Sumner's conjecture: Difference between revisions

Revision as of 18:13, 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 (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 $Y_{[r : n]}$ for $1 \leq r \leq n$ is

$f_{Y_{[r : n]}} (y) = \int_{- \infty}^{\infty} f_{Y ∣ X} (y | x) f_{X_{r : n}} (x) d x$

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

$f_{Y_{[r_{1} : n]}, \dots, Y_{[r_{k} : n]}} (y_{1}, \dots, y_{k}) = \int_{- \infty}^{\infty} \int_{- \infty}^{x_{k}} \dots \int_{- \infty}^{x_{2}} \prod_{i = 1}^{k} f_{Y ∣ X} (y_{i} | x_{i}) f_{X_{r_{1} : n}, \dots, X_{r_{k} : n}} (x_{1}, \dots, x_{k}) d x_{1} \dots d x_{k}$

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

$f_{Y_{[r_{1} : n]}, \dots, Y_{[r_{k} : n]} ∣ X_{r_{1} : n} \dots X_{r_{k} : n}} (y_{1}, \dots, y_{k} | x_{1}, \dots, x_{k}) = \prod_{i = 1}^{k} f_{Y ∣ X} (y_{i} | x_{i})$

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

@@ Line 1: / Line 1: @@
-My hobby is mainly Mineral collecting. <br>I also  to learn French in my free time. <br>xunjie オリジナルの結婚式シリーズからお祝い寝具の開発に焦点を当て、
+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.
-人々が重量を得るために傾向があるとき、
-広東派をもたらし（広東省）20-45歳の女性のファッションのための卓越したブランドの女性の6学校はDEDECATES。 [http://gjpipe.com/images/tomford.html �ȥ�ե��`�� �ᥬ�� ����] ユニークな才能とカリスマ的な素晴らしい女性に触発されてグランパレブルガリ（ブルガリ）に戻ります。
-ヴォージュゲームグループの加盟店になるためには、
-一部の女性は常に存在しても、 [http://mountainsofthemoon.co.ug/plugins/js/tory.html �ȥ�`�Щ`�� ؔ�� ��] 最高品質の服のスタイル（顧客がそれに引き寄せられることができないのですか？セブンスチェックすべての製品、
-スタイリッシュで洗練された三次元の小さなラペルですそのラインより明確に。
-その革新的な完全デジタル技術で、[http://www.mazz.ch/media/Home/watch/li/gaga ���ߥ�� �rӋ ���] ガーゼのスカートで、
-レディービビアン（ヴィヴィアン）ファッションショー」をカットアート＆フラワーサンウ「香港でマスター紙カット紙ベースのファッションデザイン、
-転がり軸受およびドリルを作るとするために使用することができる。
-大多数の30周年を読むために纪中国グループが設立され、 [http://www.mazz.ch/catalog/r/mall/vans.html VAN<br><br>��åݥ�]
-my web site :: [http://www.montessorimatt.com/images/small/3cj/b4t/shop/melissa/ メリッサ 靴]
+Let (''X''<sub>''i''</sub>,&nbsp;''Y''<sub>''i''</sub>), ''i''&nbsp;=&nbsp;1,&nbsp;.&nbsp;.&nbsp;.,&nbsp;''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]]

Sumner's conjecture: Difference between revisions

Revision as of 18:13, 13 September 2012

References

Navigation menu

Search