Comparing Bayesian, Maximum Likelihood and Classical Estimates for the Jolly-Seber Model
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2001-05-30
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In 1965 Jolly and Seber proposed a model to analyze data for open population capture-recapture studies. Despite frequent use of the Jolly-Seber model, likelihood-based inference is complicated by the presence of a number of unobservable variables that cannot be easily integrated from the likelihood. In order to avoid integration, various statistical methods have been employed to obtain meaningful parameter estimates. Conditional maximum likelihood, suggested by both Jolly and Seber, has become the standard method. Two new parameter estimation methods, applied to the Jolly-Seber Model D, are presented in this thesis. The first new method attempts to obtain maximum likelihood estimates after integrating all of the unobservable variables from the Jolly-Seber Model D likelihood. Most of the unobservable variables can be analytically integrated from the likelihood. However, the variables dealing with the abundance of uncaptured individuals must be numerically integrated. A FORTRAN program was constructed to perform the numerical integration and search for MLEs using a combination of fixed quadrature and Newton's method. Since numerical integration tends to be very time consuming, MLEs could only be obtained from capture-recapture studies with a small number of sampling periods. In order to test the validity of the MLE, a simulation experiment was conducted that obtained MLEs from simulated data for a wide variety of parameter values. Variance estimates for these MLEs were obtained using the Chapman-Robbins lower bound. These variances estimates were used to construct 90% confidence intervals with approximately correct coverage. However, in cases with few recaptures the MLEs performed poorly. In general, the MLEs tended to perform well on a wide variety of the simulated data sets and appears to be a valid tool for estimating population characteristics for open populations. The second new method employs the Gibbs sampler on an unintegrated and an integrated version of the Jolly-Seber Model D likelihood. For both version full conditional distributions are easily obtained for all parameters of interest. However, sampling from these distributions is non-trivial. Two FORTRAN programs were developed to run the Gibbs sampler for the unintegrated and the integrated likelihoods respectively. Means, medians, modes and variances were constructed from the resulting empirical posterior distributions and used for inference. Spectral density was used to construct a variance estimate for the posterior mean. Equal-tailed posterior density regions were directly calculated from the posteriors distributions. A simulation experiment was conducted to test the validity of density regions. These density regions also have approximately the proper coverage provided that the capture probability is not too small. Convergence to a stationary distribution is explored for both version of the likelihood. Often, convergence was difficult to detect, therefore a test of convergence was constructed by comparing two independent chains from both version of the Gibbs sampler. Finally, an experiment was constructed to compare these two new methods and the traditional conditional maximum likelihood estimates using data simulated from a capture-recapture experiment with 4 sampling periods. This experiment showed that there is little difference between the conditional maximum likelihood estimates and the 'true' maximum likelihood estimates when the population size is large. A second simulation experiment was conducted to determine which of the 3 estimation methods provided the 'best' estimators. This experiment was largely inconclusive as no single method routinely outperformed the others.
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PhD
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Statistics
