Moments of Maximum: Segment of AR(1)

TL;DR

This paper investigates how the expected maximum and variance of the maximum of five consecutive observations in a stationary AR(1) process depend on the serial correlation coefficient, revealing optimal correlation values for these statistics.

Contribution

It provides a detailed analysis of the moments of the maximum in a short segment of an AR(1) process, clarifying how these moments vary with the autocorrelation parameter.

Findings

Expected maximum is maximized at a specific autocorrelation value.

Variance of the maximum increases with autocorrelation.

Results extend understanding of extremal behavior in AR(1) processes.

Abstract

Let $X_{t}$ denote a stationary first-order autoregressive process. Consider five contiguous observations (in time $t$) of the series (e.g., $X_{1}, ..., X_{5}$). Let $M$ denote the maximum of these. Let $\rho$ be the lag-one serial correlation, which satisfies $|\rho| < 1$. For what value of $\rho$ is $\mathbb{E}(M)$ maximized? How does $\mathbb{V}(M)$ behave for increasing $\rho$? Answers to these questions lie in Afonja (1972), suitably decoded.