Asymptotic Properties of the Semi-Parametric Estimators of the Conditional Density for Functional Data in the Single Index Model with Missing Data at Random

Abstract

The main objective of this work is to estimate, semi-parametrically, the mode of a conditional density when the response is a real valued random variable subject to censored phenomenon and the predictor takes values in a semi-metric space. We assume that the explanatory and the response variables are linked through a single-index structure. First, we introduce a type of kernel estimator of the conditional density function when the data are supposed to be selected from an underlying stationary and ergodic process with missing at random (MAR). Under some general conditions, both the uniform almost-complete consistencies with convergence rates of the model are established. Further, the asymptotic normality of the considered model is given. As an application, the asymptotic (1−α) confidence interval of the conditional density function and the conditional mode are also presented for 0 < α < 1.

Keywords

• ergodic processes
• functional data analysis
• functional single-index process
• missing at random
• small ball probability