A BOUNDARY CROSSING PROBLEM WITH APPLICATION TO SEQUENTIAL ESTIMATION

Abstract

Let X ₁, X ₂ … 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(X ₁,…,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.

Keywords:

• Stopping time
• Sequential estimation
• Second order properties
• AMS (2000) Subject Classification
• Primary 60G40
• Secondary 62L12
• 60F05

ACKNOWLEDGMENT

The author is grateful to the Referee and the Associate Editor for their comments, which were evidently based on a very careful reading of the paper. This has lead to a much better presentation and have helped to correct many crucial errors.