## The Annals of Statistics

- Ann. Statist.
- Volume 16, Number 1 (1988), 218-235.

### Asymptotic Theory of a Test for the Constancy of Regression Coefficients Against the Random Walk Alternative

Seiji Nabeya and Katsuto Tanaka

#### Abstract

The LBI (locally best invariant) test is suggested under normality for the constancy of regression coefficients against the alternative hypothesis that one component of the coefficients follows a random walk process. We discuss the limiting null behavior of the test statistic without assuming normality under two situations, where the initial value of the random walk process is known or unknown. The limiting distribution is that of a quadratic functional of Brownian motion and the characteristic function is obtained from the Fredholm determinant associated with a certain integral equation. The limiting distribution is then computed by numerical inversion of the characteristic function.

#### Article information

**Source**

Ann. Statist., Volume 16, Number 1 (1988), 218-235.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176350701

**Digital Object Identifier**

doi:10.1214/aos/1176350701

**Mathematical Reviews number (MathSciNet)**

MR924867

**Zentralblatt MATH identifier**

0662.62098

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]

Secondary: 60J15 62F03: Hypothesis testing 62F05: Asymptotic properties of tests

**Keywords**

Asymptotic distribution Bessel function Brownian motion Fredholm determinant integral equation invariance principle LBI test limiting distribution random walk regression model

#### Citation

Nabeya, Seiji; Tanaka, Katsuto. Asymptotic Theory of a Test for the Constancy of Regression Coefficients Against the Random Walk Alternative. Ann. Statist. 16 (1988), no. 1, 218--235. doi:10.1214/aos/1176350701. https://projecteuclid.org/euclid.aos/1176350701

#### See also

- Acknowledgment of Prior Result: Seiji Nabeya, Katsuto Tanaka. Acknowledgment of Priority: Asymptotic Theory of a Test for the Constancy of Regression Coefficients Against the Random Walk Alternative. Ann. Statist., Volume 22, Number 1 (1994), 563--563.Project Euclid: euclid.aos/1176325384