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A vector statistic (cf. [[Statistics|Statistics]])  $  R = ( R _ {1} \dots R _ {n} ) $
constructed from a random observation vector  $  X = ( X _ {1} \dots X _ {n} ) $
with  $  i $-th component  $  R _ {i} = R _ {i} ( X) $, 
$  i = 1 \dots n $, 
defined by

$$ 
R _ {i}  =  \sum _ { j=  1} ^ { n }  \delta ( X _ {i} - X _ {j} ) ,
$$

where  $  \delta ( x) $
is the characteristic function (indicator function) of  $  [ 0 , + \infty ] $, 
that is,

$$ 
\delta ( x)  = \ 
\left \{ 
\begin{array}{ll}
 1  & \textrm{ if }  x \geq  0 ,  \\
 0  & \textrm{ if }  x < 0 .  \\
\end{array}

 \right .$$

The statistic  $  R _ {i} $
is called the rank of the  $  i $-th component  $  X _ {i} $, 
$  i = 1 \dots n $, 
of the random vector  $  X $. 
This definition of a rank vector is precise under the condition

$$ 
{\mathsf P} \{ X _ {i} = X _ {j} \}  =  0 ,\ \ 
i \neq j ,
$$

which automatically holds if the probability distribution of  $  X $
is defined by a density  $  p ( x) = p ( x _ {1} \dots x _ {n} ) $. 
It follows from the definition of a rank vector that, under these conditions,  $  R $
takes values in the space  $  \mathfrak R = \{ r \} $
of all permutations  $  r = ( r _ {1} \dots r _ {n} ) $
of  $  1 \dots n $
and the realization  $  r _ {i} $
of the rank  $  R _ {i} $
is equal to the number of components of  $  X $
whose observed values do not exceed the realization of the  $  i $-th component  $  X _ {i} $, 
$  i = 1 \dots n $.

Let  $  X ^ {( \cdot ) } = ( X _ {( n1)} \dots X _ {( nn)} ) $
be the vector of order statistics (cf. [[Order statistic|Order statistic]]) constructed from the observation vector  $  X $. 
Then the pair  $  ( R , X ^ {( \cdot ) } ) $
is a [[Sufficient statistic|sufficient statistic]] for the distribution of  $  X $, 
and  $  X $
itself can be uniquely recovered from  $  ( R , X ^ {( \cdot ) } ) $. 
Moreover, under the additional assumption that the density  $  p ( x) $
of  $  X $
is symmetric with respect to permutations of the arguments, the components  $  R $
and  $  X ^ {( \cdot ) } $
of the sufficient statistic  $  ( R , X ^ {( \cdot ) } ) $
are independent and

$$ 
{\mathsf P} \{ R = r \}  =  
\frac{1}{n ! }
 ,\ \ 
r \in \mathfrak R .
$$

In particular, if

$$ \tag{1 }
p ( x)  =  p ( x _ {1} \dots x _ {n} )
 =  \prod _ { i= 1} ^ { n }  f ( x _ {i} ) ,
$$

that is, the components  $  X _ {1} \dots X _ {n} $
are independent identically-distributed random variables ( $  f ( x _ {i} ) $
stands for the density of  $  X _ {i} $), 
then

$$ \tag{2 }
\left . 
\begin{array}{c}
{\mathsf P} \{ R _ {i} = k \}  =  
\frac{1}{n}
 ,\  i = 1 \dots n ,  \\
{\mathsf P} \{ R _ {i} = k , R _ {j} = m \}  =  
\frac{1}{n ( n - 1 ) }
 ,
\  i \neq j ,\  k \neq m ,  \\
{\mathsf E} \{ R _ {i} \}  =  n+ 
\frac{1}{2}
 ,\  {\mathsf D} \{ R _ {i} \}  =  
\frac{n  ^ {2} - 1 }{12}
 ,\ \ 
i = 1 \dots n ,  \\
\end{array}
 \right \}
$$

for any  $  k = 1 \dots n $.

If (1) holds, there is a joint density  $  q ( x _ {i} , k ) $, 
$  k = 1 \dots n $, 
of  $  X _ {i} $
and  $  R _ {i} $, 
defined by the formula

$$ \tag{3 }
q ( x _ {i} , k ) =
$$

$$ 
= \ 

\frac{( n - 1 ) ! }{( k - 1 ) ! ( n - k ) ! }
 [ F (
x _ {i} ) ]  ^ {k- 1} [ 1 - F ( x _ {i} ) ]  ^ {n- k }f ( x _ {i} ) ,
$$

where  $  F ( x _ {i} ) $
is the distribution function of  $  X _ {i} $. 
It follows from (2) and (3) that the conditional density  $  q ( X _ {i} \mid  R _ {i} = k ) $
of  $  X _ {i} $
given  $  R _ {i} = k $ ($  k = 1 \dots n $) 
is expressed by the formula

$$ \tag{4 }
q ( x _ {i} \mid  R _ {i} = k ) =
$$

$$ 
= \ 
{n! \over ( k - 1 ) ! ( n - k ) ! } [ F ( x _ {i} )
]  ^ {k- 1} [ 1 - F ( x _ {i} ) ]  ^ {n- k} f ( x _ {i} ) .
$$

The latter formula allows one to trace the internal connection between the observation vector  $  X $, 
the rank vector  $  R $
and the vector  $  X ^ {( \cdot ) } $
of order statistics, since (4) is just the probability density of the  $  k $-th order statistic  $  X _ {( nk) }$, 
$  k = 1 \dots n $. 
Moreover, it follows from (3) that the conditional distribution of the rank  $  R _ {i} $
is given by the formula

$$ 
{\mathsf P} \{ R _ {i} = k \mid  X _ {i} \} =
$$

$$ 
= \ 

\frac{( n - 1 ) ! }{( k - 1 ) ! ( n - k ) ! }

[ F ( X _ {i} ) ]  ^ {k- 1} [ 1 - F ( X _ {i} ) ]  ^ {n- k} .
$$

Finally, under the assumption that the moments  $  {\mathsf E} \{ X _ {i} \} $
and  $  {\mathsf D} \{ X _ {i} \} $
exist and that (1) holds, (2) and (3) imply that the correlation coefficient  $  \rho ( X _ {i} , R _ {i} ) $
between  $  X _ {i} $
and  $  R _ {i} $
is equal to

$$ 
\rho ( X _ {i} , R _ {i} )  = \ 
\sqrt {
\frac{12 ( n - 1 ) }{( n + 1 ) {\mathsf D} \{ X _ {i} \} }
 }
\int\limits _ {- \infty } ^  \infty  
x _ {i} \left [ F ( x _ {i} ) - 
\frac{1}{2}
 \right ]  d F ( x _ {i} ) .
$$

In particular, if  $  X _ {i} $
is uniformly distributed on  $  [ 0 , 1 ] $, 
then

$$ 
\rho ( X _ {i} , R _ {i} )  = \ 
\sqrt {n- \frac{1}{n+1 }} .
$$

If  $  X $
has the [[Normal distribution|normal distribution]]  $  N ( a , \sigma  ^ {2} ) $, 
then

$$ 
\rho ( X _ {i} , R _ {i} )  = \ 
\sqrt {
\frac{3 ( n - 1 ) }{\pi ( n + 1 ) }
 } ,
$$

and  $  \rho ( X _ {i} , R _ {i} ) $
does not depend on the parameters of the normal distribution.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Hoeffding,   " "Optimum"  nonparametric tests" , ''Proc. 2nd Berkeley Symp. Math. Stat. Probab. (1950)'' , Univ. California Press  (1951)  pp. 83–92</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Hájek,   Z. Sidák,   "Theory of rank tests" , Acad. Press  (1967)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  F.P. Tarasenko,   "Non-parametric statistics" , Tomsk  (1976)  (In Russian)</TD></TR></table>

{{SourceattributionEOM|Rank vector}}
[[Category:Mathematics]]