## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 30, Number 1 (2016), 127-144.

### Assigning probabilities to hypotheses in the context of a binomial distribution

Casper J. Albers, Otto J. W. F. Kardaun, and Willem Schaafsma

#### Abstract

Given is the outcome $s$ of $S\sim{\mathrm{B}}(n,p)$ ($n$ known, $p$ fully unknown) and two numbers $0<a\leq b<1$. Required are probabilities $\alpha_{<}(s)$, $\alpha_{a,b}(s)$, and $\alpha_{>}(s)$ of the hypotheses $\mathrm{H}_{<}$: $p<a$, $\mathrm{H}_{a,b}$: $a\leq p\leq b$, and $\mathrm{H}_{>}$: $p>b$, such that their sum is equal to 1. The degenerate case $a=b(=c)$ is of special interest. A method, optimal with respect to a class of functions, is derived under Neyman–Pearsonian restrictions, and applied to a case from medicine.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 30, Number 1 (2016), 127-144.

**Dates**

Received: January 2013

Accepted: September 2014

First available in Project Euclid: 19 January 2016

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1453211806

**Digital Object Identifier**

doi:10.1214/14-BJPS264

**Mathematical Reviews number (MathSciNet)**

MR3453518

**Zentralblatt MATH identifier**

1381.62046

**Keywords**

Hypothesis testing $p$-values $q$-values squared error loss weak unbiasedness

#### Citation

Albers, Casper J.; Kardaun, Otto J. W. F.; Schaafsma, Willem. Assigning probabilities to hypotheses in the context of a binomial distribution. Braz. J. Probab. Stat. 30 (2016), no. 1, 127--144. doi:10.1214/14-BJPS264. https://projecteuclid.org/euclid.bjps/1453211806