A boundary crossing problem with application to sequential estimation

Abstract

Let X₁, X₂ … be independent and identically distributed random variables with density f(x) = αx^α-1, 0 < x < 1 where α is a fixed positive number. Let N_c = inf{j ≥m: M_j≤ (j/c^β)} where M_j = max(X₁,…,X_j) and m is a fixed positive integer. We study the properties of N_c as c → ∞. As an application, we consider the problem of estimating sequentially the range of the uniform distribution and study the second order properties of an appropriate estimate.