Analytic bootstrap and kernel based methods with applications
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The bootstrap method is an appealing nonparametric technique commonly employed in many aspects of modern statistics. Under certain situations, exact analytic expressions such as the bootstrap mean and variance of L -estimates can be obtained, thus eliminating the error due to Monte Carlo resampling. This exact approach may be extended to more complex situations. In the first line of research, we develop a quantile function estimator based upon the exact bootstrap mean of a fractional order statistic given right censored data as an extension of the well-known Harrell-Davis quantile function estimator. A similar strategy to that of quantile function estimator is also employed in the context of the bootstrap bandwidth selection problem for kernel density estimation. The second line of research, inspired by the “plug-in” principle of the bootstrap, is the development of a novel kernel density estimator for a weighted average from a single population or the difference of two weighted averages from independent populations. This approach is based on utilizing the standard kernel density estimator in conjunction with classical inversion theory. We further develop a more general kernel density estimator for directly estimating the probability density and cumulative distribution function of an L -estimate via extending the inversion theory due to Knight (1985) to allow for more generalized expression including 3rd and 4th moments. The third line of research is to develop a new estimator for the r th L-moment given right censored data with an illustration of its asymptotic properties and application in distribution characterization and parameter estimation. The fourth line of research is to define a new and improved smooth population quantile function given discrete data, partly motivated by a recent AISM publication by Ma et al. (2009). The final part of this dissertation is dedicated to future work regarding the utilization and extension of the above developed methodologies.