## The Annals of Statistics

### Proper Action in Steps, with Application to Density Ratios of Maximal Invariants

Robert A. Wijsman

#### Abstract

Let $G$ be a locally compact group acting continuously on the left of a locally compact space $\mathscr{X}$. It is assumed that $G = HK$ where $H$ and $K$ are closed subgroups. It is shown that if $K$ acts properly on $\mathscr{X}$ and $H$ acts properly on $\mathscr{X}/K$, then $G$ acts properly on $\mathscr{X}$. Under a mild additional condition the converse is also true. Several examples are given to show how these results can help decide the properness of composite actions. Proper action can be used to justify the representation of the density ratio of a maximal invariant as a ratio of integrals over the group.

#### Article information

Source
Ann. Statist., Volume 13, Number 1 (1985), 395-402.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176346600

Digital Object Identifier
doi:10.1214/aos/1176346600

Mathematical Reviews number (MathSciNet)
MR773175

Zentralblatt MATH identifier
0567.62003

JSTOR
links.jstor.org

Subjects
Primary: 62H10: Distribution of statistics
Secondary: 57S99: None of the above, but in this section

#### Citation

Wijsman, Robert A. Proper Action in Steps, with Application to Density Ratios of Maximal Invariants. Ann. Statist. 13 (1985), no. 1, 395--402. doi:10.1214/aos/1176346600. https://projecteuclid.org/euclid.aos/1176346600

#### Corrections

• See Correction: Robert A. Wijsman. Correction: Proper Action in Steps, with Application to Density Ratios of Maximal Invariants. Ann. Statist., Volume 21, Number 4 (1993), 2168--2169.