exit

Mathematics   > Home   > Advances in Pure and Applied Mathematics   > Special Issue: AUS-ICMS 2020   > Article

On the excess of average squared error for data-driven bandwidths in nonparametric trend estimation

Sur l’excès de la moyenne quadratique des erreurs associées à des fenêtres adaptatives dans l’estimation non-paramétrique de la tendance


Karim Benhenni
Université Grenoble Alpes
France

Didier A. Girard
CNRS

Sana Louhichi
Université Grenoble Alpes
France



Published on 28 July 2021   DOI : 10.21494/ISTE.OP.2021.0696

Abstract

Résumé

Keywords

Mots-clés

We consider the problem of the optimal selection of the smoothing parameter $$${h}$$$ in kernel estimation of a trend in nonparametric regression models with strictly stationary errors. We suppose that the errors are stochastic volatility sequences. Three types of volatility sequences are studied: the log-normal volatility, the Gamma volatility and the log-linear volatility with Bernoulli innovations. We take the weighted average squared error (ASE) as the global measure of performance of the trend estimation using $$${h}$$$ and we study two classical criteria for selecting $$${h}$$$ from the data, namely the adjusted generalized cross validation and Mallows-type criteria. We establish the asymptotic distribution of the gap between the ASE evaluated at one of these selectors and the smallest possible ASE. A Monte-Carlo simulation for a log-normal stochastic volatility model illustrates that this asymptotic approximation can be accurate even for small sample sizes.

We consider the problem of the optimal selection of the smoothing parameter $$${h}$$$ in kernel estimation of a trend in nonparametric regression models with strictly stationary errors. We suppose that the errors are stochastic volatility sequences. Three types of volatility sequences are studied : the log-normal volatility, the Gamma volatility and the log-linear volatility with Bernoulli innovations. We take the weighted average squared error (ASE) as the global measure of performance of the trend estimation using $$${h}$$$ and we study two classical criteria for selecting $$${h}$$$ from the data, namely the adjusted generalized cross validation and Mallows-type criteria. We establish the asymptotic distribution of the gap between the ASE evaluated at one of these selectors and the smallest possible ASE. A Monte-Carlo simulation for a log-normal stochastic volatility model illustrates that this asymptotic approximation can be accurate even for small sample sizes.

Nonparametric trend estimation Kernel nonparametric models Smoothing parameter selection Average squared error Excess of average squared error Mean average squared error Mallows criterion Cross validation Generalized cross validation SV models

Nonparametric trend estimation Kernel nonparametric models Smoothing parameter selection Average squared error Excess of average squared error Mean average squared error Mallows criterion Cross validation Generalized cross validation SV models