SEQUENTIAL ESTIMATION FOR A FUNCTIONAL OF THE SPECTRAL DENSITY OF A GAUSSIAN STATIONARY PROCESS

TAKAYUKI SHIOHAMA AND MASANOBU TANIGUCHI

Department of Mathematical Science, Graduate School of Engineering Science, Osaka University,

Machikaneyama-cho 1-3, Toyonaka, Osaka 560-8531, Japan, e-mail: shiohama@sigmath.es.osaka-u.ac.jp; taniguti@ sigmath.es.osaka-u.ac.jp

(Received April 10, 2000; revised August 12, 2000)

Abstract. Integral functional of the spectral density of stationary process is an important index in time series analysis. In this paper we consider the problem of sequential point and fixed-width confidence interval estimation of an integral functional of the spectral density for Gaussian stationary process. The proposed sequential point estimator is based on the integral functional replaced by the periodogram in place of the spectral density. Then it is shown to be asymptotically risk efficient as the cost per observation tends to zero. Next we provide a sequential interval estimator, which is asymptotically efficient as the width of the interval tends to zero. Finally some numerical studies will be given.

Key words and phrases: Stationary process, spectral density, sequential point estimation, sequential interval estimation, periodogram, integral functional.