Similar statistic
A statistic having a fixed probability distribution under some compound hypothesis.
Let the statistic map the sample space
,
, into a measurable space
and consider some compound hypothesis
:
. In that case, if for any event
the probability
![]() | (*) |
one says that is a similar statistic with respect to
, or simply that it is a similar statistic. It is clear that condition (*) is equivalent to saying that the distribution of the statistic
does not vary when
runs through
. With this property in view, it is frequently said of a similar statistic that it is independent of the parameter
,
. Similar statistics play a large role in constructing similar tests, and also in solving statistical problems with nuisance parameters.
Example 1. Let be independent random variables with identical normal distribution
with
and
. Then for any
the statistic
![]() |
where
![]() |
is independent of the two-dimensional parameter .
Example 2. Let be independent identically-distributed random variables whose distribution functions belong to the family
of all continuous distribution functions on
. If
and
are empirical distribution functions constructed from the observations
and
, respectively, then the Smirnov statistic
![]() |
is similar with respect to the family .
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) |
Comments
References
[a1] | E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986) |
Similar statistic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Similar_statistic&oldid=48700