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