KWPD, KMWD, KWD and KEWD in fitting the dataset. Similarly, the probability plots displayed in Figure 7 further provide evidence that the proposed distribution (CKWPD) is more flexible for the dataset than the other four distributions (KWPD, KMWD, KWD and KEWD).

Appendix C
The R function for estimated densities of the distribution and the probability plot.