The distribution of the maximum of a first-order moving average: The discrete casex

Christopher S. Withers and Saralees Nadarajah

Communications in Statistics - Theory and Methods, 2016, vol. 45, issue 16, 4729-4744

Abstract: Extreme value theory for discrete observations is very much underdeveloped. Here, we derive exact expressions for the cumulative distribution function of Mn, the maximum of a sequence of n discrete observations from a moving average of order one. A solution appropriate for large n takes the formPrMn≤x=∑j=1Iβjxνjxn, \begin{eqnarray*} \displaystyle \Pr \left(M_n \le x\right)\ =\ \sum _{j=1}^I \beta _{jx} \ \nu _{jx}^{n}, \end{eqnarray*}where {νjx} are the eigenvalues of a certain matrix, and I depends on the number of possible values of the underlying random variables.

Date: 2016
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DOI: 10.1080/03610926.2014.927499

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