Concomitant (statistics)

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 [math]\displaystyle{ Y_{[r:n]} }[/math] for [math]\displaystyle{ 1 \le r \le n }[/math] is

[math]\displaystyle{ f_{Y_{[r:n]}}(y) = \int_{-\infty}^\infty f_{Y \mid X}(y|x) f_{X_{r:n}} (x) \, \mathrm{d} x }[/math]

If all [math]\displaystyle{ (X_i, Y_i) }[/math] are assumed to be i.i.d., then for [math]\displaystyle{ 1 \le r_1 \lt \cdots \lt r_k \le n }[/math], the joint density for [math]\displaystyle{ \left(Y_{[r_1:n]}, \cdots, Y_{[r_k:n]} \right) }[/math] is given by

[math]\displaystyle{ f_{Y_{[r_1:n]}, \cdots, Y_{[r_k:n]} }(y_1, \cdots, y_k) = \int_{-\infty}^\infty \int_{-\infty}^{x_k} \cdots \int_{-\infty}^{x_2} \prod^k_{ i=1 } f_{Y\mid X} (y_i|x_i) f_{X_{r_1:n}, \cdots, X_{r_k:n}}(x_1,\cdots,x_k)\mathrm{d}x_1\cdots \mathrm{d}x_k }[/math]

That is, in general, the joint concomitants of order statistics [math]\displaystyle{ \left(Y_{[r_1:n]}, \cdots, Y_{[r_k:n]} \right) }[/math] is dependent, but are conditionally independent given [math]\displaystyle{ X_{r_1:n} = x_1, \cdots, X_{r_k:n} = x_k }[/math] for all k where [math]\displaystyle{ x_1 \le \cdots \le x_k }[/math]. The conditional distribution of the joint concomitants can be derived from the above result by comparing the formula in marginal distribution and hence

[math]\displaystyle{ f_{Y_{[r_1:n]}, \cdots, Y_{[r_k:n]} \mid X_{r_1:n} \cdots X_{r_k:n} }(y_1, \cdots, y_k | x_1, \cdots, x_k) = \prod^k_{ i=1 } f_{Y\mid X} (y_i|x_i) }[/math]

References

• David, Herbert A.; Nagaraja, H. N. (1998). "Concomitants of Order Statistics". Order Statistics: Theory & Methods. Amsterdam: Elsevier. pp. 487–513. 
•   ; Nagaraja, H. N. (2003). Order statistics. Wiley Series in Probability and Statistics (3rd ed.). Chichester: John Wiley & Sons. p. 144. ISBN 0-471-38926-9. 
• Mathai, A. M.; Haubold, Hans J. (2008). Special Functions for Applied Scientists. Springer. ISBN 978-0-387-75893-0.