Karim Benhenni

Estimation of the regression operator from functional fixed-design with correlated errors

We consider the estimation of the regression operator r in the functional model: Y=r(x)+@e, where the explanatory variable x is of functional fixed-design type, the response Y is a real random variable and the error process @e is a second order stationary process. We construct the kernel type estimate of r from functional data curves and correlated errors. Then we study their performances in terms of the mean square convergence and the convergence in probability. In particular, we consider the cases of short and long range error processes. When the errors are negatively correlated or come from a short memory process, the asymptotic normality of this estimate is derived. Finally, some simulation studies are conducted for a fractional autoregressive integrated moving average and for an Ornstein-Uhlenbeck error processes.

Nonparametric estimation of the regression function from quantized observations

The problem of estimating the regression function for a fixed design model is considered when only quantized and correlated data are available. Moreover, repeated observations are required in order for the constructed estimator to be consistent. The asymptotic performance in terms of the mean squared error for the regression function estimator constructed from quantized observations is derived. The generated optimal bandwidth depends on the regularity of the process, the number of replications, and the number of levels of quantization. The behavior and the comparison of the performances between quantized and plain estimators are investigated through some examples.

Nonparametric Estimation of Average Growth Curve with General Nonstationary Error Process

The nonparametric estimation of the growth curve has been extensively studied in both stationary and some nonstationary particular situations. In this work, we consider the statistical problem of estimating the average growth curve for a fixed design model with nonstationary error process. The nonstationarity considered here is of a general form, and this article may be considered as an extension of previous results. The optimal bandwidth is shown to depend on the singularity of the autocovariance function of the error process along the diagonal. A Monte Carlo study is conducted in order to assess the influence of the number of subjects and the number of observations per subject on the estimation.

Bispectrum estimation for a continuous-time stationary process from a random sampling

We propose an asymptotically unbiased and consistent estimate of the bispectrum of a stationary continuous-time process X = {X(t)}t∈ℝ. The estimate is constructed from observations obtained by a random sampling of the time by {X(τk)}k∈ℤ, where {τk}k∈ℤ is a sequence of real random variables, generated from a Poisson counting process. Moreover, we establish the asymptotic normality of the constructed estimate.