Abstract

In this article we propose a weighted and mixed version of binomial and discrete Lindley distributions. The weighted distributions are usually considered for the adjustment of probability of occurrences and removal of biasedness from the data and mixed distributions are usually used in heterogeneous and over-dispersed data sets. The proposed model adjusts the probabilities by removing the bias but contrary to the over-dispersion the model handles the under dispersion issues. Moreover, various properties like failure rate, moments, and factorial moments, recurrence relation between moments, self-decomposability, infinite divisibility and limiting form of the proposed model are also studied. A simulation is conducted and estimation problem is discussed via two estimation methods like maximum likelihood and moments estimation. Performance of the proposed model over the binomial, discrete gamma, Poisson-Lindley and discrete Lindley is investigated via some test statistics and two under dispersed real data sets. We introduce a two parameter discrete distribution, so-called the ??BL distribution. The new distribution is much more flexible than the binomial, discrete Lindley distribution and could have increasing and unimodal hazard rates. We derive explicit algebraic formula for the r th moment which holds in generality for any parameter value The estimation procedures for the parameters of ??BL distribution are also derived considering two estimation methods whose convergence rate is very high as compare to the others. The MME method yields global estimates where as MLE yields local estimates. Since it is not feasible to compare these methods theoretically, we have presented an extensive simulation study in order to identify the most efficient procedure. We observed that the MLE provides a large number of local estimates whereas MME yields global one so we used this global value as a seed for finding the MLE. Moreover, we have also studied various mathematical properties including moments, ascending and descending factorial moments, negative and negative factorial moments, discrete self-decomposability, and infinite divisibility. Finally, two data-sets were analyzed for illustrative purposes