Rank statistic
A statistic (cf. Statistical estimator) constructed from a rank vector. If is the rank vector constructed from a random observation vector
, then any statistic
which is a function of
is called a rank statistic. A classical example of a rank statistic is the Kendall coefficient of rank correlation
between the vectors
and
, defined by the formula
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In the class of all rank statistics a special place is occupied by so-called linear rank statistics, defined as follows. Let be an arbitrary square matrix of order
. Then the statistic
![]() |
is called a linear rank statistic. For example, the Spearman coefficient of rank correlation , defined by the formula
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is a linear rank statistic.
Linear rank statistics are, as a rule, simple to construct from the computational point of view and their distributions are easy to find. For this reason the notion of projection of a rank statistic into the family of linear rank statistics plays an important role in the theory of rank statistics. If is a rank statistic constructed from a random vector
under a hypothesis
about its distribution, then a linear rank statistic
such that
is minimal under the condition that
is true, is called the projection of
into the family of linear rank statistics. As a rule,
approximates
well enough and the difference
is negligibly small as
. If the hypothesis
under which the components
of the random vector
are independent random variables is true, then the projection
of
can be determined by the formula
![]() | (*) |
where ,
(see [1]).
There is an intrinsic connection between and
. It is shown in [1] that the projection
of the Kendall coefficient
into the family of linear rank statistics coincides, up to a multiplicative constant, with the Spearman coefficient
; namely,
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This equality implies that the correlation coefficient between
and
is equal to
![]() |
implying that these rank statistics are asymptotically equivalent for large (cf. [2]).
References
[1] | J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967) |
[2] | M.G. Kendall, "Rank correlation methods" , Griffin (1970) |
Rank statistic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_statistic&oldid=18903