[R] order statistic of multivariate normal

[Ranjan Maitra]

I don't believe that you necessarily need to use simulation for this.
But you do need numerical integration. Here is a skeletal approach.

Calculate the density (distribution) of the order statistics of a
multivariate sample. Then since the underlying distribution is
multivariate normal, use a multivariate integration routine in R (try
the mnormt package) to get the integration part of the calculation and
proceed.

As I said before, here is the outline of an approach I would first take.
You get to work through the details:-)

Ranjan 




On Fri, 22 Mar 2013 10:48:06 -0400 li li <hannah.hlx at gmail.com> wrote:

> Yes. What I meant is "the distribution of order statistics from a
> non-iid sample of a (normal) distribution with specified sample
> covariance matrix".
>  Thanks for the idea of simulation.  I guess there is no other way
> around.
>       Hanna
> 
> 2013/3/22 Bert Gunter <gunter.berton at gene.com>
> 
> > As you suggest, Ted, it appears from the question that the OP really means
> > "order statistics of a sample of 10 from the distribution."  So what she
> > appears to want is the distribution of order statistics from a non-iid
> > sample of a (normal) distribution with specified sample covariance matrix.
> >
> > The independent case is standard first statistics course stuff, but I
> > believe this would require a 10-d integral (please correct if wrong!) for
> > non-iid.  So it would seem that simulation would be the simplest approach,
> > and, indeed, should be straightforward. E.g. the mvrnorm() function in MASS
> > could be used to simulate the samples.
> >
> > Again, corrections appreciated if I am wrong on any of this.
> >
> > -- Bert
> >
> >
> >
> >  On Fri, Mar 22, 2013 at 6:31 AM, Ted Harding <Ted.Harding at wlandres.net>wrote:
> >
> >>  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
> >> This message was sent by XFMail
> >>
> >> ______________________________________________
> >> R-help at r-project.org mailing list
> >> https://stat.ethz.ch/mailman/listinfo/r-help
> >> PLEASE do read the posting guide
> >> http://www.R-project.org/posting-guide.html<http://www.r-project.org/posting-guide.html>
> >> and provide commented, minimal, self-contained, reproducible code.
> >>
> >
> >
> >
> > --
> >
> > Bert Gunter
> > Genentech Nonclinical Biostatistics
> >
> > Internal Contact Info:
> > Phone: 467-7374
> > Website:
> >
> > http://pharmadevelopment.roche.com/index/pdb/pdb-functional-groups/pdb-biostatistics/pdb-ncb-home.htm
> >
> >
> 
> 	[[alternative HTML version deleted]]
> 
> ______________________________________________
> R-help at r-project.org mailing list
> https://stat.ethz.ch/mailman/listinfo/r-help
> PLEASE do read the posting guide http://www.R-project.org/posting-guide.html
> and provide commented, minimal, self-contained, reproducible code.
> 


-- 
Important Notice: This mailbox is ignored: e-mails are set to be
deleted on receipt. For those needing to send personal or professional
e-mail, please use appropriate addresses.

____________________________________________________________
GET FREE SMILEYS FOR YOUR IM & EMAIL - Learn more at http://www.inbox.com/smileys
Works with AIM®, MSN® Messenger, Yahoo!® Messenger, ICQ®, Google Talk™ and most webmails