A note on the multiple-recapture census

Abstract

In the field of biology the problem of estimating population parameters such as population size, death rate, birth rate, etc., for animal populations is obviously an important one. Various capture-tag-recapture models have been developed to estimate these population parameters with a minimum number of assumptions on the underlying population. One such method, the multiple-recapture census, has been the topic of many papers and is described briefly as follows. The experimenter takes a sequence of random samples a,, a2, ... , a8, say. The members of each sample a* are tagged and returned to the population before taking the next sample. Thus the members of a2, a3, ..., a, can be classified according to when, if at all, they have been captured before. Although several models have been developed from different basic assumptions, three papers in particular by Darroch (1958, 1959) and Jolly (1965) give the most general treatment of this method in the form of exact, fully stochastic models which lend themselves readily to the method of maximum-likelihood estimation. The first paper of Darroch's deals with the closed population, i.e. a population in which there is neither augmentation due to immigration (or birth) nor departure due to death (or emigration). This population is usually dealt with separately as the algebra involved is quite different from that which arises in more general populations. The main assumption made, which in fact underlies all capturerecapture models, is that marked and unmarked individuals have the same probability of being caught and as far as the author knows no test statistic has been given for testing this assumption. In ? 4 of this paper we shall consider this problem and a likelihood-ratio test statistic will be derived which gives an approximate test of the above assumption. In Darroch's second paper he derives, for a population in which there is either immigration or death (but not both), modified maximum-likelihood estimates of the population parameters that are almost unbiased and asymptotically efficient, that is efficient for a certain class of reasonable estimates. For the population in which there is both immigration and death he gives only the probability generating function for the model and derives moment estimates of the parameters. However, Jolly (1965), tackling this problem from a different viewpoint, gives a very elegant solution to the problem of finding maximumlikelihood estimates of the unknown population parameters and gives the means and variances of these estimates. In this paper we shall consider this general population with both immigration and death and set up a model which differs slightly from that of Darroch and Jolly in that certain parameters are treated as unknown constants rather than as random variables. The notation and approach to the model is essentially that of Darroch's while the method of solution is similar to that given in Seber (1962). The estimates obtained from this model will be compared briefly with those obtained by Jolly.