Practical aspects of kernel smoothing for binary regression and density estimation
This thesis explores the practical use of kernel smoothing in three areas: binary regression, density estimation and Poisson regression sample size calculations. Both nonparametric and semiparametric binary regression estimators are examined in detail, and extended to two bandwidth cases. The asymptotic behaviour of these estimators is presented in a unified way, and the practical performance is assessed using a simulation experiment. It is shown that, when using the ideal bandwidth, the two bandwidth estimators often lead to dramatically improved estimation. These benefits are not reproduced, however, when two general bandwidth selection procedures described briefly in the literature are applied to the estimators in question. Only in certain circumstances does the two bandwidth estimator prove superior to the one bandwidth semiparametric estimator, and a simple rule-of-thumb based on robust scale estimation is suggested. The second part summarises and compares many different approaches to improving upon the standard kernel method for density estimation. These estimators all have asymptotically 'better' behaviour than the standard estimator, but a small-sample simulation experiment is used to examine which, if any, can give important practical benefits. Very simple bandwidth selection rules which rely on robust estimates of scale are then constructed for the most promising estimators. It is shown that a particular multiplicative bias-correcting estimator is in many cases superior to the standard estimator, both asymptotically and in practice using a data-dependent bandwidth. The final part shows how the sample size or power for Poisson regression can be calculated, using knowledge about the distribution of covariates. This knowledge is encapsulated in the moment generating function, and it is demonstrated that, in most circumstances, the use of the empirical moment generating function and related functions is superior to kernel smoothed estimates.