AISM 53, 142-158

© 2001 ISM

## 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.

**Source**
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