Rank vector

A vector statistic (cf. Statistics) constructed from a random observation vector with -th component , , defined by

where is the characteristic function (indicator function) of , that is,

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

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

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

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

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)