Probability density estimation using new kernel functions

Abstract

This dissertation propose two new kernel functions to estimate a density unction with small biases and errors .The errors of the estimats are measured using mean squared error (MSEf (f(x,X ))), mean integrated squared error (MISE(f) and asymptotic mean integrated squared error (AMISE(f) ), and The estimates of these error measures are also given .The AMISE0(f) of the density estimates are compared to the kernel density estimates for the uniform, Epanechnikov and Gaussian kernel functions. One kernel function is derived by minimizing the sum of its absolute bias and variance and the other by minimizing the sum of the squared bias and the variance of its estimator. The kernel estimates using the proposed two new kernel functions were obtained. The estimates of these error measures are asymptotically unbiased and numerical method for comparing AMISE(f) in various populations is presented, the simulations performed using programs written in R The bandwidths used for comparing the properties of the functions are the Silverman rule of thum (SRT), the two stage direct plugin (DPI)and the solve the equation (STE) method bandwidths. A simulation study was carried out to compare the AMISE of the estimates with those of the uniform, Epanechnikov and Gaussian kernel functions. Two new kernel functions in the form of second degree polynomial with support [-1, 1] are presented. The estimats of MSE (f(x,X)) and MISE(f) were found to be asymptotically unbiased. For data with symmetric unimodal, symmetric bimodal and skewed unimodal distributions, the proposed estimates performed better than the uniform and Gaussian estimates. One of the proposed kernel estimates with an STE bandwidth performed well when data are in kurtotic unimodal and trimodal distributions with sample sizes 5 and 100. This kernel estimate also performed better than the others for data in astrongly skewed distribution. Th estimates with SRT bandwidth performed well when data are in askewed bimodal distribution with small sample size. or data inclaw, double claw and asymmetric double claw distributions the estimates with SRT bandwidth are better than the others. The same results occurred when STE bandwidth is used with large sample sizes. For data distributed as asymmetric multimodal, one of the proposed estimates with STE bandwidth performed better than the others. The other proposed kernel estimate also performed better than the uniform and Gaussian estimate.