[R] order statistic of multivariate normal

Ranjan Maitra maitra.mbox.ignored at inbox.com
Fri Mar 22 17:37:37 CET 2013 

On Fri, 22 Mar 2013 09:31:53 -0700 Bert Gunter <gunter.berton at gene.com>
wrote:

> Well...
> 
> On Fri, Mar 22, 2013 at 8:20 AM, Ranjan Maitra <
> maitra.mbox.ignored at inbox.com> wrote:
> 
> > 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.
> 
> 
> Therein lies the difficulty. Try it for the non-iid case.

I doubt this. The derivation should exactly be the same as for the iid
case, except that the multivariate density does not factorize so you
keep it as is. That is where the Genz routines in that R package help.

Ranjan

> 
> 
> Cheers,
> Bert
> 
> 
> 
> > 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
> >
> > ______________________________________________
> > 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.
> >
> 
> 
> 
> -- 
> 
> 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]]
> 


-- 
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.

____________________________________________________________
FREE ONLINE PHOTOSHARING - Share your photos online with your friends and family!
Visit http://www.inbox.com/photosharing to find out more!

More information about the R-help mailing list