Abstract

The Poisson-Shanker distribution was introduced by Shanker to model over-dispersed count data from biological sciences. The literature review reveals that some useful statistical properties, reliability properties and entropy measures of Poisson-Shanker distribution have not been studied. The main purpose of this paper is to study statistical properties, reliability properties, entropy measures which were not studied earlier. The PoissonShanker distribution has been shown to be log-concave and the two-component mixture of negative binomial distributions. The cumulative distribution function, survival function, hazard function and mean residual life function of the Poisson-Shanker distribution have been derived and their natures for varying values of parameter have been discussed. The expressions for other reliability properties including reverse hazard function, the second rate of failure, the cumulative hazard function and the Mills ratio of the Poisson-Shanker distribution have also been derived. Various entropy measures including the Shannon, Rényi and Tsallis entropy measures are obtained in a form suitable for numerical evaluation to quantify the degree of uncertainty, randomness or information content associated with the distribution. The simulation study has been presented to show the consistency of maximum likelihood estimator of the parameter of the distribution. The applications of the Poisson-Shanker distribution in automobile insurance claim frequency data have been discussed and the goodness of fit has been compared with the goodness of fit given by Poisson distribution, Poisson-Lindley distribution and Poisson-Garima distribution. 

Keywords

Poisson-Shanker distribution, reliability properties, entropy measures, simulation, automobile insurance claim frequency data