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 (1) , Didier A. Girard (1) , Sana Louhichi (1)

1 IPS - Inférence Processus Stochastiques

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 size

Mots clés: Mallows criterion; Mean average squared error; Martingale difference sequences; Bandwidth selection; Kernel smoothing; Stochastic volatility processes; Kernel nonparametric models; Cross validation; Generalized cross validation; Average squared error; Nonparametric trend estimation

Domaines: Statistiques [math.ST]

Karim Benhenni, Didier A. Girard, Sana Louhichi. On the excess of average squared error for data-driven bandwidths in nonparametric trend estimation. Advances in Pure and Applied Mathematics, 2021, 12 (N° Spécial ), pp.15-35. ⟨10.21494/ISTE.OP.2021.0696⟩. ⟨hal-02889802v2⟩