School of Mathematics and Statistics - Theses
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ItemMethods for estimating occupancy( 2014)The estimation of the probability of occupancy of a site by a species is used to monitor the distribution of that species. Occupancy models have been widely applied and several limitations have been identified. In this thesis we resolve some of these. In particular we focus on limitations of maximum likelihood estimators and the associated interval estimators, and the difficulties associated with the extension from linear to generalised additive models for the relationship between occupancy and covariates. Initially we consider in detail the basic occupancy model which includes two parameters: $\psi$ and $p$. Our primary concern is the probability that the species occupies a particular site, $\psi$. The other parameter, the detection probability $p$, is a nuisance parameter. We first derive the joint probability mass function for the sufficient statistics of occupancy which allows the exact evaluation of its mean and variance, and hence its bias. We show that estimation near the boundaries of the parameter space is difficult. For small values of detection, we show that estimation of occupancy is not possible and specify the region of the parameter space where maximum likelihood estimators exist, and give the equations for the MLEs in this region. We next demonstrate that the asymptotic variance of the estimated occupancy is underestimated, yielding interval estimators that are too narrow. Methods for constructing interval estimators are then explored. We evaluate several bootstrap-based interval estimators for occupancy. Finally, instead of the full likelihood we consider a partial likelihood approach. This gives simple closed form estimators in a basic model with only a small loss of efficiency. It greatly simplifies the inclusion of linear and nonlinear covariates by allowing the use of standard statistical software for GLM and GAM frameworks and in our simulation study there is little loss of efficiency compared to the full likelihood.
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ItemConditional inference( 1984)Conditional inference is a branch of statistical inference in which observed data is reduced using either sufficient or ancillary statistics. This often simplifies inference about the parameters. In comparison to full likelihood methods, conditional inference theory’s performance still needs validating in many areas. Some of these are the concern of this thesis. While the definition of an ancillary statistic in single parameter models is unequivocal, the presence of accessory (or nuisance) parameters in a model presents problems in defining an ancillary statistic. Statistical literature abounds with definitions of ancillarity in this case. Some of the commonest and most useful of these are discussed and shown to be interrelated. This facilitates the choice of the strongest eligible ancillary in a problem, i.e. that which offers the biggest reduction of the sample space. The Pitman-Morgan test for variance ratios in bivariate normal populations with unknown correlation coefficient is shown to be a conditional test. We condition on sufficient statistics for the accessory parameters to eliminate them. The test statistic is then derived as an ancillary statistic for the accessory parameters. When a probability model depends on a number of accessory parameters which increases with the sample size, estimation methods based on the full likelihood will often be inconsistent. Using a partial likelihood instead has been suggested. Local maximum partial likelihood estimators are shown to exist, and to be consistent and asymptotically normal under mild conditions. These results also cover conditional and marginal likelihoods, thus considerably strengthening earlier results in this area. In planning statistical inferences, it is useful to choose a sampling scheme which provides only the essential data to our inferences. Jagers’ lemma proposes very general conditions under which maximum likelihood estimation from a subset of the data is identical with that from the full data. However, the lemma is incorrect as given. We show that an additional sufficiency condition repairs the lemma. It is further shown that this lemma cannot be extended to general exponential families.