Likelihood-based Statistical Methods For The Zero-one-two Inflated Poisson Model

Usually when we deal with data statistics,it's common to meet problems that traditional statistical models can't solve,for example,count data with excess zeros,ones and twos.Such data appear in insurance,psychology,economics,and biomedicine at times.The variance of this kind of data is larger than its mean,so we usually call it the large deviation count data.We consider zero-inflated Poisson distribution and zero-and-one-inflated Poisson distribution when count data has excess zeros or excess zeros and ones simultaneously.But when there are excess twos,the existing statistical models can't work well.To model count data with excess zeros,ones and twos,we introduce a so-called zero-one-two-inflated Poisson(ZOTIP)distribution for the first time.Distributional theories,cor-responding properties and likelihood-based statistical methods for parameters of interest in the ZOTIP distribution are studied in this paper.At the same time,the thesis launches re-search on excess twos,and it consists of the following sections:We establish several equivalent stochastic representations to derive some distributional properties of the mixture distribution.The Fisher scoring and EM algorithms are developed to obtain the maximum likelihood estimates of parameters.Bootstrap confidence intervals for parameters of interest are also constructed.Testing hypotheses and simulations studies are provided,and two real data sets are used to illustrate the proposed methods.