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

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 [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,

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)