In statistics, the concept of a concomitant, also called the induced order statistic, arises when one sorts the members of a random sample according đồ sộ 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 mật độ trùng lặp từ khóa function f(x,y), then the probability mật độ trùng lặp từ khóa function of rth concomitant for is
![{\displaystyle Y_{[r:n]}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8823fb9d9f2e98870f6fc774b59f014c06a4ea2d)
![{\displaystyle f_{Y_{[r:n]}}(y)=\int _{-\infty }^{\infty }f_{Y\mid X}(y|x)f_{X_{r:n}}(x)\,\mathrm {d} x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ef69bc86168d4e75ea22f59f837cbe2e48b2d17)
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If all are assumed đồ sộ be i.i.d., then for , the joint mật độ trùng lặp từ khóa for is given by
![{\displaystyle \left(Y_{[r_{1}:n]},\cdots ,Y_{[r_{k}:n]}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a51737f8fa310ca0b4556a5cd2dc385134804cb9)
![{\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e19736904aeb7ec8e0da82c08c15d8ed7ba74a8f)
That is, in general, the joint concomitants of order statistics is dependent, but are conditionally independent given for all k where . 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})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/feecb17f88a67aa21b78e3ef69fc8b834a449611)
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References[edit]
- David, Herbert A.; Nagaraja, H. N. (1998). "Concomitants of Order Statistics". In Balakrishnan, N.; Rao, C. R. (eds.). Order Statistics: Theory & Methods. Amsterdam: Elsevier. pp. 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.
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