On Hall and Patil Recursive Kernel Density Estimator for Strong Mixing Processes

Abstract

The purpose of this paper is to extend results concerning the properties of the recursive kernel density introduced by Hall and Patil (IEEE Trans Inf Theory 40:1504–1512, 1994) for independent sequences to the case of dependence sequences. In particular, we examine the asymptotic properties of the estimator in the case of strong mixing sequences under the condition of ergodicity and geometric strong mixing of the stochastic process. We are interested in the pointwise consistency and the asymptotic distribution of the recursive estimator. The results show that the estimator is strongly consistent and asymptotically normal. Furthermore, we determine the asymptotic expression of the variance of the estimator that we compare to that of the nonrecursive Rosenblatt estimator. It turns out that, for a judicious choice of the bandwidth, the asymptotic variance of the Hall and Patil estimator is smaller than that of the nonrecursive Rosenblatt estimator. In addition, we establish asymptotic mean squared error of the estimator.

Keywords

Recursive kernel density estimator / Geometric ergodicity / Strong mixing processes / Consistence / Quadratic mean convergence / Asymptotic normality / 60G10 / 62F10 / 62G07 / 62G20 / 62M86

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School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature