Statistical Inference Based on Kernel Distribution Function Estimators

Description

This book presents a study of statistical inferences based on the kernel-type estimators of distribution functions. The inferences involve matters such as quantile estimation, nonparametric tests, and mean residual life expectation, to name just some. Convergence rates for the kernel estimators of density functions are slower than ordinary parametric estimators, which have root-n consistency. If the appropriate kernel function is used, the kernel estimators of the distribution functions recover the root-n consistency, and the inferences based on kernel distribution estimators have root-n consistency. Further, the kernel-type estimator produces smooth estimation results. The estimators based on the empirical distribution function have discrete distribution, and the normal approximation cannot be improved―that is, the validity of the Edgeworth expansion cannot be proved. If the support of the population density function is bounded, there is a boundary problem, namely the estimator does not have consistency near the boundary. The book also contains a study of the mean squared errors of the estimators and the Edgeworth expansion for quantile estimators. Preface Contents 1 Kernel Density Function Estimator 1.1 Boundary Bias Problem 1.2 Bias and Variance Reductions 1.3 Simulation Studies References 2 Kernel Distribution Function Estimator 2.1 Properties of KDFE 2.2 Bias Reduction of KDFE 2.3 Simulation Results References 3 Kernel Quantile Estimation 3.1 Quantile Estimators 3.2 Properties of Quantile Estimators 3.3 Asymptotic Properties of Quantile Estimators 3.4 Numerical Comparisons 3.4.1 Estimation of Quantiles of the Standard Normal Distribution 3.4.2 Estimation of Quantiles of the Standard Exponential Distribution 3.4.3 Estimation of Quantiles of the Gamma Distribution References 4 Mean Residual Life Estimator 4.1 Estimators of the Survival Function and the Cumulative Survival Function 4.2 Estimators of the Mean Residual Life Function 4.3 Numerical Studies References 5 Kernel-Based Nonparametric Tests 5.1 Naive Kernel Goodness-of-Fit Tests 5.2 Boundary-Free Kernel-Type Goodness-of-Fit Tests 5.2.1 Boundary-Free KDFE 5.2.2 Boundary-Free Kernel-Smoothed KS and CvM Tests 5.2.3 Numerical Results 5.3 Smoothed Nonparametric Tests and Approximation of p-Value 5.3.1 Asymptotic Properties of Smoothed Tests 5.3.2 Selection of Bandwidth and Kernel Function 5.3.3 Higher Order Approximation References