Purely Sequential and Two-Stage Fixed-Accuracy Confidence Interval Estimation Methods for Count Data from Negative Binomial Distributions in Statistical Ecology: One-Sample and Two-Sample Problems

Our main focus is on count data arising from statistical ecology. Anscombe (1949 Anscombe , F. J. ( 1949 ). The Statistical Analysis of Insect Counts Based on the Negative Binomial Distribution , Biometrics 5 : 165 – 173 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) emphasized negative binomial (NB) modeling for overdispersed count data. A large majority of existing methodologies, both sequential and nonsequential, were reviewed by Mukhopadhyay and Banerjee (2012 Mukhopadhyay , N. and Banerjee , S. ( 2012 ). Sequential Negative Binomial Problems and Statistical Ecology: Selected Review with New Directions (under submission). [Google Scholar]). We assume that the thatch parameter remains known and revisit a sequential confidence interval estimation method for an NB mean proposed by Willson and Folks (1983 Willson , L. J. and Folks , J. L. ( 1983 ). Sequential Estimation of the Mean of the Negative Binomial Distribution , Communications in Statistics-Sequential Analysis 2 : 55 – 70 .[Taylor & Francis Online] , [Google Scholar]) that may not always guarantee a positive lower confidence limit. Moreover, any postsampling adjustment of their lower confidence limit would compromise the preset confidence coefficient. In this article, we provide an appropriate resolution by estimating the NB mean with a fixed-accuracy confidence interval such that both confidence limits are positive. Our proposed purely sequential and two-stage estimation methodologies enjoy asymptotic consistency and asymptotic efficiency properties. Next, we consider estimating the ratio of two NB means under a two-sample setup assuming that the thatch parameters remain known with equal sample sizes and provide both purely sequential and two-stage methodologies. These are shown to enjoy asymptotic consistency and asymptotic efficiency properties. Extensive sets of analyses based on both simulated data and real data from potato beetle infestation are presented to highlight some of the exciting small- and moderate-sample features of the proposed estimation methodologies.