Similar statistic

A statistic having a fixed probability distribution under some compound hypothesis.

Let the statistic $ T $ map the sample space $ ( \mathfrak X , {\mathcal B} _ {\mathfrak X} , {\mathsf P} _ \theta ) $, $ \theta \in \Theta $, into a measurable space $ ( \mathfrak A, {\mathcal B} _ {\mathfrak Y} ) $ and consider some compound hypothesis $ H _ {0} $: $ \theta \in \Theta _ {0} \subseteq \Theta $. In that case, if for any event $ B \in {\mathcal B} _ {\mathfrak Y} $ the probability

$$ \tag{* } {\mathsf P} _ \theta ( T ^ {- 1} ( B)) \ \textrm{ is } \textrm{ independent } \ \textrm{ of } \theta \textrm{ for } \theta \in \Theta _ {0} , $$

one says that $ T $ is a similar statistic with respect to $ H _ {0} $, or simply that it is a similar statistic. It is clear that condition (*) is equivalent to saying that the distribution of the statistic $ T $ does not vary when $ \theta $ runs through $ \Theta _ {0} $. With this property in view, it is frequently said of a similar statistic that it is independent of the parameter $ \theta $, $ \theta \in \Theta _ {0} $. Similar statistics play a large role in constructing similar tests, and also in solving statistical problems with nuisance parameters.

Example 1. Let $ X _ {1} \dots X _ {n} $ be independent random variables with identical normal distribution $ N _ {1} ( a, \sigma ^ {2} ) $ with $ | a | < \infty $ and $ \sigma > 0 $. Then for any $ \alpha > 0 $ the statistic

$$ T = \left ( {\sum _ { i=1} ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2} } \right ) ^ {- \alpha } \sum _ { i=1} ^ { n } ( X _ {i} - \overline{X}\; ) ^ {2 \alpha } , $$

where

$$ \overline{X}\; = \frac{1}{n} \sum _ { i=1} ^ { n } X _ {i} , $$

is independent of the two-dimensional parameter $ ( a, \sigma ^ {2} ) $.

Example 2. Let $ X _ {1} \dots X _ {n+} m $ be independent identically-distributed random variables whose distribution functions belong to the family $ {\mathcal F} = \{ F ( x) \} $ of all continuous distribution functions on $ ( - \infty , + \infty ) $. If $ F _ {n} ( x) $ and $ F _ {m} ( x) $ are empirical distribution functions constructed from the observations $ X _ {1} \dots X _ {n} $ and $ X _ {n+1} \dots X _ {n+m} $, respectively, then the Smirnov statistic

$$ S _ {n,m} = \sup _ {| x| < \infty } | F _ {n} ( x) - F _ {m} ( x) | $$

is similar with respect to the family $ {\mathcal F} $.

References

[1]	J.-L. Soler, "Basic structures in mathematical statistics" , Moscow (1972) (In Russian; translated from French)
[2]	Yu.V. Linnik, "Statistical problems with nuisance parameters" , Amer. Math. Soc. (1968) (Translated from Russian)
[3]	J.-R. Barra, "Mathematical bases of statistics" , Acad. Press (1981) (Translated from French)
[a1]	E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)