The Odd Log-Logistic Poisson Inverse Rayleigh Distribution: Statistical Properties and Applications

In this work, a new extension of the Inverse Rayleigh model is proposed and studied. We derive some of its fundamental properties. We assess 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. The new model is better fit than other important competitive models based on two real data sets.


Introduction and interpretation A random variable (rv)
is said to have the Inverse Rayleigh (IR) distribution if its probability density function (PDF), cumulative distribution function (CDF) are given by where , > 0 and is the vector of parameter for baseline CDF. The corresponding PDF is given by The corresponding PDF is given by where ( ) = ( ) − 1. The survival function can be derived as , which is the CDF of the compound Poisson Inverse Rayleigh model. If the rv represents the odds ratio, the risk that the system following the lifetime will be not working at time is given by .
If we are interested in modeling the randomness of the odds ratio, , ( ) , by the exponentiated half-logistic CDF the CDF of is given by .
Furthermore, the basic motivations for using the OLLP-IR model in practice are the following: to make the kurtosis more flexible compared to the baseline Fr model; to produce a skewness for symmetrical distributions; to construct heavy-tailed distributions that are not longer-tailed for modeling real data; to generate distributions with symmetric, left-skewed, right-skewed and reversed-J shaped; to define special models with flexible types of the HRF; to provide consistently better fits than other generated models under the same baseline distribution. Although, we have stated that ∈ (0, ∞) , equation (5)

Linear representations
In this section, mixture representations for Equations (5) and (6) are obtained, firstly we have where = + (−1) ( ). Then, the OLLP-IR CDF in (5) can be written as where 0 = 0 0 and for ≥ 1 , we have The corresponding OLLP-IR density function is obtained by differentiating (7) as where ℎ +1 ( ) = 2( + 1) 2 −3 [−( + 1) 2 −2 ]is PDF of the IR distribution with scale parameter ( + 1) 1 2 . Equation (7) and (8) reveal that PDF of OLLP-IR is a linear combination of IR densities. Thereby, some properties of the proposed family such as moments and generating function can be determined by means of IR distribution.

Moments, incomplete moments and generating function
The r ℎ ordinary moment of is given by Then we obtain where is the complete gamma function. Setting = 1 in (9), we have the mean of . The last integration can be computed numerically for most parent distributions. The r ℎ incomplete moment, say ( ) , of can be expressed from (7) as where ( , ) is the incomplete gamma function.
and 1 1 [⋅,⋅,⋅] is a confluent hypergeometric function. The first incomplete moment 1 ( ) given by (9) with = 1 . The moment generating function (mgf) ( ) = ( ) of can be derived from equation (8) as Skewness and kurtosis for the OLLP-IR model can be calculated from the ordinary moments by using well-known relationships. The mean, variance, skewness and kurtosis of the OLLP-IR distribution are computed numerically using the R software and reported in Table 1. Table 1 indicate that the skewness of the OLLP-IR distribution is always positive. The kurtosis is always more than 3.

Quantile function
The OLLP-G family can easily be simulated by inverting (5) as follows: if ∼ (0,1) , then the random variable can be obtained from the baseline qf, say ( ) = −1 ( ) . In fact, the random variable has CDF (5). The effects of the shape parameters on the skewness and kurtosis can be based on quantile measures. . Then, the n ℎ moment of the reversed residual life (RRL) of becomes  (12) is the main result of this section. It reveals that the PDF of the OLLP-IR order statistics is a linear combination of IR density functions. So, several mathematical quantities of the OLLP-IR order statistics such as ordinary, incomplete and factorial moments, mean deviations and several others can be determined from those quantities of the IR distribution. Then

Maximum likelihood estimation
Here, we consider estimation of the unknown parameters of the OLLP-IR distribution by the maximum likelihood method. Let 1 , … , be a random sample from the OLLP-G distribution with a 4 × 1 parameter vector. The loglikelihood function for is given by

Simulation studies
Upon (14), we simulate the OLLP-IR model by taking =20, 50, 200, 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), biases (Bias) and mean squared errors (MSEs). Table 1 gives all simulation results. The values in Table 2 indicate that the MSEs 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: = 1.5, = 1.5 and = 2.5 and II: = 2.5, = 0.5 and = 2.5 ) when increases. This fact supports that the asymptotic normal distribution provides an adequate approximation to the finite sample distribution of the MLEs. Table 2 gives the AEs and MSEs based on 1000 simulations of the OLLP-IR distribution for some values of and when by taking = 20,50,150,500 and 1000 .   Nichols and Padgett (2006). The 2 data set consists of 63 observations of the strengths of 1.5 cm glass fibers (see Smith and Naylor (1987)), originally obtained by workers at the UK National Physical Laboratory. Unfortunately, the units of measurement are not given in the paper, we consider the AIC (Akaike Information Criterion), CAIC (Consistent Akaike Information Criterion), BIC (Bayesian information criterion) and HQIC (Hannan-Quinn information Criterion). 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. Figure 1 and 2, respectively, display the box plots and the quantile-quantile (Q-Q) plots. Tables 3 and 5 compare the OLLP-IR model with other important competitive distributions. The OLLP-IR model gives the lowest values for the AIC, BIC, HQIC and CAIC statistics (in bold values) among all competitive fitted models to these data. So, it may be considered as the best model among them. Figure 3-8, respectively, display the TTT plot, the P-P plot, estimated PDF, estimated HRF estimated CDF and Kaplan-Meier survival plot for the 1 st and 2 nd data sets. These plots reveal that the proposed distribution yields a better fit than other nested and non-nested models for both data sets. For more useful real life data sets see Elbiely and Yousof (2018

Conclusions
A new extension of the Inverse Rayleigh model is proposed and studied. Some of its fundamental properties are derived. We assessed the performance of the maximum likelihood estimators via a simulation study. The mean, variance, skewness and kurtosis of the new distribution are computed numerically using the R software. The skewness of the new distribution is always positive, the kurtosis is always more than 3. The importance of the new model is shown via two applications to real data sets. The new model is better fit than other important competitive models based on two real data sets.