## The Annals of Statistics

- Ann. Statist.
- Volume 3, Number 5 (1975), 1109-1121.

### Asymptotically Efficient Estimators for a Constant Regression with Vector- Valued Stationary Residuals

#### Abstract

Estimation of linear functions of a vector parameter $\theta$ when an observed discrete- or continuous-time vector-valued stationary process has mean value $H\theta, H$ a known matrix, is considered. Large-sample comparisons of best linear unbiased estimators and estimators based on the sample mean are made. Limits and rates of convergence of the variances of these estimators are obtained. It is shown that under general conditions there are asymptotically efficient estimators based on the sample mean, their form determined by the spectrum at the origin. Conditions under which all least squares estimators are asymptotically efficient are also given.

#### Article information

**Source**

Ann. Statist., Volume 3, Number 5 (1975), 1109-1121.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343243

**Digital Object Identifier**

doi:10.1214/aos/1176343243

**Mathematical Reviews number (MathSciNet)**

MR375697

**Zentralblatt MATH identifier**

0318.62046

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 62J05: Linear regression

**Keywords**

Linear estimation regression stationary vector-valued processes asymptotic efficiency spectral representations Moore-Penrose pseudoinverse

#### Citation

Adenstedt, Rolf K. Asymptotically Efficient Estimators for a Constant Regression with Vector- Valued Stationary Residuals. Ann. Statist. 3 (1975), no. 5, 1109--1121. doi:10.1214/aos/1176343243. https://projecteuclid.org/euclid.aos/1176343243