Chiang Mai Journal of Science

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This paper considers two new kernel estimators of a density function f (x). The errors of the estimators are measured by the mean squared error (MSE( f^{^} (x,X ~ )) and the mean integrated squared error (MISE( f^{^} )). The estimates of these error measures are also given. The estimators of MSE( f ^ (x,X ~ )) and MISE( f ^ ) are found to be asymptotically unbiased. Properties of the proposed estimators depend on the corresponding kernel functions used to derive them together with their bandwidths. The bandwidths used for comparison of  the properties are the Silverman rule of thump (SRT), two-stage direct plug-in (DPI) and the solve-the-equation (STE) bandwidths. A simulation study is carried out to compare the AMISE of  the estimates with those of  uniform, Epanechnikov and Gaussian kernel functions. For data with outlier and bimodal distributions, the proposed estimates perform better than the uniform and Gaussian estimates. One of  the proposed kernel estimates with STE bandwidth performs well when data are with a strongly skewed distribution. This estimates with SRT bandwidth performs well when data are skewed bimodal with small sample size. For data with claw distribution, the estimate with SRT bandwidth is better than the others. The same results hold when the STE bandwidth is used with large sample sizes. For data distributed as discrete comb, one of  the proposed estimates with STE bandwidth performs better than the others. Another proposed kernel estimate also performs better than the uniform and Gaussian estimate. 

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