Optimal Two-Phase Stratified Sampling for Estimation of the Age Composition of a Fish Population

Abstract

The provision of biological advice for the management of a commercial fish stock requires, among other things, an estimate of the age composition. This estimate is usually obtained from length and age information collected by sampling the commercial landings associated with that fish stock. The sampling methods used and the types of problems encountered are discussed by Quinn et al. (1983) and Stevenson (1983). The final stage of the sample design is typically a two-phase stratified sample of a large catch from a commercial vessel (Quinn et al., 1983; Stevenson, 1983; Southward, (1976). At the first phase, a large simple random sample (SRS) of size n' is selected, and each fish is assigned to a stratum consisting of all fish having lengths in a specified range. Denote by nf the number of sampled fish in length stratum i. At the second phase a SRS of size ni is chosen from stratum i (i = 1, . . ., I). Each of these ni fish is aged with nij fish found to be agej (j = 1, ... , J). The problems are: (i) estimation of IPj: j = 1, . .. , J), where P.j is the population proportion of age j fish; (ii) given n' and In': i = 1, . . ., I }, how to select Ini: i = 1, .. , I }; and (iii) for a specified cost function and given budget, how to select n'. Smith and Sedransk (1982) provide methods for (i) and (ii). For (ii), their approach is flexible in that currently used or proposed allocation schemes are shown to be optimal under appropriate specifications. In the current paper we: (i) review the Smith-Sedransk methodology, (ii) show how to determine the optimal value of n', and (iii) exhibit potential gains from using two-phase sampling. In Section 3 we give a numerical illustration of the