Statistical Problems Related to Irrational Rotations

摘要

Let $xi_i := lfloor ialpha + betarfloor - lfloor (i - 1)alpha+betarfloorquad(i=1,2,ldots,m)$ 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.