Bandwidth Selection for Recursive Kernel Density Estimators Defined by Stochastic Approximation Method

Yousri Slaoui

doi:10.1155/2014/739640

Abstract

We propose an automatic selection of the bandwidth of the recursive kernel estimators of a probability density function defined by the stochastic approximation algorithm introduced by Mokkadem et al. (2009a). We showed that, using the selected bandwidth and the stepsize which minimize the MISE (mean integrated squared error) of the class of the recursive estimators defined in Mokkadem et al. (2009a), the recursive estimator will be better than the nonrecursive one for small sample setting in terms of estimation error and computational costs. We corroborated these theoretical results through simulation study.

Highlights

The problem of automatic choice of smoothing parameters has been widely studied
We propose an automatic selection of the bandwidth of the recursive kernel estimators of a probability density function defined by the stochastic approximation algorithm introduced by Mokkadem et al (2009a)
We showed that, using the selected bandwidth and the stepsize which minimize the MISE of the class of the recursive estimators defined in Mokkadem et al (2009a), the recursive estimator will be better than the nonrecursive one for small sample setting in terms of estimation error and computational costs

Summary

Introduction

The problem of automatic choice of smoothing parameters has been widely studied. There are many reasons to use an automatic choice of smoothing. We developed a specific second generation bandwidth selection method of the recursive kernel estimators of a probability density function defined by the stochastic approximation algorithm introduced by Mokkadem et al [9]. Throughout this paper, we Πn = ∏nj=1(1 − γj); it suppose follows that f0(x) = 0 and we from (2) that one can estimate f recursively at the point x by fn (x) It was shown in Mokkadem et al [9] that the bandwidth which minimizes the MISE of fn depends on the choice of the stepsize (γn); they show in particular that the sequence (γn) = (n−1) belongs to this set, under some conditions of regularity of f, and they show that the bandwidth (hn) must equal

Assumptions and Main Results

Simulation

Conclusion