Full Likelihood Statistical Inference Of Population Size Under Zero-truncated Mixture Count Model

Capture-recapture data which is also zero-truncated count data is very common in many fields such as ecology,sociology,demography,epidemiology and health statistics.Its characteristic is that the observed frequency is a positive integer,and the individuals whose frequency is equal to 0 are often not observed.Such data include the number of rare animals captured,the number of drug addicts going to rehab,the number of drunk driving seized,and so on.Based on such data,people often care about the overall group size studied,such as the total number of rare animals,the total number of drug addicts and the total number of drunk driving people in a certain area in a certain period of time.Clarifying the group size can protect rare animals,accurately crack down on drug abuse and trafficking,and ensure good public health and social order Therefore,using zero truncation count data to estimate population size is of great significance.There are many methods of estimating population size in the literature.The earliest Lincoln-Peterson estimator assumes that individuals are homogeneous;In general,whether an individual is exposed(going to a drug treatment center or being checked for drunk driving)is related to many factors,which usually leads to different arrest probabilities for each individual.In order to characterize the heterogeneity between individuals,Bohning et al.(2005)using the zero truncated mixture Poisson distribution and mixture binomial distribution to fit the observed count data,the maximum likelihood estimation and EM algorithm of population size are proposed,and the bootstrap method is used to construct the interval estimation.However,the EM algorithm is not robust enough,and the asymptotic properties of statistics are not studied.When the sample size is large,the bootstrap confidence interval is also time-consuming.In order to test whether there is heterogeneity among individuals,goodness of fit test and AIC criterion are commonly used in the literature(Bohning et al.,2005).However,these two methods lack theoretical guarantee,and the test efficiency needs to be improved.In this paper,the population size estimation theory uner zero-truncated mixture count model is improved,and the Golden-search algorithm is proposed to enhance the EM algorithm.In order to test the heterogeneity of the zero-truncated mixture count model,this paper proposes a new EM test method,which can adaptively select the tuning parameters,and proves the large sample properties of the EM test statistics in theory.Simulation research and practical numerical analysis show that the EM algorithm based on Golden-section search is more stable,the MSE of its calculated maximum likelihood estimator is smaller,and the likelihood is more accurate than the confidence interval coverage probability;The proposed EM test can not only effectively control the first kind of error rate,but also has high efficiency,which provides a reasonable basis for practical workers to choose the model.