R help - May 2017 - Generating samples from truncated multivariate Student-t distribution

Dear Members,
I am working with 6-dimensional Student-t distribution with 4 degrees
of freedom truncated to [20; 60]. I have generated 100 000 samples
from truncated multivariate Student-t distribution using rtmvt
function from package ?tmvtnorm?. I have also calculated  mean vector
using equation (3) from attached pdf. The problem is, that after
summing all elements in one column of rtmvt result (and dividing by
100 000) I do not receive the same result as using (3) equation. Could
You tell me, what is incorrect, why there is a difference?
Yours faithfully
Czarek Kowalski
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> On May 9, 2017, at 10:09 AM, Czarek Kowalski <czarek230800 at
gmail.com> wrote:
> 
> Dear Members,
> I am working with 6-dimensional Student-t distribution with 4 degrees
> of freedom truncated to [20; 60]. I have generated 100 000 samples
> from truncated multivariate Student-t distribution using rtmvt
> function from package ?tmvtnorm?. I have also calculated  mean vector
> using equation (3) from attached pdf. The problem is, that after
> summing all elements in one column of rtmvt result (and dividing by
> 100 000) I do not receive the same result as using (3) equation. Could
> You tell me, what is incorrect, why there is a difference?
I guess the question is why you would NOT expect a difference between theory and
a sample of realizations of RV's? The result mean should still be a random
variable, night wahr?

> Yours faithfully
> Czarek Kowalski
> <truncatedT.pdf>______________________________________________
> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
> 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.
David Winsemius
Alameda, CA, USA

Of course I have expected the difference between theory and a sample
of realizations of RV's and result mean should still be a random
variable. But, for example for 4th element of mean vector: 35.31 -
34.69571 = 0.61429. It is quite big difference, nieprawda?? I have
expected that the difference would be smaller because of law of large
numbers (for 10mln samples the difference is quite similar).

On 9 May 2017 at 21:40, David Winsemius <dwinsemius at comcast.net>
wrote:
>
>> On May 9, 2017, at 10:09 AM, Czarek Kowalski <czarek230800 at
gmail.com> wrote:
>>
>> Dear Members,
>> I am working with 6-dimensional Student-t distribution with 4 degrees
>> of freedom truncated to [20; 60]. I have generated 100 000 samples
>> from truncated multivariate Student-t distribution using rtmvt
>> function from package ?tmvtnorm?. I have also calculated  mean vector
>> using equation (3) from attached pdf. The problem is, that after
>> summing all elements in one column of rtmvt result (and dividing by
>> 100 000) I do not receive the same result as using (3) equation. Could
>> You tell me, what is incorrect, why there is a difference?
>
> I guess the question is why you would NOT expect a difference between
theory and a sample of realizations of RV's? The result mean should still be
a random variable, night wahr?
>
>
>> Yours faithfully
>> Czarek Kowalski
>> <truncatedT.pdf>______________________________________________
>> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
>> 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.
>
> David Winsemius
> Alameda, CA, USA
>

It's not obvious to me that that marginal distribution of one component of a
multivariate truncated t is the corresponding univariate truncated t.

In fact, I would expect it to differ because of tail-dependence effects, e.g.
> r <- rtmvt(1e5, c(30,0), diag(2), lower=c(29,-Inf), upper=c(31, +Inf),
df=4)
> sd(r[,2])
[1] 1.191654
> r <- rtmvt(1e5, c(30,0), diag(2), lower=c(35,-Inf), upper=c(37, +Inf),
df=4)
> sd(r[,2])
[1] 3.504233

-pd
> On 9 May 2017, at 19:09 , Czarek Kowalski <czarek230800 at gmail.com>
wrote:
> 
> Dear Members,
> I am working with 6-dimensional Student-t distribution with 4 degrees
> of freedom truncated to [20; 60]. I have generated 100 000 samples
> from truncated multivariate Student-t distribution using rtmvt
> function from package ?tmvtnorm?. I have also calculated  mean vector
> using equation (3) from attached pdf. The problem is, that after
> summing all elements in one column of rtmvt result (and dividing by
> 100 000) I do not receive the same result as using (3) equation. Could
> You tell me, what is incorrect, why there is a difference?
> Yours faithfully
> Czarek Kowalski
> <truncatedT.pdf>______________________________________________
> R-help at r-project.org mailing list -- To UNSUBSCRIBE and more, see
> 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.
-- 
Peter Dalgaard, Professor,
Center for Statistics, Copenhagen Business School
Solbjerg Plads 3, 2000 Frederiksberg, Denmark
Phone: (+45)38153501
Office: A 4.23
Email: pd.mes at cbs.dk  Priv: PDalgd at gmail.com