Confidence estimation via tail functions

Abstract

This thesis advances the theory of tail functions for confidence estimation, as originally formulated by Puza and O'Neill (2006a, 2006b). The theory is extended to the case where nuisance parameters are present and to where a joint confidence set is required for two or more parameters. The use of tail functions for the confidence estimation of partially defined parameters is also explored. In the thesis, confidence sets obtained via the tail functions approach are mainly evaluated by the conditional expected volume, or the prior expected volume, when prior information is available. The optimisation procedure of confidence sets in terms of minimising the expected volume under various constraints is illustrated with normally distributed samples under the framework of non-decreasing tail functions. Comparisons are made with the approach to confidence estimation in Brown, Casella and Hwang (1995). The thesis also further develops the tail functions approach to confidence estimation for parameters of discrete distributions, in particular, the geometric and negative binomial. A review of the theory of Pratt (1961) on confidence estimation via the likelihood function precedes an examination of various weight functions in relation to minimisation of expected volume. A new weight function is proposed and the associated confidence intervals studied. Early in the thesis, the class of piecewise-constant tail functions is introduced and subsequently used to help solve the difficult equations that arise in later chapters.