BIAS‐CORRECTED NONPARAMETRIC SPECTRAL ESTIMATION

Abstract

Abstract. The theory of nonparametric spectral density estimation based on an observed stretch X1,…, XN from a stationary time series has been studied extensively in recent years. However, the most popular spectral estimators, such as the ones proposed by Bartlett, Daniell, Parzen, Priestley and Tukey, are plagued by the problem of bias, which effectively prohibits ✓N‐convergence of the estimator. This is true even in the case where the data are known to be m‐dependent, in which case ✓N‐consistent estimation is possible by a simple plug‐in method. In this report, an intuitive method for the reduction in the bias of a nonparametric spectral estimator is presented. In fact, applying the proposed methodology to Bartlett's estimator results in bias‐corrected estimators that are related to kernel estimators with lag‐windows of trapezoidal shape. The asymptotic performance (bias, variance, rate of convergence) of the proposed estimators is investigated

in particular, it is found that the trapezoidal lag‐window spectral estimator is ✓N‐consistent in the case of moving‐average processes, and ✓(N/log/N)‐consistent in the case of autoregressive moving‐average processes. The finite‐sample performance of the trapezoidal lag‐window estimator is also assessed by means of a numerical simulation. Copyright © 1995, Wiley Blackwell. All rights reserved