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The oscillatory behaviour of moving averages

Published online by Cambridge University Press:  24 October 2008

P. A. P. Moran
P. A. P. Moran
Affiliation:
Institute of StatisticsOxford
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Extract

The theory of graduation discusses methods of obtaining a smoothed series of values of a function from a given empirical set of values. This is usually done by replacing each observation by a weighted average of it and neighbouring observations. Thus if {xi} (i = 0, ± 1, …) is a sequence of values, we can replace them by the series

where the A's are constants which are chosen in some suitable manner. If we regard the xi as the sum of a functional part fi and an error term εi we may attempt to choose the A's in such a way that the functional part is reproduced as well as possible, e.g. that

for some desired type of function. If this is so and the error terms εi are independently distributed with zero mean and finite standard deviation σ, we find that

where the ηi are a series of random variables which are no longer independent of each other but which have zero mean and a standard deviation σ1 such that

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 46 , Issue 2 , April 1950 , pp. 272 - 280
Copyright
Copyright © Cambridge Philosophical Society 1950

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References

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