Asymptotic results for the regression function estimate on continuous time stationary and ergodic data

Abstract

This paper is devoted to the study of asymptotic properties of the regression function kernel estimate in the setting of continuous time stationary and ergodic data. More precisely, considering the Nadaraya–Watson type estimator, say m̂ T (x), of the l-indexed regression function m(x) =𝔼 (l(Y)|X = x) built upon continuous time stationary and ergodic data (X t , Y t )0≤t≤T , we establish its pointwise and uniform, over a dilative compact set, convergences with rates. Notice that the ergodic setting covers and completes various situations as compared to the mixing case and stands as more convenient to use in practice.

Trapezoidal rule and sampling designs for the nonparametric estimation of the regression function in models with correlated errors

“…where ϕ x,h (t) = 1 h K x−t h for a kernel K and a bandwidth h. For details on the Kernel estimation of the regression function based on continuous observations see Blanke and Bosq [9] and Didi and Louani [10].…”

Optimal asymptotic MSE of kernel regression estimate for continuous time processes with missing at random response

“…Several examples to satisfy these conditions are given in Laïb and Louani (2010) for discrete time functional data process. Some examples are also given to satisfy this condition in Didi and Louani (2014) for the case where the observations (X t , Y t ) are sampled from an ergodic continuous times process taking values in R d × R space.…”

“…, it is easy to prove that the sequence (f jδ,(j−1)δ (x)) j≥1 is stationary and ergodic (see, Didi and Louani (2014)). ( A2)-(iv) is an usual condition when dealing with functional data.…”

“…One may refer to the monograph by Bosq (1998) and the references theirin. Some of these results are extended by Didi and Louani (2014) and Bouzebda and Didi (2017) to the case where the underlying real-valued process is stationary and ergodic. Chaouch and Laïb (2019) have obtained an explicit upper bound of the asymptotic mean square error (AMSE) of kernel regression estimator when the data are sampled from a real-valued continuous time process with a missing at random response.…”

Some asymptotic properties of kernel regression estimators of the mode for stationary and ergodic continuous time processes

“…It is worth noticing that the ergodicity is implied by all mixing conditions, being weaker than all of them. Further motivations to consider ergodic data are discussed in Laib and Louani [ 48 , 49 ], Didi and Louani [ 27 ], Bouzebda et al [ 12 ], Bouzebda et al [ 8 ], Bouzebda and Didi [ 9 – 11 ] and Krebs [ 46 ], in some of these references the definitions of the ergodic property of continuous time processes are given. In the present work, we do not assume anything beyond ergodicity of the underlying process.…”