Two-stage estimation of mean in a negative binomial distribution with applications to Mexican Bean Beetle data

Working with insect counts, Anscombe (1949) emphasized negative binomial modeling by introducing a parameterization involving µ(>0) and kappa(>0). The parameters µ, kappa stood for average ¿infestation¿ and ¿clumping, ¿ respectively. Assuming that kappa was known, Willson and Folks (1983) adopted purely sequential sampling to estimate µ, whereas Mukhopadhyay and Diaz (1985) 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 kappa 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 kappa. 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 kappa is known or unknown. When kappa 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.