The Annals of Statistics

On the Information Contained in Additional Observations

Lucien Le Cam

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Abstract

Let $\{X_j; j = 1, 2, \cdots\}$ be independent identically distributed random variables whose individual distribution $p_\theta$ is indexed by a parameter $\theta$ in a set $\Theta$. For two integers $m < n$ the experiment $\mathscr{E}_n$ which consists in observing the first $n$ variables is more informative than $\mathscr{E}_m$. Two measures of the supplementary information are described. One is the deficiency $\delta (\mathscr{E}_m, \mathscr{E}_n)$ introduced by this author. Another is a number $\eta(\mathscr{E}_m, \mathscr{E}_n)$ called "insufficiency" and related to previous arguments of Wald (1943). Relations between $\delta$ and $\eta$ are described. One defines a dimensionality coefficient $D$ for $\Theta$ and obtains a bound of the type $\eta(\mathscr{E}_m, \mathscr{E}_n) \leqq \lbrack 2D(n - m)/n\rbrack^{\frac{1}{2}}.$ Examples show that $\delta(\mathscr{E}_m, \mathscr{E}_n)$ may stay bounded away from zero in infinite dimensional cases, even if $m \rightarrow \infty$ and $n = m + 1$.

Article information

Source
Ann. Statist., Volume 2, Number 4 (1974), 630-649.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342753

Digital Object Identifier
doi:10.1214/aos/1176342753

Mathematical Reviews number (MathSciNet)
MR436400

Zentralblatt MATH identifier
0286.62004

JSTOR
links.jstor.org

Keywords
6230 Experiments estimates information sufficiency

Citation

Cam, Lucien Le. On the Information Contained in Additional Observations. Ann. Statist. 2 (1974), no. 4, 630--649. doi:10.1214/aos/1176342753. https://projecteuclid.org/euclid.aos/1176342753


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