Open Access
2023 Efficient density estimation in an AR(1) model
Anton Schick, Wolfgang Wefelmeyer
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Electron. J. Statist. 17(2): 2880-2911 (2023). DOI: 10.1214/23-EJS2166


This paper studies a class of plug-in estimators of the stationary density of an autoregressive model with autoregression parameter 0<ϱ<1. These use two types of estimator of the innovation density, a standard kernel estimator and a weighted kernel estimator with weights chosen to mimic the condition that the innovation density has mean zero. Bahadur expansions are obtained for this class of estimators in L1, the space of integrable functions. These stochastic expansions establish root-n consistency in the L1-norm. It is shown that the density estimators based on the weighted kernel estimators are asymptotically efficient if an asymptotically efficient estimator of the autoregression parameter is used. Here asymptotic efficiency is understood in the sense of the Hájek–Le Cam convolution theorem.


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Anton Schick. Wolfgang Wefelmeyer. "Efficient density estimation in an AR(1) model." Electron. J. Statist. 17 (2) 2880 - 2911, 2023.


Received: 1 March 2023; Published: 2023
First available in Project Euclid: 10 November 2023

Digital Object Identifier: 10.1214/23-EJS2166

Primary: 62G07 , 62G20
Secondary: 62M10

Keywords: Bahadur expansions in L1 , Hadamard differentiability , kernel density estimation

Vol.17 • No. 2 • 2023
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