The reproducing kernel Hilbert space approach in nonparametric regression problems with correlated observations
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The reproducing kernel Hilbert space approach in nonparametric regression problems with correlated observations D. Benelmadani1 · K. Benhenni1 · S. Louhichi1 Received: 28 February 2019 / Revised: 31 July 2019 © The Institute of Statistical Mathematics, Tokyo 2019
Abstract In this paper, we investigate the problem of estimating the regression function in models with correlated observations. The data are obtained from several experimental units, each of them forms a time series. Using the properties of the reproducing kernel Hilbert spaces, we construct a new estimator based on the inverse of the autocovariance matrix of the observations. We give the asymptotic expressions of its bias and its variance. In addition, we give a theoretical comparison between this new estimator and the popular one proposed by Gasser and Müller, we show that the proposed estimator has an asymptotically smaller variance than the classical one. Finally, we conduct a simulation study to investigate the performance and the robustness of the proposed estimator and to compare it to the Gasser and Müller’s estimator in a finite sample set. Keywords Nonparametric regression · Correlated observations · Growth curve · Reproducing kernel Hilbert space · Projection estimator · Asymptotic normality
1 Introduction One of the situations that statisticians encounter in their studies is the estimation of a whole function based on partial observations of this function. For instance, in
Electronic supplementary material The online version of this article (https://doi.org/10.1007/s10463019-00733-3) contains supplementary material, which is available to authorized users.
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K. Benhenni [email protected] D. Benelmadani [email protected] S. Louhichi [email protected]
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Laboratoire Jean Kuntzmann (CNRS 5224), Université Grenoble Alpes, 700 Avenue Centrale, 38401 Saint-Martin-d’Hères, France
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pharmacokinetics one wishes to estimate the concentration time of some injected medicine in the organism, based on the observations of the concentration from blood tests over a period of time. In statistical terms, one wants to estimate a function, say g, relating two random variables: the explanatory variable X and the response variable Y , without any parametric restrictions on the function g. The statistical model often used is the following: Yi = g(X i ) + εi where (X i , Yi )1≤i≤n are n independent replicates of (X , Y ) and {εi , i = 1, . . . , n} are centered random variables (called errors). The most intensively treated model has been the one in which (εi )1≤i≤n are independent errors and (X i )1≤i≤n are fixed within some domain. We mention the works of Priestly and Chao (1972), Benedetti (1977) and Gasser and Müller (1979) among others. However, the independence of the observations is not always a realistic assumption. For instance, the growth curve models are usually used in the case of longitudinal data, where the same experimental unit is being observed on multiple p
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