Poisson Exponential Power Distribution: Properties and Application

In this study, we have established a new three-parameter Poisson Exponential Power distribution using the Poisson-G family of distribution. We have presented the mathematical and statistical properties of the proposed distribution including probability density function, cumulative distribution function, reliability function, hazard rate function, quantile, the measure of skewness, and kurtosis. The parameters of the new distribution are estimated using the maximum likelihood estimation (MLE) method, and constructed the asymptotic confidence intervals also the Fisher information matrix is derived analytically to obtain the variance-covariance matrix for MLEs. All the computations are performed in R software. The potentiality of the proposed distribution is revealed by using some graphical methods and statistical tests taking a real dataset. We have empirically proven that the proposed distribution provided a better fit and more flexible in comparison with some other lifetime distributions.


I. INTRODUCTION
The exponential distribution is the most frequently used distribution due to the existence of simple elegant closedform solutions to many survival analysis problems. The failure rate of the exponential distribution is constant but in real practice, the failure rates are not always constant. Hence in some situations, it seems to be inadequacy and unrealistic. For this, some modifications are desirable to make exponential distribution more flexible. In recent, a new class of models has been introduced based on the adjustment of exponential distribution. Gupta and Kundu (1999) introduced the generalized exponential (GE) distribution, this extended family can accommodate data with increasing and decreasing failure rate functions, Nadarajah and Kotz (2006) have introduced a generalization referred to as the beta exponential distribution generated from the logit of a beta random variable. There are lots of lifetime models which are obtained by compounding with Zero truncated Poisson distribution some of them are as follows, Kus (2007) has introduced the two-parameter exponential Poisson (EP) distribution by compounding exponential distribution with zero truncated Poisson distribution with a decreasing failure rate. The While Barreto-Souza and Cribari-Neto (2009) have introduced generalized EP distribution having the decreasing or increasing or upside-down bathtub shaped failure rate. This is the generalization of the distribution proposed by Kus (2007) adding a power parameter to this distribution.
Following a similar approach, Percontini   In this study, we propose a new distribution based on the exponential power distribution has introduced by (Srivastava & Kumar, 2011) to analyze the software reliability data having the shape of decreasing, increasing, jshaped, and bathtub-shaped failure rate function for different values of the parameters. The CDF and PDF of the exponential power distribution are respectively as The different sections of this study are arranged as follows; in Section 2 we present the new distribution Poisson exponential power (PEP) with its mathematical and statistical properties. We comprehensively discuss the maximum likelihood estimation method in Section 3. In Section 4 using a real dataset, we present the estimated values of the model parameters and their corresponding asymptotic confidence intervals and fisher information matrix. Besides, we have illustrated the different test criteria to assess the goodness of fit of the proposed model. Some concluding remarks are presented in Section 5.

II. THE POISSON EXPONENTIAL POWER (PEP) DISTRIBUTION
Alkarni and Oraby (2012) have introduced a new lifetime class with a decreasing failure rate which is obtained by compounding truncated Poisson distribution and a lifetime distribution, where the compounding procedure follows the same way that was previously carried out by (Adamidis & Loukas, 1998). Let () Gx and () gx be the baseline cumulative distribution function and probability density function respectively then the Poisson family with CDF and PDF may be expressed as, Substituting (4) and (5) in (6) and (7) then the Poisson exponential power distribution can be defined as, Let X be a nonnegative random variable representing the survival time of an item or component or a system of some population. The random variable X is said to follow the PEP distribution And its corresponding probability density function is

Reliability function:
The reliability function   Rt , which is the probability of an item not failing up to time t, is defined by     The survival /reliability function of the Poisson exponential power distribution is given by

The hazard rate function (HRF)
The hazard rate function for the PEP distribution can be defined as, "Poisson Exponential Power Distribution: Properties and Application" 2154 Ramesh Kumar Joshi 1 , IJMCR Volume 08 Issue 11 November 2020 In Figure 1 we have presented the graph for PDF and hazard function for PEP distribution for different values of the parameters. From Figure 1 (left panel), the density function of the PEP distribution can bear different shapes according to the values of the parameters. Figure 1 (right panel) demonstrates the increasing, decreasing, the j-shaped, and constant shape of the hazard rate.

The quantile function of PEP distribution
According to Hyndman and Fan (1996), the value of the p th quantile can be obtained by solving the following equation,

Median of PEP distribution
The median of X from the PEP distribution is simply obtained by replacing p = 0.5 in equation (12) Skewness The likelihood function of the PEP using the PDF in equation (9) is given by: It is easy to deals with natural logarithm, hence let   , , l    be log-likelihood function, where H is the Hessian matrix. The Newton-Raphson algorithm to maximize the likelihood produces the observed information matrix. Therefore, the variance-covariance matrix is given by, Hence from the asymptotic normality of MLEs, approximate 100(1-α) % confidence intervals for  ,  and  can be constructed as,

IV. APPLICATION WITH A REAL DATASET
In this section, we illustrate the applicability of the PEP model using a real dataset used by former researchers. We have taken 100 observations on breaking the stress of carbon fibers (in Gba) used by (Nichols & Padgett, 2006 The plots of profile log-likelihood function for the parameters α, β, and λ have been displayed in Figure 2 and noticed that the ML estimates can be uniquely determined.  Table 1 we have demonstrated the MLE's with their standard errors (SE) and 95% confidence interval for α, β, and  .  The Q-Q plots and P-P plot of PEP distribution are displayed in Figure 3. It is observed that the distribution fits the data excellently.
We have fitted the PEP distribution and some selected distributions which are as follows,

A. Weibull extension (WE) distribution
The probability density function of Weibull extension (WE) distribution (Tang et al., 2003)

D. Exponential power (EP) distribution:
The probability density function Exponential power (EP) distribution (Smith & Bain, 1975 where α and λ are the shape and scale parameters respectively.
The negative log-likelihood value and the value of AIC, BIC, CAIC and HQIC are presented in Table 2. We conclude that the proposed model produces a better fit to the data taken than other models.  The histogram and the fitted density functions are displayed in Figure 4 which compares the distribution function for the different models with the empirical distribution function that produces the same. Therefore, the given data sets illustrate the proposed distribution gets better fit and more reliable results from other alternatives. In Table 3 we have displayed the value of the test statistics the Kolmogorov-Simnorov (KS), the Anderson-Darling (AD) and the Cramer-Von Mises (CVM) statistics and their corresponding p-value of different models. The result verifies that the proposed model has the minimum value of the test statistic and higher p-value hence we conclude that the Poisson exponential power distribution is better in the view of goodness-of-fit.

V. CONCLUSION
In this study, we have presented a new expansion of the exponential power model called Poisson exponential power (PEP) distribution. Some statistical and mathematical properties of the PEP model have been discussed. From the graphical analysis of PDF and HRF, the proposed model is versatile and increasing, decreasing and upside bathtub hazard function. We have calculated the maximum likelihood estimates of the model parameters and the corresponding confidence intervals and information matrix of the MLE's. We have also illustrated the application of PEP distribution using a real data set and found quite useful and behaves better in terms of fitting as compared to some selected models. It may be an alternative model for practitioners in the area of theory and applied statistics.