The best truncation point for the estimated spectral density function of a stationary time series

Abstract

In many applications of time series analysis, the scientist estimates the spectral density function of the process. One type of spectral density estimator is obtained by using the periodogram representation of the spectral density function. However, if the estimator is to be consistent, a weight function which satisfies certain conditions must be added to the periodogram estimator. If the weight function is a truncated function, the estimator of the spectral density function is labeled a truncated estimator. For this type of estimator, a truncation point can be chosen which yields a minimum mean square estimator among all truncated estimators.

In this study we are concerned with two problems; i) finding consistent estimators of the spectral density function, and ii) determining the best truncation point. Two different types of weight functions, both of which give consistent estimators of the spectral density function, are presented in this dissertation. Both of the weight functions have related truncation functions.

For the problem of determining the best truncation function one particular weight function is considered. Using this weight function, the asymptotic forms of the variance and the bias of the estimator are found. It is discovered that the best truncation function is dependent on one term of the bias of the estimator; this term is denoted by C(m(T)). There are three different forms of C(m(T)); each of these three forms can be used to obtain the best truncation function.

There are several different types of covariance functions for a time series. The asymptotic relative efficiency of the spectral density estimator is dependent on the covariance function of the process. Using the best truncation point for a given weight function the asymptotic relative efficiency is derived for two covariance functions.

The proposed method of finding the best truncation point is applied to other types of weight functions. Finally, the problem of estimating the best truncation function when the covariance function is unknown is discussed.