Some Improvement on Convergence Rates of Kernel Density Estimator

Xiaoran Xie, Jingjing Wu

Department of Mathematics and Statistics, University of Calgary, Calgary, Canada.

DOI: 10.4236/am.2014.511161                

Abstract

In this paper two kernel density estimators are introduced and investigated. In order to reduce bias, we intuitively subtract an estimated bias term from ordinary kernel density estimator. The second proposed density estimator is a geometric extrapolation of the first bias reduced estimator. Theoretical properties such as bias, variance and mean squared error are investigated for both estimators. To observe their finite sample performance, a Monte Carlo simulation study based on small to moderately large samples is presented.

Keywords

Kernel Density Estimation, Geometric Extrapolation, Bias Reduction, Mean Squared Error, Convergence Rate

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Farrell, R.H. (1972) On the Best Obtainable Asymptotic Rates of Convergence in Estimation of a Density Function at a Point. The Annals of Mathematics and Statistics, 43, 170-180. http://dx.doi.org/10.1214/aoms/1177692711
[2]	Terrell, G.R. and Scott, D.W. (1980) On Improving Convergence Rates for Nonnegative Kernel Density Estimators. The Annals of Statistics, 8, 1160-1163. http://dx.doi.org/10.1214/aos/1176345153
[3]	Hall, P. and Marron, J.S. (1988) Choice of Kernel Order in Density Estimation. The Annals of Statistics, 16, 161-173. http://dx.doi.org/10.1214/aos/1176350697
[4]	Abramson, I.S. (1982) On Bandwidth Variation in Kernel Estimates—A Square Root Law. The Annals of Statistics, 10, 1217-1223. http://dx.doi.org/10.1214/aos/1176345986
[5]	Samiuddin, M. and El-Sayyad, G.M. (1990) On Nonparametric Kernel Density Estimates. Biometrica, 77, 865-874. http://dx.doi.org/10.1093/biomet/77.4.865
[6]	El-Sayyad, G.M., Samiuddin, M. and Abdel-Ghaly, A.A. (1992) A New Kernel Density Estimate. Journal of Nonparametric Statistics, 3, 1-11. http://dx.doi.org/10.1080/10485259308832568
[7]	Cheng, M.Y., Choi, E., Fan, J. and Hall, P. (2000) Skewing Methods for Two-Parameter Locally Parametric Density Estimation. Bernoulli, 6, 169-182. http://dx.doi.org/10.2307/3318637
[8]	Marron, J.S. and Ruppert, D. (1992) Transformations to Reduce Boundary Bias in Kernel Density Estimation. Journal of the Royal Statistical Society: Series B, 4, 653-671.
[9]	Ruppert, D. and Cline, D.B.H. (1994) Bias Reduction in Kernel Density Estimation by Smoothed Empirical Transformations. The Annals of Statistics, 22, 185-210. http://dx.doi.org/10.1214/aos/1176325365
[10]	Jones, M.C., Linton, O. and Nielsen, J.P. (1995) A Simple Bias Reduction Method for Density Estimation. Biometrica, 82, 327-338. http://dx.doi.org/10.1093/biomet/82.2.327
[11]	Kim, C., Kim, W. and Park, B.U. (2003) Skewing and Generalized Jackknifing in Kernel Density Estimation. Communications in Statistics: Theory and Methods, 32, 2153-2162. http://dx.doi.org/10.1081/sta-120024473
[12]	Mynbaev, K. and Martins-Filho, C. (2010) Bias Reduction in Kernel Density Estimation via Lipschitz Condition. Journal of Nonparametric Statistics, 22, 219-235. http://dx.doi.org/10.1080/10485250903266058
[13]	Srihera, R. and Stute, W. (2011) Kernel Adjusted Density Estimation. Statistics and Probability Letters, 81, 571-579. http://dx.doi.org/10.1016/j.spl.2011.01.013
[14]	Cheng, M.Y., Peng, L. and Wu, S.H. (2007) Reducing Variance in Univariate Smoothing. The Annals of Statistics, 35, 522-542. http://dx.doi.org/10.1214/009053606000001398
[15]	Kim, J. and Kim, C. (2013) Reducing the Mean Squared Error in Kernel Density Estimation. Journal of the Korean Statistical Society, 42, 387-397. http://dx.doi.org/10.1016/j.jkss.2012.12.003
[16]	Rosenblatt, M. (1956) Remarks on Some Nonparametric Estimates of a Density Function. The Annals of Mathematical Statistics, 27, 832-837. http://dx.doi.org/10.1214/aoms/1177728190
[17]	Parzen, E. (1962) On Estimation of a Probability Density Function and the Mode. The Annals of Mathematical Statistics, 33, 1065-1076. http://dx.doi.org/10.1214/aoms/1177704472
[18]	Silverman, B.W. (1986) Density Estimation for Statistics and Data Analysis. Chapman & Hall, London. http://dx.doi.org/10.1007/978-1-4899-3324-9
[19]	Wand, M.P. and Jones, M.C. (1995) Kernel Smoothing. Chapman & Hall, London. http://dx.doi.org/10.1007/978-1-4899-4493-1