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 (Xi, Yi), i = 1,. . ., n be a random sample from a bivariate distribution. If the sample is ordered by the Xi, then the Y-variate associated with Xr:n will be denoted by Y[r:n] and termed the concomitant of the rth 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 rth concomitant $$Y_{[r:n]}$$ for $$1 \le r \le n $$ is

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

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 $$\left(Y_{[r_1:n]}, \cdots, Y_{[r_k:n]} \right)$$ is given by

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

That is, in general, the joint concomitants of order statistics $$\left(Y_{[r_1:n]}, \cdots, Y_{[r_k:n]} \right)$$ 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

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