Two-Stage Estimation of Mean in a Negative Binomial Distribution with Applications to Mexican Bean Beetle Data

Working with insect counts, 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 modeling by introducing a parameterization involving μ(>0) and κ(>0). The parameters μ, κ stood for average “infestation” and “clumping, ” respectively. Assuming that κ was known, Willson and Folks (1983 Willson , L. J. , and Folks , J. L. ( 1983 ). Sequential Estimation of the Mean of the Negative Binomial Distribution , Sequential Analysis 2 : 55 – 70 .[Taylor & Francis Online] , [Google Scholar]) adopted purely sequential sampling to estimate μ, whereas Mukhopadhyay and Diaz (1985 Mukhopadhyay , N. , and Diaz , J. (1985). Two-Stage Sampling for Estimating the Mean of a Negative Binomial Distribution, Sequential Analysis 4:1–18.[Taylor & Francis Online] , [Google Scholar]) developed a two-stage methodology because of its operational convenience. We first prove a new striking result (Theorem 2.1) that claims the asymptotic second-order efficiency property of the two-stage procedure. In order to handle the case when κ is unknown, we develop a new approach (section 3) for evaluating estimators of μ. We control a new criterion, namely the integrated coefficient of variation (ICV), by averaging the CV with respect to a weight function for κ. A two-stage methodology is proposed, and both first- and second-order properties are highlighted (Theorems 3.1–3.3). We summarize findings from extensive sets of simulations of the two-stage methodologies both when κ is known or unknown. When κ is unknown, the robustness of the proposed methodology with respect to choices of a weight function is critically examined. In the end, both methodologies are applied to four sets of Mexican bean beetle data with encouraging findings.