Standard kernel density estimation is subject to bias that can mask
structure by flattening peaks and filling in troughs in the density.
We explore the potential of bias reduction by modification, and
propose a new type of weighted kernel density estimator in which the
weights are defined by the proportion of informations around the
given values. The free parameters of these methods are optimized
with respect to a set of sample performance criteria. Technical
aspects of the implementation of our estimators are discussed and
its performance at finite sample size is analysed with experiments
of simulation. The results are very encouraging, and suggest that
our modified method is capable of obtaining significant bias
reduction over the traditional method.

Contents
1 Introduction 4
2 Theoretical formulations 5
2.1 Main formulations . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Estimation of probability density function . . . . . . . . . . . 7
2.3 Estimation of hazard rate function . . . . . . . . . . . . . . . 8
2.4 Estimation of regression function . . . . . . . . . . . . . . . . 10
3 Simulation Studies 11
3.1 Variables and parameters . . . . . . . . . . . . . . . . . . . . . 11
3.2 Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.1 Probability Density Function . . . . . . . . . . . . . . 11
3.2.2 Hazard Rate function . . . . . . . . . . . . . . . . . . . 12
3.2.3 Regression function . . . . . . . . . . . . . . . . . . . . 13
4 Simulation Result 13
4.1 Estimating Probability Density Function . . . . . . . . . . . 14
4.1.1 Normal distribution . . . . . . . . . . . . . . . . . . . 14
4.1.2 Exponential distribution . . . . . . . . . . . . . . . . 16
4.2 Estimating hazard rate function . . . . . . . . . . . . . . . . 19
4.2.1 Normal distribution . . . . . . . . . . . . . . . . . . . 19
4.2.2 Exponential distribution . . . . . . . . . . . . . . . . 21
4.3 Estimating Regression function . . . . . . . . . . . . . . . . 24
5 Extrapolation Method 27
5.1 Estimation of probability density function . . . . . . . . . . . 27
5.2 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . . . 31
5.3 Computational Results . . . . . . . . . . . . . . . . . . . . . . 33
5.3.1 Normal Distribution . . . . . . . . . . . . . . . . . . . 33
5.3.2 Exponential Distribution . . . . . . . . . . . . . . . . . 35
6 Conclusion 37
7 Appendix 38
7.1 The plots on Normal distributions . . . . . . . . . . . . . . . . 38
7.2 The plots on Exponential distributions . . . . . . . . . . . . . 41
7.3 The plots on hazard rate from Normal distributions . . . . . . 44
7.4 The plots on hazard rate from Exponential distributions . . . 47
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