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. ).
|||J. Hájek, Z. Sidák, "Theory of rank tests" , Acad. Press (1967)|
|||M.G. Kendall, "Rank correlation methods" , Griffin (1970)|
Rank statistic. M.S. Nikulin (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Rank_statistic&oldid=18903