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February 2022 On bandwidth selection problems in nonparametric trend estimation under martingale difference errors
Karim Benhenni, Didier A. Girard, Sana Louhichi
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Bernoulli 28(1): 395-423 (February 2022). DOI: 10.3150/21-BEJ1347


In this paper, we are interested in the problem of smoothing parameter selection in nonparametric curve estimation under dependent errors. We focus on kernel estimation and the case when the errors form a general stationary sequence of martingale difference random variables where neither linearity assumption nor “all moments are finite” are required. We compare the behaviors of the smoothing bandwidths obtained by minimizing either the unknown average squared error, the theoretical mean average squared error, a Mallows-type criterion adapted to the dependent case and the family of criteria known as generalized cross validation (GCV) extensions of the Mallows’ criterion. We prove that these three minimizers and those based on the GCV family are first-order equivalent in probability. We give also a normal asymptotic behavior of the gap between the minimizer of the average squared error and that of the Mallows-type criterion. This is extended to the GCV family. Finally, we apply our theoretical results to a specific case of martingale difference sequence, namely the Auto-Regressive Conditional Heteroscedastic (ARCH(1)) process. A Monte-Carlo simulation study, for this regression model with ARCH(1) process, is conducted.


We thank the Editor Marcus Reiss, the Associate Editor and two reviewers for their accurate and constructive comments. This led to a significant improvement of the original manuscript, in particular in the extent of the results, both theoretical and experimental, we present here. This paper was developed in the framework of Grenoble Alpes Data Institute (ANR-15-IDEX-02).


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Karim Benhenni. Didier A. Girard. Sana Louhichi. "On bandwidth selection problems in nonparametric trend estimation under martingale difference errors." Bernoulli 28 (1) 395 - 423, February 2022.


Received: 1 March 2020; Revised: 1 October 2020; Published: February 2022
First available in Project Euclid: 10 November 2021

Digital Object Identifier: 10.3150/21-BEJ1347

Keywords: ARCH(1) , average squared error , Cross validation , generalized cross validation , kernel nonparametric models , Mallows criterion , Martingale difference sequences , mean average squared error , Nonparametric trend estimation , smoothing parameter selection

Rights: Copyright © 2022 ISI/BS


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Vol.28 • No. 1 • February 2022
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