Fuzzy and randomized confidence intervals and p-values. 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.

References in zbMATH (referenced in 17 articles , 1 standard article )

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  1. Habiger, Joshua D.: Multiple test functions and adjusted $p$-values for test statistics with discrete distributions (2015)
  2. Thulin, Måns: On split sample and randomized confidence intervals for binomial proportions (2014)
  3. Bodenhofer, Ulrich; Krone, Martin; Klawonn, Frank: Testing noisy numerical data for monotonic association (2013)
  4. Doebler, Anna; Doebler, Philipp; Holling, Heinz: Optimal and most exact confidence intervals for person parameters in item response theory models (2013)
  5. Bickel, David R.: The strength of statistical evidence for composite hypotheses: inference to the best explanation (2012)
  6. Parchami, Abbas; Taheri, S.Mahmoud; Mashinchi, Mashaallah: Testing fuzzy hypotheses based on vague observations: a $p$-value approach (2012)
  7. Piterbarg, Leonid I.: Parameter estimation from small biased samples: fuzzy sets vs statistics (2011)
  8. Terán, Pedro: Centrality as a gradual notion: a new bridge between fuzzy sets and statistics (2011)
  9. Thompson, Elizabeth: MCMC in the analysis of genetic data on related individuals (2011)
  10. Parchami, Abbas; Taheri, S.Mahmoud; Mashinchi, Mashaallah: Fuzzy $p$-value in testing fuzzy hypotheses with crisp data (2010)
  11. Geyer, Charles J.: Likelihood inference in exponential families and directions of recession (2009)
  12. Kulinskaya, Elena; Lewin, Alex: On fuzzy familywise error rate and false discovery rate procedures for discrete distributions (2009)
  13. Pires, Ana M.; Amado, Conceição: Interval estimators for a binomial proportion: comparison of twenty methods (2008)
  14. Agresti, Alan; Gottard, Anna: Nonconservative exact small-sample inference for discrete data (2007)
  15. Krishnamoorthy, K.; Peng, Jie: Some properties of the exact and score methods for binomial proportion and sample size calculation (2007)
  16. Nguyen, Hung T.; Wu, Berlin: Random and fuzzy sets in coarse data analysis (2006)
  17. Geyer, Charles J.; Meeden, Glen D.: Fuzzy and randomized confidence intervals and p-values (2005)