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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.
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| Let (''X''<sub>''i''</sub>, ''Y''<sub>''i''</sub>), ''i'' = 1, . . ., ''n'' be a random sample from a bivariate distribution. If the sample is ordered by the ''X''<sub>''i''</sub>, then the ''Y''-variate associated with ''X''<sub>''r'':''n''</sub> will be denoted by ''Y''<sub>[''r'':''n'']</sub> and termed the '''concomitant''' of the ''r''<sup>th</sup> [[order statistic]].
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| 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''<sup>''th''</sup> '''concomitant''' <math>Y_{[r:n]}</math> for <math>1 \le r \le n </math> is
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| <math> f_{Y_{[r:n]}}(y) = \int_{-\infty}^\infty f_{Y \mid X}(y|x) f_{X_{r:n}} (x) \, \mathrm{d} x</math>
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| If all <math> (X_i, Y_i) </math> are assumed to be i.i.d., then for <math>1 \le r_1 < \cdots < r_k \le n</math>, the joint density for <math>\left(Y_{[r_1:n]}, \cdots, Y_{[r_k:n]} \right)</math> is given by
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| <math>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 </math>
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| That is, in general, the joint concomitants of order statistics <math>\left(Y_{[r_1:n]}, \cdots, Y_{[r_k:n]} \right)</math> is dependent, but are conditionally independent given <math>X_{r_1:n} = x_1, \cdots, X_{r_k:n} = x_k</math> for all ''k'' where <math>x_1 \le \cdots \le x_k</math>. The conditional distribution of the joint concomitants can be derived from the above result by comparing the formula in [[marginal distribution]] and hence
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| <math>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)</math>
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| ==References==
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| * {{cite book | last1 = David | first1 = Herbert A. | last2 = Nagaraja |first2 = H. N. | chapter = Concomitants of Order Statistics | title = Order Statistics: Theory & Methods | editor-last1 = Balakrishnan | editor-first1 = N. | editor-last2 = Rao | editor-first2 = C. R. | publisher = Elsevier | location = Amsterdam | year = 1998 | page = 487 - 513}}
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| * {{cite book | zbl=1053.62060 | last1=David | first1=Herbert A. | last2=Nagaraja | first2=H. N. | title=Order statistics | edition=3rd | series=Wiley Series in Probability and Statistics | location=Chichester | publisher=John Wiley & Sons | year=2003 | isbn=0-471-38926-9 | page=144 |author-mask=2 }}
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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 }}
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| [[Category:Probability and statistics]]
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