Purely sequential bounded-risk point estimation of the negative binomial means under various loss functions: Multi-sample problems

ABSTRACT

A negative binomial (NB) distribution is often used to model over dispersed count data arising from areas including ecology, agriculture, health, etc. We have designed purely sequential bounded-risk methodologies to (1) simultaneously estimate an unknown k-vector of NB means and (2) estimate the difference in means of two independent NB populations under different forms of loss functions including customary and modified Linex loss as well as squared error loss. We have developed point estimation techniques both when (i) the thatch parameters τ_is are assumed known or unknown and (ii) the sample sizes are equal or unequal. The proposed methodologies are shown to satisfy interesting desirable properties including first-order asymptotic efficiency and first-order asymptotic risk efficiency. Summaries are provided from extensive sets of simulations showing encouraging performances of the proposed methodologies for both small and moderate sample sizes. These are followed by illustrations obtained by implementing each estimation strategy using real data from statistical ecology: Raptor count data of different species of raptors at the Hawk Mountain sanctuary in Pennsylvania.

KEYWORDS:

• Customary Linex loss
• CV approach
• first-order asymptotic efficiency
• first-order asymptotic risk efficiency
• Raptor count data
• simulations
• squared error loss
• statistical ecology

SUBJECT CLASSIFICATIONS:

• 62L12
• 62L05
• 92D40
• 62P12

Acknowledgments

We are grateful to Gigi Romano, the communications specialist at the Hawk Mountain sanctuary, Pennsylvania, who gave us permission to use the raptor count data found on their webpage (www.hawkmountain.org). We heartily thank Gigi Romano and others who have made these data available for our illustrations. We express sincere gratitude to an Associate Editor and the reviewers for sharing with us their candid and helpful comments.