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 $Y_{[r:n]}$ for $1\le r\le n$ is

$f_{Y_{[r:n]}}(y)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{f}_{Y\mid X}(y|x)f_{X_{r:n}}(x)\mathrm{d}x$

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

$f_{Y_{[r_1:n]},\cdots ,Y_{[r_k:n]}}(y_1,\cdots ,y_k)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{x_k}{\int }}\cdots \underset{-\mathrm{\infty }}{\overset{x_2}{\int }}\prod _{i=1}^k{f}_{Y\mid X}(y_i|x_i)f_{X_{r_1:n},\cdots ,X_{r_k:n}}(x_1,\cdots ,x_k)\mathrm{d}{x}_1\cdots \mathrm{d}{x}_k$

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

$f_{Y_{[r_1:n]},\cdots ,Y_{[r_k:n]}\mid X_{r_1:n}\cdots X_{r_k:n}}(y_1,\cdots ,y_k|x_1,\cdots ,x_k)=\prod _{i=1}^k{f}_{Y\mid X}(y_i|x_i)$

References

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