## The Annals of Statistics

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

#### 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

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