## Annals of Statistics

### Comparison of Linear Normal Experiments

#### 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

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