Statistical problems related to irrational rotations

摘要

Let ξi := 「iα + β」-「(i - 1)α + β」 (i = 1, 2, . . . ,m) be random variables as functions of β in the probability space [0, 1) with the Lebesgue measure, where α ∈ [0, 1] is considered to be an unknown parameter which we want to estimate from the observation ξ:= ξ1, ξ2 . . . ξm. Let an observation ξ be given, which is a finite Sturmian sequence. We determine the likelihood function Pα(ξ ) as a function of parameter α, and obtain the maximum likelihood estimator ?α(ξ) 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+· · ·+ξm)/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.