## The Annals of Statistics

### Deficiencies Between Linear Normal Experiments

Anders Rygh Swensen

#### Abstract

Let $X_1, \cdots, X_n$ be independent and normally distributed variables, 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 $k \times n$ matrix with known coefficients and $\beta = (\beta_1, \cdots, \beta_k)'$ is an unknown vector. $\sigma$ may be known or unknown. Denote the experiment obtained by observing $X_1, \cdots, X_n$ by $\mathscr{E}_A.$ Let $A$ and $B$ be matrices of dimension $n_A \times k$ and $n_B \times k.$ The deficiency $\delta(\mathscr{E}_A, \mathscr{E}_B)$ is computed when $\sigma$ is known and for some cases, including the case $BB' - AA'$ positive semidefinite and $AA'$ nonsingular, also when $\sigma$ is unknown. The technique used consists of reducing to testing a composite hypotheses and finding a least favorable distribution.

#### Article information

Source
Ann. Statist., Volume 8, Number 5 (1980), 1142-1155.

Dates
First available in Project Euclid: 12 April 2007

https://projecteuclid.org/euclid.aos/1176345151

Digital Object Identifier
doi:10.1214/aos/1176345151

Mathematical Reviews number (MathSciNet)
MR585712

Zentralblatt MATH identifier
0445.62007

JSTOR