Sumner's conjecture

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

$f_{Y_{[r:n]}}(y)=\int _{-\infty }^{\infty }f_{Y\mid X}(y|x)f_{X_{r:n}}(x)\,\mathrm {d} x$

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

$f_{Y_{[r_{1}:n]},\cdots ,Y_{[r_{k}:n]}}(y_{1},\cdots ,y_{k})=\int _{-\infty }^{\infty }\int _{-\infty }^{x_{k}}\cdots \int _{-\infty }^{x_{2}}\prod _{i=1}^{k}f_{Y\mid X}(y_{i}|x_{i})f_{X_{r_{1}:n},\cdots ,X_{r_{k}:n}}(x_{1},\cdots ,x_{k})\mathrm {d} x_{1}\cdots \mathrm {d} x_{k}$

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

$f_{Y_{[r_{1}:n]},\cdots ,Y_{[r_{k}:n]}\mid X_{r_{1}:n}\cdots X_{r_{k}:n}}(y_{1},\cdots ,y_{k}|x_{1},\cdots ,x_{k})=\prod _{i=1}^{k}f_{Y\mid X}(y_{i}|x_{i})$

References

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• 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.
  
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• 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