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On sequential fixed-width confidence intervals for the mean and second-order expansions of the associated coverage probabilities. (English) Zbl 0926.62069

Summary: 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 Y. S. Chow and H. Robbins [Ann. Math. Stat. 36, 457-462 (1965; Zbl 0142.15601)]. However, for this new procedure, the coverage probability is shown to be at least \((1-\alpha)+ Ad^2+o(d^2)\) as \(d\to 0\) 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 A. Csenki [Scand. Actuarial. J. 1980, 107-111 (1980; Zbl 0431.62054)] and N. Mukhopadhyay [Commun. Stat., Theory Methods A10, 2231-2244 (1981; Zbl 0468.62077)] 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 variables are \(O_P(\tau^*)\). 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.

MSC:

62L12 Sequential estimation
62G15 Nonparametric tolerance and confidence regions
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