THE CONVERGENCE RATE OF SEQUENTIAL FIXED‐WIDTH CONFIDENCE INTERVALS FOR REGULAR FUNCTIONALS

Summary Let X 1 X 2 ,.be i.i.d. random variables and let U n = (n r) ‐1 S̀ (n,r) h ( X i1 ,., X ir ,) be a U ‐statistic with EU n = v, v unknown. Assume that g( X 1 ) = E [h( X 1 ,., X r ) ‐ v | X 1 ]has a strictly positive variance s̀ 2 . Further, let a be such that φ(a) ‐ φ(‐a) =α for fixed α, 0 < α < 1, where φ is the standard normal d.f., and let S 2 n be the Jackknife estimator of n Var U n . Consider the stopping times N(d) = min { n: S 2 n : + n ‐1 ≤ 2 a ‐2 },d > 0, and a confidence interval for v of length 2 d ,of the form I n,d = [U n ,‐d, U n + d ]. We assume that Var U n is unknown, and hence, no fixed sample size method is available for finding a confidence interval for v of prescribed width 2d and prescribed coverage probability α Turning to a sequential procedure, let I N(d),d be a sequence of sequential confidence intervals for v. The asymptotic consistency of this procedure, i.e. lim d → 0 P(v ∈ I N(d),d ) =α follows from Sproule (1969). In this paper, the rate at which | P(v ∈ I N(d),d ) converges to α is investigated. We obtain that | P(v ∈ I N(d),d ) ‐ α| = 0 (d 1/2‐(1+k)/2(1+m) ), d → 0, where K = max {0,4 ‐ m}, under the condition that E|h(X 1 , X r )| m < ∞ m > 2. This improves and extends recent results of Ghosh & DasGupta (1980) and Mukhopadhyay (1981).