Discrimination between Gamma and Rayleigh Models

The two popular life testing models are considered to verify whether one can be an alternative to other. The motivation for this study is as follows. It is well known that the cumulative distribution function of Rayleigh distribution can be analytically inverted where as it is not so with Gamma distribution. Generally analytical inversion of cumulative distribution function would be advantageous in the study of problems of inference. “Whether this advantage can be explored in assessing the discrimination or otherwise of the two models” is studied in this paper.


Introduction
The cumulative distribution function and probability density function of Gamma distribution with shape parameter 2 and Rayleigh distribution (Weibull with shape parameter 2) are given by Eqs. (1-4): .
and . 0 , and Here σ is called the scale parameter in the above models. We see in most of the text books that the frequency curves of the two distributions look alike. The cumulative distribution function of Rayleigh distribution is analytically invertible whereas that of Gamma distribution is an incomplete Gamma function extensively tabulated. This advantage of Rayleigh distribution is the motivation to study whether it can be an alternative to Gamma with shape parameter 2. Some of the authors working in this direction are of Gupta and Kundu (2003) for the Weibull and the generalised exponential distributions, Gupta and Kundu (2004) for the Gamma and generalised exponential distributions, Kundu and Manglick (2004) for the Weibull and log-normal distributions, Kundu (2005) for normal and Laplace distributions,  for the lognormal and the generalised exponential distributions,  for the Gamma and log-normal distributions, Kundu and Raqab (2007) for the log-normal and generalised Rayleigh distributions, Dey and Kundu (2009) for log-normal, Weibull and generalised exponential distributions, Dey and Kundu (2010) for the log-normal and log-logistic distributions, Dey and Kundu (2012a) for the Weibull and log-normal distributions for type -II censored data, Dey and Kundu (2012b) for bivariate generalised exponential and bivariate Weibull distributions,  for generalised exponential distribution and some life test models based on population quantiles,  for half normal and half logistic distributions, Rao and Kantam (2014) for log-logistic and Rayleigh distributions, Kantam et al. (2014a) for likelihood ratio test for linear failure rate and exponential distributions, Kantam et al. (2014b) for linear failure rate and Rayleigh distributions, Prasad and Kantam (2015) for the test procedure between probability models and Kantam and Ravikumar (2015) for Burr type XII and Weibull-exponential distributions. We made an attempt in that direction through likelihood ratio criterion.
The likelihood ratio criterion is narrated in Section-2. The adopted simulation study results and conclusions are given in Section-3 and Section-4.

Likelihood Ratio Criterion
In our study Gamma 2 is taken as null population and Rayleigh is taken as an alternative population.
i.e., H0: A given sample   12 , ,..., n x x x belongs to Gamma 2 model. If λ is negative then the null hypothesis that the sample is from Gamma is accepted. In other words for a sample that is generated from Gamma, a negative value of λ yields correct decision otherwise it will give wrong decision.
Therefore λ can be used as a criterion for testing the above H0 vs. H1. Hence the percentiles of λ are essential for this purpose. However we adopted a different simulation study to get the proportion of correct decision vs. a wrong decision as given in Section 3.

Simulation Study
Samples of size 5, 10, 15, 20 and 25 are generated from Gamma 2 and at those samples the σ of Gamma 2 as well as Rayleigh distribution is estimated by ML method. The respective likelihood and the LR criterion λ are evaluated. The proportion of negative and positive λ's are computed out of the used simulation runs. The details are given in the following These results reveal that when sample size is more than 15 Rayleigh cannot be an alternative to Gamma 2. In small samples that are of size 5, 10 Rayleigh can be used as a reasonable alternative to Gamma 2. Specifically over increased simulation runs for sample sizes 5, 10 we observe the following Table 2. This further shows a fifty-fifty choice of Gamma 2 vs. Rayleigh up to a sample of size 10.

Conclusion
We therefore conclude that in samples of size as small as up to 10. Gamma 2 and Rayleigh are equally likely models for a sample generated from either of the two. For other large samples they make a distinction.