## The Annals of Statistics

- Ann. Statist.
- Volume 5, Number 4 (1977), 709-721.

### Prediction Functions and Mean-Estimation Functions for a Time Series

Lawrence Peele and George Kimeldorf

#### Abstract

Let $T \subseteqq I$ be sets of real numbers. Let $\{Y(t): t \in I\}$ be a real time series whose mean is an unknown element of a known class of functions on $I$ and whose covariance kernel $k$ is assumed known. For each fixed $s \in I, Y(s)$ is predicted by a minimum mean square error unbiased linear predictor $\hat{Y}(s)$ based on $\{Y(t): t \in T\}$. If $\hat{y}(s)$ is the evaluation of $\hat{Y}(s)$ given a set of observations $\{Y(t) = g(t): t \in T\}$, then the function $\hat{y}$ is called a prediction function. Mean-estimation functions are defined similarly. For certain prediction and estimation problems, characterizations are derived for these functions in terms of the covariance structure of the process. Also, relationships between prediction functions and spline functions are obtained that extend earlier results of Kimeldorf and Wahba (Sankhya Ser. A 32 173-180).

#### Article information

**Source**

Ann. Statist., Volume 5, Number 4 (1977), 709-721.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176343894

**Digital Object Identifier**

doi:10.1214/aos/1176343894

**Mathematical Reviews number (MathSciNet)**

MR436515

**Zentralblatt MATH identifier**

0365.62086

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 41A15: Spline approximation

**Keywords**

Prediction functions mean-estimation functions spline functions time series

#### Citation

Peele, Lawrence; Kimeldorf, George. Prediction Functions and Mean-Estimation Functions for a Time Series. Ann. Statist. 5 (1977), no. 4, 709--721. doi:10.1214/aos/1176343894. https://projecteuclid.org/euclid.aos/1176343894