[R] order statistic of multivariate normal

[(Ted Harding)]

On 22-Mar-2013 13:02:25 li li wrote:
> Thank you all for the reply.
> 
> One example of my question is as follows.
> 
> Suppose X1, ..., X10 has multivariate normal distribution
> and X(1), ..., X(10) are the corresponding order statistics.
> 
> My question is that whether there is a R function that would
> help compute the c which satisfies
> P(X(4) <c)=beta.
> Here beta is a known constant between 0 and 1.
> 
> Thank you.
>    Hanna

The basic question which needs to be answered (which has been hinted
at in earlier replis) is: How do you define "order statistic" for
multivariate observations?

For example, here is a sample of 10 (to 3 d.p.) from a bivariate
normal distribution:

          [,1]   [,2]
   [1,]  1.143 -0.396
   [2,] -0.359 -0.217
   [3,] -0.391 -0.601
   [4,] -0.416 -1.093
   [5,] -1.810 -1.499
   [6,] -0.367 -0.636
   [7,] -2.238  0.563
   [8,]  0.811  1.230
   [9,]  0.082  0.174
  [10,] -1.359 -0.364

Which one of these 10 rows is X(4)?

There is an alternative interpretation of your question:

  "Suppose X1, ..., X10 has multivariate normal distribution
  and X(1), ..., X(10) are the corresponding order statistics."

This could mean that the vector (X1,...,X10) has a multivariate
normal distribution with 10 dimensions, and, for a single vector
(X1,...,X10) drawn from this distribution, (X(1), ..., X(10))
is a vector consisting of these same values (X1,...,X10), but
in increasing order.

Is that what you mean?

Hoping this helps,
Ted.


-------------------------------------------------
E-Mail: (Ted Harding) <Ted.Harding at wlandres.net>
Date: 22-Mar-2013  Time: 13:31:31
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