Optimal bandwidth selection for recursive Gumbel kernel density estimators

Yousri Slaoui

doi:10.1515/demo-2019-0020

Abstract

Abstract In this paper, we propose a data driven bandwidth selection of the recursive Gumbel kernel estimators of a probability density function based on a stochastic approximation algorithm. The choice of the bandwidth selection approaches is investigated by a second generation plug-in method. Convergence properties of the proposed recursive Gumbel kernel estimators are established. The uniform strong consistency of the proposed recursive Gumbel kernel estimators is derived. The new recursive Gumbel kernel estimators are compared to the non-recursive Gumbel kernel estimator and the performance of the two estimators are illustrated via simulations as well as a real application.

Highlights

In probability theory and statistics, the Gumbel distribution introduced by [11, 12] is one of the most used parametric model in extreme value theory
In this paper, we propose a data driven bandwidth selection of the recursive Gumbel kernel estimators of a probability density function based on a stochastic approximation algorithm
The new recursive Gumbel kernel estimators are compared to the non-recursive Gumbel kernel estimator and the performance of the two estimators are illustrated via simulations as well as a real application

Summary

Introduction

In probability theory and statistics, the Gumbel distribution (called Generalized Extreme Value distribution Type-I) introduced by [11, 12] is one of the most used parametric model in extreme value theory. We explicit the choice of the optimal bandwidth (hn) through a plug-in method, based on the minimization the Mean Weighted Integrated Squared Error MWISE of our proposed recursive Gumbel kernel density estimators fnGum de ned in (1). We developed a plug-in bandwidth selector based on the minimization of the MWISE of the proposed recursive Gumbel kernel density estimator by using the function f (x) as a weight function. 2. Table 3 provides the estimation of the two unknown quantities I and I before given the optimal bandwidth of the four considered methods respectively, we use the MWISE to compare the four considered approaches, we infer that the Gumbel kernel estimators outperformed the Gaussian kernel estimators with a particular preference for using the recursive version.

Conclusion

A Proofs

Proof of Proposition 1

Proof of Theorem 1