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

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

If all $(X_i, Y_i)$ are assumed to be i.i.d., then for ${\displaystyle 1\leq r_{1}<\cdots <r_{k}\leq n}$, the joint density for ${\displaystyle \left(Y_{[r_{1}:n]},\cdots ,Y_{[r_{k}:n]}\right)}$ is given by

${\displaystyle f_{Y_{[r_{1}:n]},\cdots ,Y_{[r_{k}:n]}}(y_{1},\cdots ,y_{k})=\int _{-\infty }^{\infty }\int _{-\infty }^{x_{k}}\cdots \int _{-\infty }^{x_{2}}\prod _{i=1}^{k}f_{Y\mid X}(y_{i}|x_{i})f_{X_{r_{1}:n},\cdots ,X_{r_{k}:n}}(x_{1},\cdots ,x_{k})\mathrm {d} x_{1}\cdots \mathrm {d} x_{k}}$

That is, in general, the joint concomitants of order statistics ${\displaystyle \left(Y_{[r_{1}:n]},\cdots ,Y_{[r_{k}:n]}\right)}$ is dependent, but are conditionally independent given ${\displaystyle X_{r_{1}:n}=x_{1},\cdots ,X_{r_{k}:n}=x_{k}}$ for all k where ${\displaystyle x_{1}\leq \cdots \leq x_{k}}$. The conditional distribution of the joint concomitants can be derived from the above result by comparing the formula in marginal distribution and hence

${\displaystyle f_{Y_{[r_{1}:n]},\cdots ,Y_{[r_{k}:n]}\mid X_{r_{1}:n}\cdots X_{r_{k}:n}}(y_{1},\cdots ,y_{k}|x_{1},\cdots ,x_{k})=\prod _{i=1}^{k}f_{Y\mid X}(y_{i}|x_{i})}$

References

• 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.