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ON SEQUENTIAL FIXED-WIDTH CONFIDENCE INTERVALS

FOR THE MEAN AND SECOND-ORDER EXPANSIONS

OF THE ASSOCIATED COVERAGE PROBABILITIES

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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-*alpha*)+*Ad*^{2}+*o*(*d*^{2}) as *d* --> 0 where (1-*alpha*) 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-*alpha*)
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 *tau*^{*}. 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}(*tau*^{*}). 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.

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