ON SEQUENTIAL FIXED-WIDTH CONFIDENCE INTERVALS FOR THE MEAN AND SECOND-ORDER EXPANSIONS OF THE ASSOCIATED COVERAGE PROBABILITIES

NITIS MUKHOPADHYAY AND SUJAY DATTA

Department of Statistics, University of Connecticut, Storrs, CT 06269, U.S.A.

(Received August 15, 1994; revised February 13, 1995)

Abstract. In order to construct fixed-width (2d) confidence intervals for the mean of an unknown distribution function F, a new purely sequential sampling strategy is proposed first. The approach is quite different from the more traditional methodology of Chow and Robbins (1965, Ann. Math. Statist., 36, 457–462). However, for this new procedure, the coverage probability is shown (Theorem 2.1) to be at least (1 − α) + Ad^{2} + o(d^{2}) as d → 0 where (1 − α) is the preassigned level of confidence and A is an appropriate functional of F, under some regularity conditions on F. The rates of convergence of the coverage probability to (1 − α) obtained by Csenki (1980, Scand. Actuar. J., 107–111) and Mukhopadhyay (1981, Comm. Statist. Theory Methods, 10, 2231–2244) were merely O(d^{1/2−q}), with 0 < q < 1/2, under the Chow-Robbins stopping time τ^{∗}. It is to be noted that such considerable sharpening of the rate of convergence of the coverage probability is achieved even though the new stopping variable is O_{P} (τ^{∗}). An accelerated version of the stopping rule is also provided together with the analogous second-order characteristics. In the end, an example is given for the mean estimation problem of an exponential distribution.

Key words and phrases: Distribution-free, fixed-width confidence intervals, confidence level, second-order expansions, purely sequential, accelerated sequential, Markov inequality.