# 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

### 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.