Abstract
This M.Sc. thesis focuses on improving the convergence rates of kernel density estimators. Firstly, a bias reduced kernel density estimator is introduced and investigated. In order to reduce bias, we intuitively subtract an estimated bias term from ordinary kernel density estimator.
Theoretical properties such as bias, variance and mean squared error are investigated
for this estimator and comparisons with ordinary kernel density estimator and location-scale kernel density estimator are made. Compared with the ordinary density estimator, this estimator has reduced bias and mean squared error (MSE) of the order O(h^3) and O(n^(-6/7))，respectively, and even further reduced orders O(h^4) and O(n^(-8
9)) respectively when the kernel
is chosen symmetric. Secondly, we propose a geometric extrapolation of the location-scale
kernel estimator and a geometric extrapolation of the bias reduced kernel estimator introduced above. Similarly, we investigate their theoretical properties and compare them with the geometric extrapolation of ordinary kernel estimator. These results show that among the three geometric extrapolated kernel estimators, the one based on bias reduced kernel estimator has smallest bias and MSE. The geometric extrapolation of bias reduced kernel estimator can improve the convergence rates of bias and MSE to O(h^6) and O(n^(-12/13)) respectively
for symmetric kernels. The geometric extrapolation of location-scale kernel estimator can reduce bias and MSE of location-scale kernel estimator, however it has bias and MSE with a slower rate than those of the geometric extrapolation of ordinary kernel estimator. In order to assess nite sample performances of the proposed estimators, Monte Carlo simulation studies based on small to moderately large samples are carried out. Finally, an analysis
of the old faithful geyser data are presented to demonstrate the proposed methods. Both
the simulation studies and the real data analysis consolidate our theoretical findings.
