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''/sub> 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 $Y_$ for $1 \le r \le n$ is

$f_(y) = \int_^\infty f_(y, x) f_ (x) \, \mathrm x$

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 $\left(Y_, \cdots, Y_ \right)$ is given by

$f_(y_1, \cdots, y_k) = \int_^\infty \int_^ \cdots \int_^ \prod^k_ f_ (y_i, x_i) f_(x_1,\cdots,x_k)\mathrmx_1\cdots \mathrmx_k$

That is, in general, the joint concomitants of order statistics $\left(Y_, \cdots, Y_ \right)$ is dependent, but are conditionally independent given $X_ = x_1, \cdots, X_ = x_k$ 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

$f_(y_1, \cdots, y_k , x_1, \cdots, x_k) = \prod^k_ f_ (y_i, x_i)$

References

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* {{cite book , title = Special Functions for Applied Scientists , first1 = A. M. , last1 = Mathai , first2 = Hans J. , last2 = Haubold , publisher = Springer , year = 2008 , isbn = 978-0-387-75893-0

Theory of probability distributions