On the convergence rate of fixed-width sequential confidence intervals

Abstract

Abstract Let X f1, X f2, ... be a sequence of i.i.d. random variables with mean µ and variance σ2∈ (0, ∞). Define the stopping times N(d)=min {n:n −1 Σ n i=1} (X i&#x2212;X n)2+n −1⩽nd 2/a 2}, d>0, where X n =n −1 Σ n i=1} Xi and (2π)−½ ∫ a −a exp (−u 2/2) du=α ∈(0,1). Chow and Robbins (1965) showed that the sequence In,d =[Xn −d, X n + d], n=1,2, ... is an asymptotic level -α fixed-width confidence sequence for the mean, i.e. limd→0 P(µ∈IN(d),d )=α. In this note we establish the convergence rate P(µ∈IN(d),d )=α + O(d½−δ) under the condition E|X1|3+ϰ+5/(28) < ∞ for some δ ∈ (0, ½) and ϰ−0. The main tool in the proof is a result of Landers and Rogge (1976) on the convergence rate of randomly selected partial sums.