Asymptotically Optimal Bandwidth Selection for Kernel Density Estimators from Randomly Right-Censored Samples.

Variable bandwidth kernel density estimators increase the window width at low densities and decrease it where data concentrate. This represents an improvement over the fixed bandwidth kernel density estimators. In this article, we explore the use of one implementation of a variable kernel estimator in conjunction with several rules and procedures for bandwidth selection applied to several real datasets. The considered examples permit us to state that when working with tens or a few hundreds of data observations, least-squares cross-validation bandwidth rarely produces useful estimates; with thousands of observations, this problem can be surpassed. Optimal bandwidth and biased cross-validation (BCV), in general, oversmooth multimodal densities. The Sheather–Jones plug-in rule pro-duced bandwidths that behave slightly better in this respect. The Silverman test is considered as a very sophisticated and safe procedure to estimate the number of modes in univariate distributions; however, similar results could be obtained with the Sheather–Jones rule, but at a much lower computational cost. As expected, the variable bandwidth kernel density estimates showed fewer modes than those chosen by the Silverman test, especially those distributions in which multimodality was caused by several noisy minor modes. More research on the subject is needed.

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Optimal Bandwidth Selection for Kernel Density Functionals Estimation

Journal of Probability and Statistics ◽

10.1155/2015/242683 ◽

2015 ◽

Vol 2015 ◽

pp. 1-21 ◽

Cited By ~ 31

Author(s):

Su Chen

Keyword(s):

Density Estimation ◽

Bandwidth Selection ◽

Kernel Density ◽

Least Square ◽

Density Functionals ◽

Selection Methods ◽

Optimal Bandwidth ◽

Location Parameters ◽

The Mean ◽

Selection For

The choice of bandwidth is crucial to the kernel density estimation (KDE) and kernel based regression. Various bandwidth selection methods for KDE and local least square regression have been developed in the past decade. It has been known that scale and location parameters are proportional to density functionals∫γ(x)f2(x)dxwith appropriate choice ofγ(x)and furthermore equality of scale and location tests can be transformed to comparisons of the density functionals among populations.∫γ(x)f2(x)dxcan be estimated nonparametrically via kernel density functionals estimation (KDFE). However, the optimal bandwidth selection for KDFE of∫γ(x)f2(x)dxhas not been examined. We propose a method to select the optimal bandwidth for the KDFE. The idea underlying this method is to search for the optimal bandwidth by minimizing the mean square error (MSE) of the KDFE. Two main practical bandwidth selection techniques for the KDFE of∫γ(x)f2(x)dxare provided: Normal scale bandwidth selection (namely, “Rule of Thumb”) and direct plug-in bandwidth selection. Simulation studies display that our proposed bandwidth selection methods are superior to existing density estimation bandwidth selection methods in estimating density functionals.

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