## The Annals of Statistics

- Ann. Statist.
- Volume 2, Number 2 (1974), 367-373.

### Comparison of Linear Normal Experiments

Ole Havard Hansen and Erik N. Torgersen

#### Abstract

Consider independent and normally distributed random variables $X_1,\cdots, X_n$ such that $0 < \operatorname{Var} X_i = \sigma^2; i = 1,\cdots, n$ and $E(X_1,\cdots, X_n)' = A'\beta$ where $A'$ is a known $n \times k$ matrix and $\beta = (\beta_1,\cdots, \beta_k)'$ is an unknown column matrix. (The prime denotes transposition.) The cases of known and totally unknown $\sigma^2$ are considered simultaneously. Denote the experiment obtained by observing $X_1,\cdots, X_n$ by $\mathscr{E}_A$. Let $A$ and $B$ be matrices of, respectively, dimensions $n_A \times k$ and $n_B \times k$. Then, if $\sigma^2$ is known, (if $\sigma^2$ is unknown) $\mathscr{E}_A$ is more informative than $\mathscr{E}_B$ if and only if $AA' - BB'$ is nonnegative definite (and $n_A \geqq n_B + \operatorname{rank} (AA' - BB'))$.

#### Article information

**Source**

Ann. Statist., Volume 2, Number 2 (1974), 367-373.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176342672

**Digital Object Identifier**

doi:10.1214/aos/1176342672

**Mathematical Reviews number (MathSciNet)**

MR370847

**Zentralblatt MATH identifier**

0289.62011

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62B15: Theory of statistical experiments

Secondary: 62K99: None of the above, but in this section

**Keywords**

Informational inequality invariant kernels normal models

#### Citation

Hansen, Ole Havard; Torgersen, Erik N. Comparison of Linear Normal Experiments. Ann. Statist. 2 (1974), no. 2, 367--373. doi:10.1214/aos/1176342672. https://projecteuclid.org/euclid.aos/1176342672