The Poisson Burr X Inverse Rayleigh Distribution And Its Applications

A new flexible extension of the inverse Rayleigh model is proposed and studied. Some of its fundamental statistical properties are derived. We assessed the performance of the maximum likelihood method via a simulation study. The importance of the new model is shown via three applications to real data sets. The new model is much better than other important competitive models.


Introduction and physical motivation
The well-known inverse Rayleigh (IR) model is considered as a distribution for a life time random variable (r.v.). The IR distribution has many applications in the area of reliability studies. Voda (1972) proved that the distribution of lifetimes of several types of experimental (Exp) units can be approximated by the IR distribution and studied some properties of the maximum likelihood estimation (MLE) of the its parameter. Mukerjee and Saran (1984) studied the failure rate of an IR distribution. Aslam  In this paper we propose and study a new extension of the IR distribution using the zero truncated Poisson (ZTP) distribution. Suppose that a system has subsystems functioning independently at a given time where has ZTP distribution with parameter . It is the conditional probability distribution of a Poisson-distributed r.v., given that the value of the r.v. is not zero. The probability mass function (PMF) of is given by Note that for ZTP r.v., the expected value ( | ) and variance ( | ) are, respectively, given by Suppose that the failure time of each subsystem has the Burr X IR (BX-IR( , ) for short) defined by the CDF and PDF given by respectively, where > 0 is a scale parameter and > 0 is the shape parameter. Let denote the failure time of the ith subsystem and let = min{ 1 , 2 , ⋯ , }. Then the conditional CDF of given is the corresponding PDF is . The PBX-IR density can be right-skewed, unimodal and symmetric (see Figure   1) whereas the PBX-IR HRF can be unimodal then bathtub, increasing and bathtub (see Figure 2).  This article is organized as follows: In Section 2, we derive some mathematical properties of the new model. Maximum likelihood method for the model parameters is addressed in Section 3. Sections 4 presents the simulation studies. In Section 5, The potentiality of the proposed model is illustrated by means of three real data sets. Section 6 provides some concluding remarks.

Mathematical properties 2.1 Useful expansions
Upon the power series the PDF in (6) can be written as ( , , ) If | | < 1 and > 0 is a real non-integer, the following power series holds Via applying the power series to the term Then consider the series expansion Applying the expansion in (11) to (10) This can be written as

Moments
The ℎ ordinary moment of , say ′ , follows from (12)

Incomplete moments
The ℎ incomplete moment of is defined by Based on (12) .
The moment generating function of , say ( ) = (exp( )), is obtained from (12) as  , which represents the expected additional life length for the system which is alive at age .

Moments reversed residual life (MRRL) functions
The

Estimation
Consider a random sample from your PBX-IR, then the log likelihood function can be expressed as = log2 + log + log + log2 + 2 log − log[

Simulation studies
Upon (14), we simulate the PBX-IR model by taking =20, 50, 150, 500 and 1000. For each sample size, we evaluate the ML estimations (MLEs) of the parameters using the optim function of the R software (see the R code in the Appendix). Then, we repeat this process 1000 times and compute the averages of the estimates (AEs) and mean squared errors (MSEs). Table 1 gives all simulation results. The values in Table 1 indicate that the MSEs of ˆ, ˆ and ˆ decay toward zero when increases for all settings of , and , as expected under first-under asymptotic theory. The AEs of the parameters tend to be closer to the true parameter values (I: = 0.5, = 1.5 and = 2.5 and II: = 1.5, = 0.5 and = 1.5) when increases. This fact supports that the asymptotic normal distribution provides an adequate approximation to the finite sample distribution of the MLEs. Table 1 gives the AEs and MSEs based on 1000 simulations of the PBX-IR distribution for some values of and when by taking = 20,50,150,500 and 1000.

Real data modeling
This section presents two applications of the new distribution using real data sets. We          where l denotes the log-likelihood function evaluated at the MLEs, is the number of model parameters and is the sample size. The model with minimum values for these statistics could be chosen as the best model to fit the data. All results are obtained using the R PROGRAM. Tables 2, 4 and 6 compare the PBX-IR model with other important competitive distributions. The PBX-IR model gives the lowest values for the AIC, BIC, HQIC and CAIC statistics (in bold values) among all fitted models to these data. So, it may be considered as the best model among them. Figure 3, 4 and 5, respectively, display the plots of estimated density for the proposed model and estimated CDF of the new model for the three data sets. These plots reveal that the proposed distribution yields a better fit than other nested and non-nested models for both data sets.

Conclusions
A new flexible extension of the IR model is proposed and studied. Some of its fundamental statistical properties are derived such as quantile, moments, incomplete moments and moment generating function. We assessed the performance of the maximum likelihood estimators via a simulation study. The importance of the new model is shown via two applications to real data sets.