The New Weibull-Pareto Distribution

A new distribution, the New Weibull-Pareto, is defined and studied. Various properties of the distributions are obtained and the method of maximum likelihood used to estimate the parameters of the distribution. The usefulness of the distribution has also been demonstrated by applying it to real life data.


Introduction
Many lifetime data used for statistical analysis follows a particular statistical distribution.Knowledge of the appropriate distribution that any phenomenon follows, greatly improves the sensitivity, power and efficiency of the statistical tests associated with it.Several distributions exist for modeling these lifetime data however, some of these lifetime data do not follow these existing distributions or are inappropriately described by them.This therefore creates room for developing new distributions which could better describe some of these phenomena and therefore provide greater flexibility in the modeling of lifetime data.As a result, umpteen of distributions have been developed and studied by researchers.

New Weibull-Pareto Distribution (NWPD)
Let  be a random variable from a Pareto distribution with its cumulative distribution function (cdf) for  ≥  given by where  > 0 is a scale parameter and  > 0 is the shape parameter.The probability density function (pdf) corresponding to (1) is The NWPD has a cdf of the form where () is the survival function of the Pareto distribution and is given by () = 1 −  1 (; , ) while  2 () is the pdf of a Weibull distribution and is given by where  > 0,  > 0, and  > 0.

𝛼𝑘
If we let  =   and  = , then the cdf of the NWPD can be written as The pdf is obtained by finding the first derivate of (4) with respect to .Thus the pdf is where 0 <  < ∞,  > 0,  > 0 and  > 0.
Lemma 2.1.The limit of the pdf of the NWPD, (), as  → ∞ is 0 and as  → 0 is 0. This can be proved by taking the limit of (5) as  → ∞ and as  → 0.
Proof.Lemma 2.2.The limit of the cdf of the NWPD, (), as  → ∞ is 1 and as  → 0 is 0. This can be proved by taking the limit of (4) as  → ∞ and as  → 0. Proof.

Survival and hazard functions
The NWPD can be a useful characterization of the survival time of a given system because of it analytical structure.The survival function is given by () = 1 − ().Thus using (4), Thus using ( 5) and ( 6), another characteristics of interest of a random variable is the hazard function defined by Thus using ( 5) and ( 6), From the hazard function the following can be observed: i.
If  = 1, the failure rate is constant and given by: ℎ() =   This makes the NWPD suitable for modeling systems or components with constant failure rate. ii.
If  > 1, the hazard is an increasing function of , which makes the NWPD suitable for modeling components that wears faster with time. iii.
If  < 1, the hazard is a decreasing function of , which makes the NWPD suitable for modeling components that wears slower with time.
Figure 3 is the plot of the hazard function of the NWPD for different values of the parameters of the distribution.

Moments
If  is a random variable distributed as a NWPD, then the  ℎ non-central moment is given by: Proof.

𝐸(𝑋
Therefore the variance is given by

Incomplete moments
If  is a random variable distributed as a NWPD with parameters ,  and , the  ℎ incomplete moment of  is given by: Proof.
() = ∫ By definition Using Taylor series

Quantile function and simulation
Let (), 0 <  < 1 denote the quantile function for the NWPD.Then () is given by In particular, the distribution of the median is: Let  be a uniform variate on the unit interval (0,1).Thus by means of the inverse transformation method, we consider the random variable  given by: This follows the NWPD.

Skewness and Kurtosis
In this study, the quantile based measures of skewness and kurtosis was employed due to non-existence of the classical measures in some cases.The Bowley's measure of skewness based on quartiles is given by where (. ) represents the quantile function.

Mode and Mean deviations
The mode of the NWPD is obtained by finding the first derivate of ln () with respect to  and equating it to zero.That is  ln ()  = 0. Therefore the mode at  =  0 is given by The mean deviation about the mean and the median are useful measures of variation for a population.Let  = () and  be the mean and median of the NWPD respectively.The mean deviation about the mean is The mean deviation from the median is The pdf of the  ℎ order statistic of the NWPD is The pdf of the largest order statistic  () is therefore and the pdf of the smallest order statistic  ( 1) is given by

Parameter Estimation and inference
In this section, the maximum likelihood estimation and inference of the parameters of the NWPD have been discussed.Let ~NWPD(, , ) and let (, , )  be the vector of the model parameters.The log-likelihood function for (, , )  is given by The log-likelihood function , is The score functions are given by: The maximum likelihood estimators ( ̂,  ̂,  ̂) are obtained by equating (15) to zero and solving for the non-linear system of equations iteratively.In order to compute the standard errors and asymptotic confidence interval, the large sample approximation in which the maximum likelihood estimator of a parameter can be treated as being approximately multivariate normal was used.Thus as  → ∞, the asymptotic distribution of the maximum likelihood estimators ( ̂,  ̂,  ̂) is given by

Application
In this section we demonstrated that the NWPD is useful in modeling real life situation.The newly proposed distribution was used to model the exceedances of flood peaks (in m 3 /s) of the Wheaton River near Carcross in Yukon Territory, Canada.The data consists of 72 exceedances for the year 1958-1984, rounded to one decimal place as shown in Table 1.Recently, Merovci and Puka (2014), and Bourguignon et al. (2013) analysed this data using the Transmuted Pareto (TP) distribution and Kumaraswamy Pareto (Kw-P) distribution respectively, demonstrating the superiority of their distributions over the weibull and Pareto distributions.We therefore fitted the NWPD to this data and compared our results to theirs.

Conclusion
This article defined a New Weibull-Pareto Distribution (NWPD) and studied various properties of the distribution.The moments, deviations from the mean and median, mode, survival function, hazard function and the maximum likelihood estimates of the parameters, have been investigated.The application of the new distribution has also been demonstrated with real life data.The results, compared with other known distributions, revealed that the NWPD provides a better fit for modeling real life data.

Figure 1
Figure 1 is the plot of the pdf of the NWPD for the different values of the parameters of the distribution.

Figure 2
Figure 2 is the plot of the survival function of the NWPD for different values of the parameters of the distribution.Another characteristics of interest of a random variable is the hazard function defined by ℎ() = () ()

Table 2
displays the Maximum Likelihood Estimates (MLEs) of the model parameters.Since for the Pareto distribution  ≥ , the MLE of  is the first-order statistic.From Table2, it was obvious that the NWPD provides a better fit compared to the other candidate models since it has the lowest value of −2, Akaike Information Criterion (AIC),