Extended Poisson Exponential Distribution

A new mixture of Modified Exponential (ME) and Poisson distribution has been introduced in this paper. Taking the Maximum of Modified Exponential random variable when the sample size follows a zero truncated Poisson distribution we have derived the new distribution, named as Extended Poisson Exponential distribution. This distribution possesses increasing and decreasing failure rates. The Poisson-Exponential, Modified Exponential and Exponential distributions are special cases of this distribution. We have also investigated some mathematical properties of the distribution along with Information entropies and Order statistics of the distribution. The estimation of parameters has been obtained using the Maximum Likelihood Estimation procedure. Finally we have illustrated real data applications of our distribution.


Introduction
In lifetime data analysis problems the most widely used distribution is exponential distribution. The popularity of the exponential distribution lies in its nice and simple properties. It provides closed form results through simple mechanism in reliability and lifetime testing problems. One property of Exponential distribution is that it supports a constant failure rate. The problem arises in situations when a given data set does not possess a constant failure rate then we cannot use exponential distribution to model the data. Different modifications of the exponential distribution, supporting non-constant failure rates, have been introduced by several authors to handle this problem.
Furthermore to have a better fit to the data and to attain more accuracy, two different distributions can be compounded together to generate a new distribution. Various techniques are used to compound distributions. The distribution of sum of independent random variates or the distribution of minimum or maximum of the random variates, when the sample size is fixed and when it is random, provides us with a new distribution. This technique has been widely used by several authors to provide new modifications of the exponential distribution. Using this technique Adamidis and Loukas (1998) introduced Exponential-Geometric (EG) distribution. Adamidis et. al. (2005) introduced a new extension of the Exponential-Geometric (EG) distribution by compounding the Exponential and Modified Extreme value distributions and called the new distribution the Extended Exponential-Geometric (EEG) distribution. Kuss (2007) introduced the Exponential-Poisson (EP) distribution. Tahmasbi and Rezaei (2008) introduced the In this paper we have introduced a new probability distribution to model the life time data. In section 2 we have obtained the density function and distribution function of our model along with the survival and hazard rate. In section 3 we have derived several properties of our distribution incuding Quantile function, Characteristic function, Information entropies, Mean deviations and Order statistics. In section 4 we have discussed the Maximum Likelihood Estimates along with the associated large sample inference. In last section we have used three real data sets to illustrate the application of our distribution.

Density Function, Distribution Function and Hazard Function
Here we have derived the density function, distribution function survival and hazard functions of the Extended Poisson Exponential distribution.
Marshall and Olkin (1997) introduced a parameterization scheme to get a new family of distributions, given as: The density given in (1) is named as the "Extended Poisson Exponential (EPE) distribution".
In the density of Extended Poisson Exponential distribution substituting a=1 we get the density of Poisson-Exponential distribution. When λ approaches to zero the density of EPE distribution converges to the Modified Exponential distribution. Also when a=1 and λ approaches to zero the EPE distribution converges to the simple exponential distribution.
Giving different values to the parameters of the EPE distribution we have the shape of our density function given in Figure 1. From the figure we can observe that the density is reversed J shaped for small values of the shape parameters a and λ. And for larger values of these shape parameters the curve becomes bell shaped.
Now we have the survival function of our distribution as: Since for small values of shape parameters "a" and "λ" failure rate is first decreasing and then becomes constant. For large values of these shape parameters the failure rate is first increasing and then becomes constant.

Properties of the Distribution
In this section we have derived different properties of the EPE distribution such as Quantile function, Characteristic function and Information entropies. We have also derived Mean deviations and the density of i th order statistics.

Quantile Function
The quantile function of the distribution, obtained using (2) is given as:

Characteristic Function
Further the characteristic function of the distribution can be obtained as: We get: The characteristic function of the EPE distribution turns out to be:

Information Entropies
The Rényi entropy of any distribution is defined as: Simplifying using the exponential series expansion, substitution (6) and result (7) we get: Hence the final expression for Rényi entropy is: The Shannon entropy is defined as: For the EPE distribution we get: To get E(X) we proceed:  Substituting (14) in equation (13) we get E(X) given as: Using (12) along with exponential series expansion, substitution (6) and result (7), equation (18) Hence substituting (15), (17) and (19) in (10) we get the Shannon entropy as:

Mean Deviations
The mean deviation about mean and median can be obtained using: Hence the mean deviations after using (21)

Order Statistics
The density of ith order statistic in a random sample of size "n" say Y 1 ,…, Y n , is: Where F(y) is the distribution function and f(y) is the density function.
Substituting the density function (1) and the distribution function (2) of EPE density we get: Equating these expressions to zero and solving the equations we can get the Maximum Likelihood Estimates (MLEs) of the distribution. Here we do not have any closed form expressions for MLEs so numerical maximization through iterative procedures have to be used to get the MLEs.
In order to draw large sample inferences for the distribution we make use of the fact that ( )( ̂ ) D   ( ) as n approaches to infinity, when the suitable regularity conditions are satisfied. Here I is the identity matrix and ( ) is the Choleski decomposition of the Fisher"s Information matrix ( ), i.e. ( ) ( ) ( ). (Cox & Hinkley, 1974) The asymptotic variance-covariance matrix of the MLEs is the inverse of the Fisher"s Information matrix which can be estimated by the inverse of the observed information matrix. And these estimates can be used to construct the asymptotic confidence intervals and to draw inferences about the parameters asymptotically.
The Fisher"s Information matrix is:

Real Data Application
Extended Poisson-Exponential distribution has been fitted to real data sets in order to determine whether it provides a good fit or not. Comparison of the new model with Poisson Exponential distribution has been made for three different data sets.

Data set 1:
The data employed consists of 63 observations on "the strength of 1.5 cm glass fibers, measured at the National Physical Laboratory, England". The data set is given as: ). For the given hypothesis the value of Likelihood ratio test statistic is 25.98732 with corresponding P-value 0.0001. Hence at this P-value we reject the null hypothesis at the usual level of significance and conclude that the EPE distribution provides a better fit than the PE distribution for the given data set.
Again we have tested the performance of PE distribution against the EPE distribution using the Likelihood Ratio test. The value of test statistic is 7.6286 and the corresponding P-value is 0.0057. Hence we can state that the EPE distribution provided a significantly better fit than PE distribution to the remission times of Bladder cancer patients.

Concluding Remarks
The Extended Poisson Exponential distribution can be used to model data with increasing and decreasing failure rate from Engineering and Biostatistics like the breaking strength of materials and remission time of diseases. This distribution cannot be used to model data having bathtub failure rate. However it can be used if data have either increasing or decreasing failure rate. Furthermore there is scope for further mathematical development like the Bayesian estimation of the parameters of the model and their comparison with the Maximum Likelihood estimates.