Re: [R] order statistic of multivariate normal

[Ranjan Maitra]

On Fri, 22 Mar 2013 09:31:53 -0700 Bert Gunter 
wrote:

> Well...
> 
> On Fri, Mar 22, 2013 at 8:20 AM, Ranjan Maitra <
> maitra.mbox.igno...@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  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 
> > >
> > > > 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.hard...@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)  > > >> > 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) 
> > > >> Date: 22-Mar-2013  Time: 13:31:31
> > > >> This message was sent by XFMail
> > > >>
> > > >> __
> > > >> R-help@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
> > > > Gen

Re: [R] order statistic of multivariate normal

[Bert Gunter]

Well...

On Fri, Mar 22, 2013 at 8:20 AM, Ranjan Maitra <
maitra.mbox.igno...@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.


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  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 
> >
> > > 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.hard...@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)  > >> > 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) 
> > >> Date: 22-Mar-2013  Time: 13:31:31
> > >> This message was sent by XFMail
> > >>
> > >> __
> > >> R-help@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@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, repr

Re: [R] order statistic of multivariate normal

[li li]

Thank you!
Hanna

2013/3/22 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  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 
> >
> > > 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.hard...@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)  > >> > 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) 
> > >> Date: 22-Mar-2013  Time: 13:31:31
> > >> This message was sent by XFMail
> > >>
> > >> __
> > >> R-help@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]]
> >
> > __
> > R-help@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

Re: [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  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 
> 
> > 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 
> > 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)  >> > 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) 
> >> Date: 22-Mar-2013  Time: 13:31:31
> >> This message was sent by XFMail
> >>
> >> __
> >> R-help@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]]
> 
> __
> R-help@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.
> 


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Re: [R] order statistic of multivariate normal

[li li]

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 

> 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 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) > > 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) 
>> Date: 22-Mar-2013  Time: 13:31:31
>> This message was sent by XFMail
>>
>> __
>> R-help@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
>
>

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Re: [R] order statistic of multivariate normal

[Bert Gunter]

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 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)  > 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) 
> Date: 22-Mar-2013  Time: 13:31:31
> This message was sent by XFMail
>
> __
> R-help@r-project.org mailing list
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> PLEASE do read the posting guide
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> 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

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Re: [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)  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) 
Date: 22-Mar-2013  Time: 13:31:31
This message was sent by XFMail

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Re: [R] order statistic of multivariate normal

[li li]

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) 

> Hello,
>
>
> Em 21-03-2013 21:42, Albyn Jones escreveu:
>
> R^n for n > 1 is not an ordered set.
>>
>
> Theorem. All sets are well ordered.
> (This theorem is equivalent to the Axiom of Choice.)
>
> Rui Barradas
>
>
>> albyn
>>
>> On Thu, Mar 21, 2013 at 02:32:44PM -0700, David Winsemius wrote:
>>
>>>
>>> On Mar 21, 2013, at 1:44 PM, li li wrote:
>>>
>>> Hi all,
 Is there an R function that computes the probabilty or quantiles of
 order statistics of multivariate normal?
   Thank you.

>>>
>>> There is an mvtnorm package. I don't know what you mean by "quantiles of
>>> order statistics of multivariate normal", though.
>>> --
>>> David Winsemius
>>> Alameda, CA, USA
>>>
>>> __**
>>> R-help@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.
>>>
>>>
>>
> __**
> R-help@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.
>

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Re: [R] order statistic of multivariate normal

[Rui Barradas]

Hello,


Em 21-03-2013 21:42, Albyn Jones escreveu:

R^n for n > 1 is not an ordered set.


Theorem. All sets are well ordered.
(This theorem is equivalent to the Axiom of Choice.)

Rui Barradas


albyn

On Thu, Mar 21, 2013 at 02:32:44PM -0700, David Winsemius wrote:


On Mar 21, 2013, at 1:44 PM, li li wrote:


Hi all,
Is there an R function that computes the probabilty or quantiles of
order statistics of multivariate normal?
  Thank you.


There is an mvtnorm package. I don't know what you mean by "quantiles of order 
statistics of multivariate normal", though.
--
David Winsemius
Alameda, CA, USA

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Re: [R] order statistic of multivariate normal

[Albyn Jones]

R^n for n > 1 is not an ordered set.

albyn

On Thu, Mar 21, 2013 at 02:32:44PM -0700, David Winsemius wrote:
> 
> On Mar 21, 2013, at 1:44 PM, li li wrote:
> 
> > Hi all,
> > Is there an R function that computes the probabilty or quantiles of
> > order statistics of multivariate normal?
> >  Thank you.
> 
> There is an mvtnorm package. I don't know what you mean by "quantiles of 
> order statistics of multivariate normal", though. 
> -- 
> David Winsemius
> Alameda, CA, USA
> 
> __
> R-help@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.
> 

-- 
Albyn Jones
Reed College
jo...@reed.edu

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Re: [R] order statistic of multivariate normal

[David Winsemius]

On Mar 21, 2013, at 1:44 PM, li li wrote:

> Hi all,
> Is there an R function that computes the probabilty or quantiles of
> order statistics of multivariate normal?
>  Thank you.

There is an mvtnorm package. I don't know what you mean by "quantiles of order 
statistics of multivariate normal", though. 
-- 
David Winsemius
Alameda, CA, USA

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[R] order statistic of multivariate normal

[li li]

Hi all,
 Is there an R function that computes the probabilty or quantiles of
order statistics of multivariate normal?
  Thank you.
 Hanna

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