Negative Binomial Three Parameter Lindley Distribution and Its Properties

Abstract

Many researchers have proposed mixed distributions as one of the most important methods for obtaining new probability distributions. Several studies have shown that mixed Negative Binomial distributions fits count data better than Poisson and Negative Binomial distribution itself. In this paper, we introduce a mixed distribution by mixing the distributions of negative binomial and three Parameter Lindley distribution. This new distribution has a thick tail and may be considered as an alternative for fitting count data with over dispersion. The parameters of the new distribution are estimated using MLE method and properties studied. Special cases of the new distribution and also identified. A simulation study carried out shows that the ML estimators give the parameter estimates close to the parameter when the sample is large, that is, the bias and variance of the parameter estimates decrease with increase in sample size showing the consistent nature of the new compound distribution. The study also compares the performance of the new distribution over distributions of Poisson, Negative Binomial, Negative Binomial oneParameter Lindley Distribution, Negative Binomial two Parameter distribution, three parameter Lindley distribution using a real count over dispersed dataset and the results shows that Negative Binomial three parameter Lindley distribution gave the smallest Kolmogorov Smirnov test statistic, AIC and BIC as compared to other distributions, hence the new distribution provided a better fit compared to other distributions under study for fitting over dispersed count data.