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A Nonparametric Prewhitened Covariance Estimator[Note 1. Actually, their method is really more closely related to ...]
Author(s) -
XIAO ZHIJIE,
LINTON OLIVER
Publication year - 2002
Publication title -
journal of time series analysis
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.576
H-Index - 54
eISSN - 1467-9892
pISSN - 0143-9782
DOI - 10.1111/1467-9892.00263
Subject(s) - mathematics , estimator , mean squared error , bias of an estimator , minimum variance unbiased estimator , kernel density estimation , statistics , consistent estimator , autocorrelation , nonparametric statistics , minimum mean square error , kernel (algebra) , efficient estimator , combinatorics
This paper proposes a new nonparametric spectral density estimator for time series models with general autocorrelation. The conventional nonparametric estimator that uses a positive kernel has mean squared error no better than n −4/5 . We show that the best implementation of our estimator has mean squared error of order n −8/9 , provided there is sufficient smoothness present in the spectral density. This is, of course, achieved by bias reduction; however, unlike most other bias reduction methods, like the kernel method with higher‐order kernels, our procedure ensures a positive definite estimate. Our method is a generalization of the well‐known prewhitening method of spectral estimation; we argue that this can best be interpreted as multiplicative bias reduction. Higher‐order expansions for the proposed estimator are derived, providing an improved bandwidth choice that minimizes the mean squared error to the second order. A simulation study shows that the recommended prewhitened kernel estimator reduces bias and mean squared error in spectral density estimation.