## The Annals of Statistics

- Ann. Statist.
- Volume 2, Number 3 (1974), 520-527.

### Regression Designs in Autoregressive Stochastic Processes

Jaroslav Hajek and George Kimeldorf

#### Abstract

This paper extends some recent results on asymptotically optimal sequences of experimental designs for regression problems in stochastic processes. In the regression model $Y(t) = \beta f(t) + X(t), 0 \leqq t \leqq 1$, the constant $\beta$ is to be estimated based on observations of $Y(t)$ and its first $m - 1$ derivatives at each of a set $T_n$ of $n$ distinct points. The function $f$ is assumed known as is the covariance kernel of $X(t)$, a zero-mean $m$th order autoregressive process. Under certain conditions, we derive a sequence $\{T_n\}$ of experimental designs which are asymptotically optimal for estimating $\beta$.

#### Article information

**Source**

Ann. Statist., Volume 2, Number 3 (1974), 520-527.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342711

**Digital Object Identifier**

doi:10.1214/aos/1176342711

**Mathematical Reviews number (MathSciNet)**

MR356412

**Zentralblatt MATH identifier**

0282.62067

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62K05: Optimal designs

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

**Keywords**

Experimental design asymptotically optimal designs autoregressive stochastic processes

#### Citation

Hajek, Jaroslav; Kimeldorf, George. Regression Designs in Autoregressive Stochastic Processes. Ann. Statist. 2 (1974), no. 3, 520--527. doi:10.1214/aos/1176342711. https://projecteuclid.org/euclid.aos/1176342711