Problems in prediction
Andrew Rieck · ANU Open Research (Australian National University) · 2001
identifying canonical forms are, for example, a one-sided prediction interval (assuming the predictand is a random variable), or a prediction region obtained by profiling a function which depends on the sample X and a.The type of canonical form determines how the probability mass a should be shared, in an asymptotic sense for a nominal «-level prediction region, throughout the probability space.An outline of this thesis is as follows:Chapter 2 defines and reviews the methods used to construct a nominal «-level prediction region or interval for a parametric population.One method uses pivotal transformations to construct an exact «-level prediction region for a location-scale population.Other methods rely on the definition of a predictive likelihood, function or density which are analogous, in certain regards, to the conditional probability density function of the predictand given the sample, except that they depend on the sample and « only and not on any unknown population parameters.Chapter 3 defines and reviews the methods used to construct a nominal «-level prediction in terval for a nonparametric population.Firstly, methods based on an independent and identically distributed sample are discussed.These include a Studentised method and quantile estimation methods for a predictand and the percentileT and accelerated bias-correction methods for a pre dictand statistic.Secondly, methods are proposed for regression and structural models.Chapter 4 defines quantile estimation methods used to construct nominal a-level prediction intervals for a nonparametric population.The quantile estimates are constructed from interpolation among quantiles of the empirical distribution function (or equivalently, interpolation among order statistics).Two forms of predictive interval calibration, the jackknife and smoothed bootstrap, are investigated.Chapter 5 investigates, for small sample sizes, the numerical properties of coverage error for prediction intervals defined in Chapter 4. The convergence of numerical approximations is also considered.INotes on N otation.The transpose of a vector v and a matrix M are denoted by vT and M T, respectively.The trace, determinant, and inverse of a matrix M are denoted by trM , det M, and M~l , respectively.