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Time Series Prediction Functions Based on Imprecise Observations

Lawrence Peele and George Kimeldorf
The Annals of Statistics
Vol. 7, No. 4 (Jul., 1979), pp. 801-811
Stable URL: http://www.jstor.org/stable/2958927
Page Count: 11
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Time Series Prediction Functions Based on Imprecise Observations
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Abstract

Let $T \subseteq I$ be sets of real numbers. Let {Y(t): t∈ I} be a real time series whose covariance kernel is assumed known and positive definite. The mean is assumed either to be known or to be an unknown member of a known class of functions on I. For each fixed s∈ I, Y(s) is predicted by a minimum mean square error unbiased linear predictor Ŷ(s) based on {Y(t): t∈ T}. If ŷ(s) is the evaluation of Ŷ(s) given that the sample path for {Y(t): t∈ T} is an unknown element of a known collection of functions on T, then ŷ(s) is a prediction for Y(s) and the function ŷ is called a prediction function. Mean-estimation functions are defined similarly. For certain prediction problems based on imprecise observations, characterizations are obtained for these functions in terms of the covariance structure of the process. For a particular prediction problem ŷ is shown to be a spline function interpolating a convex set.

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