#
Preprint: Fuzzy and Randomized Confidence Intervals and P-values

University of Minnesota,
Twin Cities
School of Statistics

## Paper (revised 20 May 2005), to appear in *Statistical Science*

**Title:**
Fuzzy and Randomized Confidence Intervals and P-values

**Authors:**
Charles J. Geyer and Glen D. Meeden

**Affiliation:**
School of Statistics, University of Minnesota

**Abstract:**
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,
called fuzzy `P`-values or abstract randomized `P`-values,
that also inherits
the same exactness and optimality.

**Key words and phrases:**
Confidence interval, P-value,
hypothesis test, uniformly most powerful unbiased (UMP and UMPU),
fuzzy set theory, randomized test.

Complete text of the current version of the paper
as PostScript or as PDF.

The design document for the R package contains
some mathematical arguments that were cut from the
second version of the paper for reasons of length.
It is available as PDF
in the R package.

For purely historical reasons, earlier versions of the
paper are still available

They contain nothing of interest that is not also in either the
current version or the design document for the R package, except that
the second version contains two figures cut from the third.