A Poisson Fishing Model

Abstract

A fishing model of Starr, Wardrop, and Woodroofe is related to the sequential search model of Cozzolino. The latter is generalized to allow an arbitrary joint distribution of capture times and fish sizes. Implications to the foraging models of Oaten and Green and to debugging software are indicated. 0. Introduction. The central theme of this paper is a result in sequential analysis that has application to a wide variety of problems. These problems have appeared in papers dealing with sequential estimation in statistics, estimation of the number of species, the fishing problem, the proofreading problem, auditing, foraging, search, etc. Authors in different areas are not always aware of each other, and so often recompute the basic results again. This is partly because the basic assumptions required in the different areas necessarily differ in signficant ways. Yet the main result in Section 4 of this paper would be of interest in all these areas. Since the basic assumption of the result is that a certain parameter has a Poisson distribution, it seems appropriate to use the model of Starr, Woodroofe and others as a fishing problem. 1. The Fishing Problem. One of the first papers to deal with the fishing problem was Starr (1974). The main result of this paper is easy to state. There are m fish in a lake, where m is known. The capture time of fish j if one fishes indefinitely is T j. We assume T 1 ,. .. , T m are i.i.d. exponential with hazard rate r. Let K(t) denote the number of fish caught by time t , so that (1) K(t) = m j=1 I(T j ≤ t), where I denotes the indicator function. There is a constant cost of time, so the payoff for stopping at time t is (2) Y t = K(t) − ct.