Non-parametric estimation of population size from capture–recapture data when the capture probability depends on a covariate

Summary.  In capture–recapture experiments the capture probabilities may depend on individual covariates such as an individual's weight or age. Typically this dependence is modelled through simple parametric functions of the covariates. Here we first demonstrate that misspecification of the model can produce biased estimates and subsequently develop a non-parametric procedure to estimate the functional relationship between the probability of capture and a single covariate. This estimator is then incorporated in a Horvitz–Thompson estimator to estimate the size of the population. The resulting estimators are evaluated in a simulation study and applied to a data set on captures of the Mountain Pygmy Possum.     

Continuous-time capture-recapture models with time variation and behavioural response

This paper develops a likelihood-based inference procedure for continuous-time capture-recapture models. The first-capture and recapture intensities are assumed to be in constant proportion but may otherwise vary arbitrarily through time. The full likelihood is partitioned into two factors, one of which is analogous to the likelihood in a special type of multiplicative intensity model arising in failure time analysis. The remaining factor is free of the non-parametric nuisance parameter and is easily maximized. This factor provides an estimator of population size and an asymptotic variance under a counting process framework. The resulting estimation procedure is shown to be equivalent to that derived from a martingale-based estimating function approach. Simulation results are presented to examine the performance of the proposed estimators.     

Measurement errors in continuous-time capture–recapture models

The extent of bias due to measurement errors is an important problem in the context of regression and survival analysis. While research in these areas has been extensive and fruitful, investigations into the effect of measurement errors in capture–recapture models have been very limited. The contributions of this paper are to understand the effects of measurement errors in continuous-time capture–recapture models and to propose new methods to circumvent their impact. We derive asymptotic variance formulas for each method and assess their finite sample properties via simulation studies.     

APPLICATION OF LAPLACE'S BOUNDARY-MODE APPROXIMATIONS TO ESTIMATE SPECIES AND SHARED SPECIES RICHNESS

The Laplace method for approximating integrals is a useful technique in a number of research fields. This paper shows that it also has interesting applications in biological and ecological statistical inferences. When sample abundance or replicated incidence (i.e., presence or absence) records of each species are available, the expected low‐order frequency counts in heterogeneous communities can be approximated by the Laplace method when the species discovery or detection probabilities are bounded from below by a constant. The approximation formulae as applied to one community can then be used to derive estimators of species richness and to examine their performance. The approach is also extended to obtain simple and new estimators for the number of shared species in two communities. The replicated species incidence data recorded by competing teams of the Hong Kong Big Bird Race for the years 1999 and 2000 are used to estimate the number of resident birds in Hong Kong and to illustrate the method of estimation.     

A Note on the Inverse Birthday Problem With Applications

The classical birthday problem considers the probability that at least two people in a group of size N share the same birthday. The inverse birthday problem considers the estimation of the size N of a group given the number of different birthdays in the group. In practice, this problem is analogous to estimating the size of a population from occurrence data only. The inverse problem can be solved via two simple approaches including the method of moments for a multinominal model and the maximum likelihood estimate of a Poisson model, which we present in this study. We investigate properties of both methods and show that they can yield asymptotically equivalent Wald-type interval estimators. Moreover, we show that these methods estimate a lower bound for the population size when birth rates are nonhomogenous or individuals in the population are aggregated. A simulation study was conducted to evaluate the performance of the point estimates arising from the two approaches and to compare the performance of seven interval estimators, including likelihood ratio and log-transformation methods. We illustrate the utility of these methods by estimating: (1) the abundance of tree species over a 50-hectare forest plot, (2) the number of Chlamydia infections when only the number of different birthdays of the patients is known, and (3) the number of rainy days when the number of rainy weeks is known. Supplementary materials for this article are available online.     

APPLICATION OF SEMIPARAMETRIC REGRESSION MODELS IN THE ANALYSIS OF CAPTURE-RECAPTURE EXPERIMENTS

If the capture probabilities in a capture‐recapture experiment depend on covariates, parametric models may be fitted and the population size may then be estimated. Here a semiparametric model for the capture probabilities that allows both continuous and categorical covariates is developed. Kernel smoothing and profile estimating equations are used to estimate the nonparametric and parametric components. Analytic forms of the standard errors are derived, which allows an empirical bias bandwidth selection procedure to be used to estimate the bandwidth. The method is evaluated in simulations and is applied to a real data set concerning captures of Prinia flaviventris, which is a common bird species in Southeast Asia.