Sequential negative binomial problems and statistical ecology: A selected review with new directions

Count data is abundant in entomology, more broadly, in statistical ecology. In 1949, Frank Anscombe pioneered the role of negative binomial (NB) modeling while working with insect count data. Since then, the spectrum of available research methods has grown immensely in more than past sixty years in involving count data modeled by a NB distribution. NB distribution also finds extensive use in agriculture, insect infestation, soil and weed science, etc. In this paper we have used a real dataset on potato beetle infestation (Beall, 1939) to illustrate smooth data collection under various sequential inferential procedures to draw important and practical conclusions.

We begin by selectively reviewing a majority of influential research methods for a number of formulations and their executions in the context of fixed-sample-size inferential procedures (Section  2). Subsequently, we elaborate on purely sequential and two-stage sampling methodologies for data collection (Sections  3 Sequential inferential problems: tests of hypotheses, 4 Sequential inferential problems: estimation). In Section  5, we summarize some major results with their interpretations including large-sample first- and second-order properties as appropriate. The illustrations of all the sequential inferential procedures on the real dataset gives interesting insights (Section  6). We also propose a number of selected directions for future research of substantial nature (Section  7). Finally, our own R codes are provided in the Appendix.