Concomitant (statistics)

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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 (XiYi), 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)


  • David, Herbert A.; Nagaraja, H. N. (1998). "Concomitants of Order Statistics". In Balakrishnan, N.; Rao, C. R. Order Statistics: Theory & Methods. Amsterdam: Elsevier. p. 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. Zbl 1053.62060. 
  • Mathai, A. M.; Haubold, Hans J. (2008). Special Functions for Applied Scientists. Springer. ISBN 978-0-387-75893-0.