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A structure determined by a statistic that is independent of a sufficient statistic. The concept was introduced by [[Neyman, Jerzy|J. Neyman]] (see [[#References|[1]]]) in connection with the problem of constructing similar tests (cf. [[Similar test|Similar test]]) in the theory of statistical hypothesis testing, and the term  "Neyman structure"  is used when referring to the structure of a statistical test if its critical function has Neyman structure. Suppose that in the realization of a random variable  $  X $
taking values in a sample space  $  ( \mathfrak X , \mathfrak B , P _  \theta  ) $, 
$  \theta \in \Theta $, 
it is required to verify a composite hypothesis  $  H _ {0} $: 
$  \theta \in \Theta _ {0} \subset  \Theta $
and that for the family  $  \{ {P _  \theta  } : {\theta \in \Theta _ {0} } \} $
there exists a [[Sufficient statistic|sufficient statistic]]  $  T $
with distribution in the family  $  \{ {P _ {e}  ^ {T} } : {e \in \Theta _ {0} } \} $. 
Then any statistical test of level  $  \alpha $
intended for testing  $  H _ {0} $
has Neyman structure if its critical function  $  \phi $
satisfies the condition:

$$ \tag{1 }
{\mathsf E} \{ \phi ( X) \mid  T = t \}  =  \alpha
$$

almost everywhere with respect to the measure  $  P _  \theta   ^ {T} $, 
$  \theta \in \Theta _ {0} $. 
Evidently, if a statistical test has Neyman structure, then it is similar (cf. [[Similar test|Similar test]]) relative to the family  $  \{ {P _  \theta  } : {\theta \in \Theta _ {0} } \} $, 
since

$$ 
{\mathsf E} _  \theta  \{ \phi ( X) \}  = \ 
{\mathsf E} _  \theta  \{ {\mathsf E} \{ \phi ( X) \mid  T = t \} \}  =  \alpha
$$

for all  $  \theta \in \Theta _ {0} $.

The validity of (1) essentially reduces the problem of testing the composite hypothesis  $  H _ {0} $
to that of testing  $  H _ {0} $
as a simple hypothesis for every fixed value  $  t $
of the sufficient statistic  $  T $.

Example. Suppose that two independent random variables  $  X _ {1} $
and  $  X _ {2} $
are subject to Poisson laws with unknown parameters  $  \lambda _ {1} $
and  $  \lambda _ {2} $(
cf. [[Poisson distribution|Poisson distribution]]) and that the hypothesis  $  H _ {0} $: 
$  \lambda _ {1} = \lambda _ {2} $
is to be tested against the alternative  $  H _ {1} $: 
$  \lambda _ {1} \neq \lambda _ {2} $. 
Thanks to the independence of  $  X _ {1} $
and  $  X _ {2} $
the statistic  $  T = X _ {1} + X _ {2} $
is subject to the Poisson law with parameter  $  \lambda _ {1} + \lambda _ {2} $
and the conditional distributions of  $  X _ {1} $
and  $  X _ {2} $
under the condition  $  T = t $
are binomial with parameters  $  t $, 
$  \lambda _ {1} / ( \lambda _ {1} + \lambda _ {2} ) $
and  $  t $, 
$  \lambda _ {2} / ( \lambda _ {1} + \lambda _ {2} ) $, 
respectively, that is,

$$ \tag{2 }
{\mathsf P} \{ X _ {i} = k \mid  T = t \}  = \ 
\left ( \begin{array}{c}
t \\
 k 
\end{array}
 \right )
\left (

\frac{\lambda _ {i} }{\lambda _ {1} + \lambda _ {2} }

\right )  ^ {k}
\left (
1 - 
\frac{\lambda _ {i} }{\lambda _ {1} + \lambda _ {2} }

\right )  ^ {t-} k ,
$$

$$ 
k  =  0 \dots t .
$$

When  $  H _ {0} $
is valid, then  $  T $
is sufficient for the unknown common value  $  \lambda = \lambda _ {1} = \lambda _ {2} $, 
and from (2) it follows that when  $  H _ {0} $
holds, then the conditional distribution of  $  X _ {1} $
for a fixed value of the sufficient statistic  $  T = t $
is binomial with parameters  $  t $
and  $  1 / 2 $, 
that is, under  $  H _ {0} $,

$$ 
{\mathsf P} \{ X _ {1} = k \mid  T = t \}  = \ 
\left ( \begin{array}{c}
t \\
 k 
\end{array}
 \right )
\left ( {
\frac{1}{2}
 } \right )  ^ {t} ,\ \ 
k = 0 \dots t .
$$

Thus, in this case the problem of testing the composite hypothesis  $  H _ {0} $
reduces to that of testing the simple hypothesis  $  H _ {0}  ^ {t} $, 
according to which the conditional distribution of  $  X _ {1} $(
for a fixed sum  $  X _ {1} + X _ {2} = t $) 
is binomial with parameters  $  t $
and  $  1 / 2 $. 
For testing  $  H _ {0}  ^ {t} $
one can use, for example, the [[Sign test|sign test]].

The concept of a Neyman structure is of great significance in the problem of testing composite statistical hypotheses, since among the tests having Neyman structure there frequently is a [[Most-powerful test|most-powerful test]]. E. Lehmann and H. Scheffé have shown that a statistical test for testing a composite hypothesis  $  H _ {0} $: 
$  \theta \in \Theta _ {0} $
has Neyman structure relative to a sufficient statistic  $  T $
if and only if the family  $  \{ {P _  \theta   ^ {T} } : {\theta \in \Theta _ {0} } \} $
induced by  $  T $
is boundedly complete. On the basis of the concept of a Neyman structure general methods have been worked out for the construction of similar tests. See [[Distributions, complete family of|Distributions, complete family of]]; [[Similar test|Similar test]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  J. Neyman,   "Current problems of mathematical statistics" , ''Proc. Internat. Congress Mathematicians (Amsterdam, 1954)'' , '''1''' , Noordhoff &amp; North-Holland  (1957)  pp. 349–370</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  E.L. Lehmann,   "Testing statistical hypotheses" , Wiley  (1986)</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  Yu.V. Linnik,   "Statistical problems with nuisance parameters" , Amer. Math. Soc.  (1968)  (Translated from Russian)</TD></TR>
</table>