# Rank statistic

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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 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 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 ).

There is an intrinsic connection between and . It is shown in  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, This equality implies that the correlation coefficient between and is equal to implying that these rank statistics are asymptotically equivalent for large (cf. ).

How to Cite This Entry:
Rank statistic. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Rank_statistic&oldid=18903
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article