Local Linear Modelling of the Conditional Distribution Function for Functional Ergodic Data

Abstract

The focus of functional data analysis has been mostly on independent functional observations. It is therefore hoped that the present contribution will provide an informative account of a useful approach that merges the ideas of the ergodic theory and the functional data analysis by using the local linear approach. More precisely, we aim, in this paper, to estimate the conditional distribution function (CDF) of a scalar response variable given a random variable taking values in a semimetric space. Under the ergodicity assumption, we study the uniform almost complete convergence (with a rate), as well as the asymptotic normality of the constructed estimator. The relevance of the proposed estimator is verified through a simulation study.

Kolmogorov Entropy for Convergence Rate in Incomplete Functional Time Series: Application to Percentile and Cumulative Estimation in High Dimensional Data

“…|F j−1 ) = V JK (y | x) in probability. (27)For(26), we use Lemma 5 of[72] to conclude|IE(η n,j |F j−1 )| = nψ(λ K ) nIE(Γ 1 Ker 1 |IE(B j (J j − CDF(y | x))Γ j Ker j |F j−1 )| ≤ C nψ(λ K )(λ b 1 K + λ b 2 J ) |F j−1 )) 2 = O a.co. (nψ(λ K )(λ b 1 K + λ b 2 J ) 2 ).…”