Statistics – Methodology
Scientific paper
2008-02-04
Statistics
Methodology
15 A4 pages. Version 4 corrects (3.8). Version 3 expands Section 2. Version 2 corrected recurrence relation (2.5)
Scientific paper
We give the distribution of $M_n$, the maximum of a sequence of $n$ observations from a moving average of order 1. Solutions are first given in terms of repeated integrals and then for the case where the underlying independent random variables have an absolutely continuous density. When the correlation is positive, $$ P(M_n %\max^n_{i=1} X_i \leq x) =\ \sum_{j=1}^\infty \beta_{jx} \nu_{jx}^{n} \approx B_{x} \nu_{1x}^{n} $$ where %$\{X_i\}$ is a moving average of order 1 with positive correlation, and $\{\nu_{jx}\}$ are the eigenvalues (singular values) of a Fredholm kernel and $\nu_{1x}$ is the eigenvalue of maximum magnitude. A similar result is given when the correlation is negative. The result is analogous to large deviations expansions for estimates, since the maximum need not be standardized to have a limit. % there are more terms, and $$P(M_n
Nadarajah Saralees
Withers Christopher S.
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