Number of Sign Changes: Segment of AR(1)

TL;DR

This paper investigates the variance of the number of sign changes in a stationary AR(1) process, comparing theoretical predictions with model simulations for small sample sizes.

Contribution

It provides a focused analysis of the variance of sign changes in AR(1) processes, extending previous work on expected number and higher moments.

Findings

The theoretical variance aligns well with model simulations for small n.

The study confirms the applicability of existing formulas for variance in AR(1) processes.

Results suggest potential for improved understanding of sign change variability in time series.

Abstract

Let $X_{t}$ denote a stationary first-order autoregressive process. Consider $n$ contiguous observations (in time $t$) of the series (e.g., $X_{1}, ..., X_{n}$). Let its mean be zero and its lag-one serial correlation be $\rho$, which satisfies $|\rho| < 1$. Rice (1945) proved that $(n-1) \arccos(\rho)/\pi$ is the expected number of sign changes. A corresponding formula for higher-order moments was proposed by Nyberg, Lizana & Ambj\"ornsson (2018), based on an independent interval approximation. We focus on the variance only, for small $n$, and see a promising fit between theory and model.