# [R] Generating samples from truncated multivariate Student-t distribution

David Winsemius dwinsemius at comcast.net
Tue May 9 23:33:04 CEST 2017

```> On May 9, 2017, at 2:05 PM, Czarek Kowalski <czarek230800 at gmail.com> wrote:
>
> I have already posted that in attachement - pdf file.

I see that now. I failed to scroll to the 3rd page.

> I am posting
> plain text here:
>
>> library(tmvtnorm)
>> meann = c(55, 40, 50, 35, 45, 30)
>> covv = matrix(c(  1, 1, 0, 2, -1, -1,
>                    1, 16, -6, -6, -2, 12,
>                    0, -6, 4, 2, -2, -5,
>                    2, -6, 2, 25, 0, -17,
>                   -1, -2, -2, 0, 9, -5,
>                   -1, 12, -5, -17, -5, 36), 6, 6)
> df = 4
> lower = c(20, 20, 20, 20, 20, 20)
> upper = c(60, 60, 60, 60, 60, 60)
> X1 <- rtmvt(n=100000, meann, covv, df, lower, upper)
>
>
>> sum(X1[,1]) / 100000
> [1] 54.98258
> sum(X1[,2]) / 100000
> [1] 40.36153
> sum(X1[,3]) / 100000
> [1] 49.83571
> sum(X1[,4]) / 100000
> [1] 34.69571      # "4th element of mean vector"
> sum(X1[,5]) / 100000
> [1] 44.81081
> sum(X1[,6]) / 100000
> [1] 31.10834
>
> And corresponding results received using equation (3) from pdf file:
> [54.97,
> 40,
> 49.95,
> 35.31, #  "4th element of mean vector"
> 44.94,
> 31.32]
>

I get similar results for the output from your code,

My 100-fold run of your calculations were:

meansBig <- replicate(100, {Xbig <- rtmvt(n=100000, meann, covv, df, lower, upper)
colMeans(Xbig)} )

describe(meansBig[4,])  # describe is from Hmisc package

meansBig[4, ]
n  missing distinct     Info     Mean      Gmd      .05      .10      .25
100        0      100        1     34.7  0.01954    34.68    34.68    34.69
.50      .75      .90      .95
34.70    34.72    34.72    34.73

lowest : 34.65222 34.66675 34.66703 34.66875 34.67566
highest: 34.72939 34.73012 34.73051 34.73742 34.74441

So agree, 35.31 is outside the plausible range of an RV formed with that package, but I don't have any code relating to your calculations from theory.

Best;
David.

> On 9 May 2017 at 22:17, David Winsemius <dwinsemius at comcast.net> wrote:
>>
>>> On May 9, 2017, at 1:11 PM, Czarek Kowalski <czarek230800 at gmail.com> wrote:
>>>
>>> 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, nieprawdaż? I have
>>> expected that the difference would be smaller because of law of large
>>> numbers (for 10mln samples the difference is quite similar).
>>
>> I for one have no idea what is meant by a "4th element of mean vector". So I have now idea what to consider "big". I have found that my intuitions about multivariate distributions, especially those where the covariate structure is as complex as you have assumed, are often far from simulated results.
>>
>> I suggest you post some code and results.
>>
>> --
>> David.
>>
>>
>>>
>>> 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
>>>>> and provide commented, minimal, self-contained, reproducible code.
>>>>
>>>> David Winsemius
>>>> Alameda, CA, USA
>>>>
>>
>> David Winsemius
>> Alameda, CA, USA
>>

David Winsemius
Alameda, CA, USA

```