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Fuzzy and randomized confidence intervals and p-values. (English) Zbl 1130.62319

Summary: The optimal hypothesis tests for the binomial distribution and some other discrete distributions are uniformly most powerful (UMP) one-tailed and UMP unbiased (UMPU) two-tailed randomized tests. Conventional confidence intervals are not dual to randomized tests and perform badly on discrete data at small and moderate sample sizes. We introduce a new confidence interval notion, called fuzzy confidence intervals, that is dual to and inherits the exactness and optimality of UMP and UMPU tests. We also introduce a new P-value notion, called fuzzy P-values or abstract randomized P-values, that also inherits the same exactness and optimality.

MSC:

62F25 Parametric tolerance and confidence regions
62F03 Parametric hypothesis testing
62F99 Parametric inference

Software:

ump; R
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References:

[1] Agresti, A. and Coull, B. A. (1998). Approximate is better than “exact” for interval estimation of binomial proportions. Amer. Statist. 52 119–126. JSTOR: · Zbl 04546791
[2] Blyth, C. R. and Hutchinson, D. W (1960). Table of Neyman—shortest unbiased confidence intervals for the binomial parameter. Biometrika 47 381–391. JSTOR: · Zbl 0104.13001
[3] Brown, L. D., Cai, T. T. and DasGupta, A. (2001). Interval estimation for a binomial proportion (with discussion). Statist. Sci. 16 101–133. · Zbl 1059.62533
[4] Casella, G. (2001). Comment on “Interval estimation for a binomial proportion,” by Brown, Cai and DasGupta (2001). Statist. Sci. 16 120–122.
[5] Geyer, C. J. and Meeden, G. D. (2004). ump: An R package for UMP and UMPU tests. Available at www.stat.umn.edu/geyer/fuzz/.
[6] Klir, G. J., St. Clair, U. H. and Yuan, B. (1997). Fuzzy Set Theory: Foundations and Applications . Prentice Hall, Upper Saddle River, NJ. · Zbl 0907.04002
[7] Lehmann, E. L. (1959). Testing Statistical Hypotheses . Wiley, New York (2nd ed., Wiley, 1986; Springer, 1997). · Zbl 0089.14102
[8] R Development Core Team (2004). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Available at www.R-project.org.
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