Abundance Estimation Using The Zero-truncated One-inflated Geometric Regression Model

Abundance estimation is of great importance in all aspects of our society.In the field of ecology,biologists can effectively monitor invasive species and assess the extinction risk of endangered species if they know the population size of species.In the field of sociology,policymakers can make relevant policies to maintain social stability if they know the population size of people with certain characteristics or behaviors such as drug users or drunk drivers.However,it is impossible for people to obtain the information of all individuals in practice,so researchers often collect data by sampling methods such as the capture-recapture method,and then estimate the size of the population based on the obtained count data such as the frequency of being captured.Researchers have proposed a variety of models to estimate population size based on the capture-recapture data.In current literature,Most researchers estimate the population size by using conditional likelihood and Horvitz-Thompson estimator.However,in the case of limited samples,the estimation based on full empirical likelihood is usually better than that based on conditional likelihood.Besides,the maximum empirical likelihood estimator is more effective than Horvitz-Thompson estimator,and the confidence interval based on empirical likelihood ratio has more accurate coverage probabilities than that based on conditional likelihood.In this paper,we propose a zero-truncated one-inflated geometric regression model by incorporating covariates into the existing models related to geometric distribution through link function.Based on the new model,the maximum empirical likelihood estimator and the empirical likelihood ratio confidence interval of the population size are proposed.The asymptotic normality of the maximum empirical likelihood estimator and the asymptotic properties of the empirical likelihood ratio statistics are proved.In addition,in order to test whether there exists one inflation in the observed data,this paper proposes a score test based on full likelihood function,and proves the asymptotic properties of the test statistics.Finally,the effectiveness of the proposed method is verified by numerical simulation and real data analysis.