Statistical Problems Related to Irrational Rotations

Abstract

Let $\xi_i := \lfloor i\alpha + \beta\rfloor - \lfloor (i - 1)\alpha+\beta\rfloor\quad(i=1,2,\ldots,m)$ be random variables as functions of β in the probability space [0,1) with the Lebesgue measure, where $\alpha \in [0,1]$ is considered to be an unknown parameter which we want to estimate from the observation ξ :=ξ₁, ξ₂...ξ _m . Let an observation ξ be given, which is a finite Sturmian sequence. We determine the likelihood function P _α(ξ) as a function of parameter α, and obtain the maximum likelihood estimator $\hat{\alpha}(\xi)$ as the relative frequency of 1s in a minimal cycle of ξ, where a factor η of ξ is called a minimal cycle if ξ is a factor of η^∞ and η has the minimum length among them. We also obtain a minimum sufficient statistics. The sample mean (ξ₁ + ξ₂ + ... + ξ _m )/m which is an unbiased estimator of α is not admissible if m=6 or m ≥ 8 since it is not based on the minimum sufficient statistics.