Concomitant (statistics) explained

In statistics, the concept of a concomitant, also called the induced order statistic, arises when one sorts the members of a random sample according to corresponding values of another random sample.

Let (X_i, Y_i), i = 1, . . ., n be a random sample from a bivariate distribution. If the sample is ordered by the X_i, then the Y-variate associated with X_r:n will be denoted by Y_{[''r'':''n'']} and termed the concomitant of the r^th order statistic.

Suppose the parent bivariate distribution having the cumulative distribution function F(x,y) and its probability density function f(x,y), then the probability density function of r^th concomitant

Y _{[r:n]} for 1 \le r \le n isf _{Y _{[r:n]}} (y) = \int ^{inf ty} _{-inf ty} f _{Y} (y|x) f _{X _{r:n}} (x) d x

If all

(X _{i,} Y _{i)} are assumed to be i.i.d., then for 1 \le r _{1} < … < r _{k} \le n , the joint density for \left(Y _{[r _{1:n]}} , … , Y _{[r _{k:n]}} \right) is given byf _{Y _{[r _{1:n]}} , … , Y _{[r _{k:n]}}} (y _{1,} … , y _{k)} = \int ^{inf ty} _{-inf ty} \int ^{x _{k}} _{-inf ty} … \int ^{x _{2}} _{-inf ty} \prod ^{k} _{i =1} f _{Y\mid} (y _{i|x} _{i)} f _{X _{r _{1:n}} , … , X _{r _{k:n}}} (x _{1, … ,x} _{k)dx} _{1 …} d x _{k}

That is, in general, the joint concomitants of order statistics

\left(Y _{[r _{1:n]}} , … , Y _{[r _{k:n]}} \right) is dependent, but are conditionally independent given X _{r _{1:n}} = x _{1,} … , X _{r _{k:n}} = x _{k} for all k where x _{1} \le … \le x _{k} . The conditional distribution of the joint concomitants can be derived from the above result by comparing the formula in marginal distribution and hencef _{Y _{[r _{1:n]}} , … , Y _{[r _{k:n]}} \mid X _{r _{1:n}} … X _{r _{k:n}}} (y _{1,} … , y _{k} | x _{1,} … , x _{k)} = \prod ^{k} _{i =1} f _{Y\mid} (y _{i|x} _{i)}

References

• Book: David . Herbert A. . Nagaraja . H. N. . Concomitants of Order Statistics . Order Statistics: Theory & Methods . Balakrishnan . N. . Rao . C. R. . Elsevier . Amsterdam . 1998 . 487–513.
• Book: 1053.62060 . David . Herbert A. . Nagaraja . H. N. . Order statistics . 3rd . Wiley Series in Probability and Statistics . Chichester . John Wiley & Sons . 2003 . 0-471-38926-9 . 144 . 2 .
• Book: Special Functions for Applied Scientists . A. M. . Mathai . Hans J. . Haubold . Springer . 2008 . 978-0-387-75893-0 .