Number of Sign Changes: Segment of AR(1)

09/05/2019

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by   Steven Finch, et al.

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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 ρ, which satisfies |ρ| < 1. Rice (1945) proved that (n-1) arccos(ρ)/π is the expected number of sign changes. A corresponding formula for higher-order moments was proposed by Nyberg, Lizana Ambjörnsson (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.