Rank vector
A vector statistic (cf. Statistics) constructed from a random observation vector
with
-th component
,
, defined by
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where is the characteristic function (indicator function) of
, that is,
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The statistic is called the rank of the
-th component
,
, of the random vector
. This definition of a rank vector is precise under the condition
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which automatically holds if the probability distribution of is defined by a density
. It follows from the definition of a rank vector that, under these conditions,
takes values in the space
of all permutations
of
and the realization
of the rank
is equal to the number of components of
whose observed values do not exceed the realization of the
-th component
,
.
Let be the vector of order statistics (cf. Order statistic) constructed from the observation vector
. Then the pair
is a sufficient statistic for the distribution of
, and
itself can be uniquely recovered from
. Moreover, under the additional assumption that the density
of
is symmetric with respect to permutations of the arguments, the components
and
of the sufficient statistic
are independent and
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In particular, if
![]() | (1) |
that is, the components are independent identically-distributed random variables (
stands for the density of
), then
![]() | (2) |
for any .
If (1) holds, there is a joint density ,
, of
and
, defined by the formula
![]() | (3) |
![]() |
where is the distribution function of
. It follows from (2) and (3) that the conditional density
of
given
(
) is expressed by the formula
![]() | (4) |
![]() |
The latter formula allows one to trace the internal connection between the observation vector , the rank vector
and the vector
of order statistics, since (4) is just the probability density of the
-th order statistic
,
. Moreover, it follows from (3) that the conditional distribution of the rank
is given by the formula
![]() |
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Finally, under the assumption that the moments and
exist and that (1) holds, (2) and (3) imply that the correlation coefficient
between
and
is equal to
![]() |
In particular, if is uniformly distributed on
, then
![]() |
If has the normal distribution
, then
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and does not depend on the parameters of the normal distribution.
References
[1] | W. Hoeffding, " "Optimum" nonparametric tests" , Proc. 2nd Berkeley Symp. Math. Stat. Probab. (1950) , Univ. California Press (1951) pp. 83–92 |
[2] | J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967) |
[3] | F.P. Tarasenko, "Non-parametric statistics" , Tomsk (1976) (In Russian) |
Rank vector. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_vector&oldid=48436