## The Annals of Statistics

### A Note on Two-Sided Confidence Intervals for Linear Functions of the Normal Mean and Variance

#### Abstract

The confidence sets for linear functions $\mu + \lambda\sigma^2$ of the mean $\mu$ and variance $\sigma^2$ of a normal distribution, defined in terms of the uniformly most powerful unbiased level $\alpha$ tests of hypotheses of form $H_0(\lambda, m): \mu + \lambda\sigma^2 = m$ against the two-sided alternative $H_1(\lambda, m): \mu + \lambda\sigma^2 \neq m$ for $-\infty < m < \infty$, for fixed $\alpha$ and $\lambda$, are shown to be intervals if the number of degrees of freedom for estimating $\sigma^2$ is $\geqq 2$.

#### Article information

Source
Ann. Statist., Volume 1, Number 5 (1973), 940-943.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176342514

Digital Object Identifier
doi:10.1214/aos/1176342514

Mathematical Reviews number (MathSciNet)
MR345294

Zentralblatt MATH identifier
0273.62021

JSTOR