Communications for Statistical Applications and Methods

  • 제31권4호
  • /
  • Pages.427-440
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  • 2024
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  • 2287-7843(pISSN)
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  • 2383-4757(eISSN)
한국통계학회 (The Korean Statistical Society)
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On the Bayes risk of a sequential design for estimating a mean difference

  • Sangbeak Ye (Methods Center, University of Tubingen) ;
  • Kamel Rekab (Department of Mathematics and Statistics, University of Missouri Kansas City)
  • 투고 : 2024.01.01
  • 심사 : 2024.03.18
  • 발행 : 2024.07.31
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초록

The problem addressed is that of sequentially estimating the difference between the means of two populations with respect to the squared error loss, where each population distribution is a member of the one-parameter exponential family. A Bayesian approach is adopted in which the population means are estimated by the posterior means at each stage of the sampling process and the prior distributions are not specified but have twice continuously differentiable density functions. The main result determines an asymptotic second-order lower bound, as t → ∞, for the Bayes risk of a sequential procedure that takes M observations from the first population and t - M from the second population, where M is determined according to a sequential design, and t denotes the total number of observations sampled from both populations.

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참고문헌

  1. Ash R and Doleans-Dade C (2000). Probability and Measure Theory, Academic Press, New York.
  2. Bickel PJ and Yahav JA (1969). Some contributions to the asymptotic theory of Bayes solutions, Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete, 11, 257-276. https://doi.org/10.1007/BF00531650
  3. Benkamra Z, Terbeche M, and Tlemcani M (2015). Nearly second-order three-stage design for estimating a product of several Bernoulli proportions, Journal of Statistical Planning and Inference, 167, 90-101. https://doi.org/10.1016/j.jspi.2015.05.006
  4. Diaconis P and Ylvisaker D (1979). Conjugate priors for exponential families, The Annals of Statistics, 7, 269-281.
  5. Lehman EL (1959). Testing Statistical Hypothesis, John Wiley & Sons, Inc, New York.
  6. Martinsek AT (1983). Second order approximation to the risk of a sequential procedure, The Annals of Statistics, 11, 827-836.
  7. Rekab K (1989). Asymptotic efficiency in sequential designs for estimation, Sequential Analysis, 8, 269-280. https://doi.org/10.1080/07474948908836181
  8. Rekab K (1990). Asymptotic efficiency in sequential designs for estimation in the exponential family case, Sequential Analysis, 9, 305-315. https://doi.org/10.1080/07474949008836212
  9. Rekab K (1992). A nearly optimal two stage procedure, Communications in Statistics-Theory and Methods, 21, 197-201. https://doi.org/10.1080/03610929208830772
  10. Rekab K and Tahir M (2004). An asymptotic second-order lower bound for the Bayes risk of a sequential procedure, Sequential Analysis, 25, 451-464. https://doi.org/10.1081/SQA-200027055
  11. Shapiro CP (1985). Allocation schemes for estimating the product of positive parameters, Journal of the American Statistical Association, 80, 449-454. https://doi.org/10.1080/01621459.1985.10478139
  12. Song X and Rekab K (2017). Asymptotic efficiency in sequential designs for estimating product of k means in the exponential family case, Journal of Applied Mathematics, 4, 50-69.
  13. Terbeche M (2000). Sequential designs for estimation, Florida Institute of Technology.
  14. Woodroofe M (1985). Asymptotic local minimaxity in sequential point estimation, Annals of Statistics, 13, 676-688.
  15. Woodroofe M and Hardwick J (1990). Sequential allocation for estimation problem with ethical costs, Annals of Statistics, 18, 1358-1377.