Sumner's conjecture: Difference between revisions
Jump to navigation
Jump to search
Deltahedron (talk | contribs) m →References: zbl |
en>Giftlite m +s |
||
| Line 1: | Line 1: | ||
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''<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]]. | |||
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 | |||
<math> f_{Y_{[r:n]}}(y) = \int_{-\infty}^\infty f_{Y \mid X}(y|x) f_{X_{r:n}} (x) \, \mathrm{d} x</math> | |||
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 | |||
<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> | |||
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 | |||
<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> | |||
==References== | |||
* {{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}} | |||
* {{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 }} | |||
* {{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 }} | |||
[[Category:Probability and statistics]] | |||
Revision as of 19:13, 13 September 2012
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 (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 density function f(x,y), then the probability density function of rth concomitant for is
![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)
If all are assumed to be i.i.d., then for , the joint density 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)
References
- 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534