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Small Sample Inference for Two-way Capture Recapture Experiments

Louis-Paul Rivest, Mamadou Yauck

Abstract

The properties of the generalized Waring distribution defined on the non negative integers are reviewed. Formulas for its moments and its mode are given. A construction as a mixture of negative binomial distributions is also presented. Then we turn to the Petersen model for estimating the population size $N$ in a two-way capture recapture experiment. We construct a Bayesian model for $N$ by combining a Waring prior with the hypergeometric distribution for the number of units caught twice in the experiment. Credible intervals for $N$ are obtained using quantiles of the posterior, a generalized Waring distribution. The standard confidence interval for the population size constructed using the asymptotic variance of Petersen estimator and .5 logit transformed interval are shown to be special cases of the generalized Waring credible interval. The true coverage of this interval is shown to be bigger than or equal to its nominal converage in small populations, regardless of the capture probabilities. In addition, its length is substantially smaller than that of the .5 logit transformed interval. Thus the proposed generalized Waring credible interval appears to be the best way to quantify the uncertainty of the Petersen estimator for populations size.

Small Sample Inference for Two-way Capture Recapture Experiments

Abstract

The properties of the generalized Waring distribution defined on the non negative integers are reviewed. Formulas for its moments and its mode are given. A construction as a mixture of negative binomial distributions is also presented. Then we turn to the Petersen model for estimating the population size in a two-way capture recapture experiment. We construct a Bayesian model for by combining a Waring prior with the hypergeometric distribution for the number of units caught twice in the experiment. Credible intervals for are obtained using quantiles of the posterior, a generalized Waring distribution. The standard confidence interval for the population size constructed using the asymptotic variance of Petersen estimator and .5 logit transformed interval are shown to be special cases of the generalized Waring credible interval. The true coverage of this interval is shown to be bigger than or equal to its nominal converage in small populations, regardless of the capture probabilities. In addition, its length is substantially smaller than that of the .5 logit transformed interval. Thus the proposed generalized Waring credible interval appears to be the best way to quantify the uncertainty of the Petersen estimator for populations size.
Paper Structure (8 sections, 1 theorem, 17 equations, 3 figures, 2 tables)

This paper contains 8 sections, 1 theorem, 17 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The generalized Waring distribution
  3. New inference methods for Petersen's model
  4. Some Bayesian estimation procedures
  5. Approximate confidence intervals for $N$
  6. Some examples
  7. The expected length and the coverage of the GWD credible interval for $N$
  8. Discussion

Key Result

Proposition 1

If $Y$ has a $GWD(1,b,c)$ distribution and $m$ is a positive integer, then

Figures (3)

  • Figure 1: Prior and posterior distributions for $N$ corresponding to the fourth example in Table \ref{['tab:Seb82']}. Lower and upper bounds for the 95% credible interval are represented by vertical lines.
  • Figure 2: Expected coverage, as a function of $p_1, p_2$, of three 95 % credible/confidence intervals for $N=20$ and $N=50$: the Tlogit interval and the GWD intervals for $\ell=2,3$.
  • Figure 3: Boxplots for the relative lengths of Tlogit confidence intervals and the corresponding GWD credible intervals, with $\ell=2$, at the 95% confidence level. This representation only visualizes the distribution of the relative lengths when the interval length is less than 50 for both methods.

Theorems & Definitions (1)

  • Proposition 1