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July, 1977 Prediction Functions and Mean-Estimation Functions for a Time Series
Lawrence Peele, George Kimeldorf
Ann. Statist. 5(4): 709-721 (July, 1977). DOI: 10.1214/aos/1176343894


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).


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Lawrence Peele. George Kimeldorf. "Prediction Functions and Mean-Estimation Functions for a Time Series." Ann. Statist. 5 (4) 709 - 721, July, 1977.


Published: July, 1977
First available in Project Euclid: 12 April 2007

zbMATH: 0365.62086
MathSciNet: MR436515
Digital Object Identifier: 10.1214/aos/1176343894

Primary: 62M10
Secondary: 41A15

Keywords: mean-estimation functions , Prediction functions , Spline functions , time series

Rights: Copyright © 1977 Institute of Mathematical Statistics


Vol.5 • No. 4 • July, 1977
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