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Science Forum Index » Statistics - Math Forum » sums of squares of Student-t variates
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| Beliavsky |
 Posted: Thu May 01, 2008 12:41 pm |
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The sums of the squares of independent normal deviates follow the chi-
squared distribution. What about the sums of squares of independent
Student-t deviates -- does this distribution have a name? I can
simulate the distribution, of course.
Research has found that the Student-t distribution fits daily stock
market log returns much better than the normal distribution, because
daily returns have excess kurtosis. I am studying the distribution of
stock market variance and volatility over N-day samples. |
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| David Jones |
| Posted: Fri May 02, 2008 10:09 am |
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Beliavsky wrote:
Quote: The sums of the squares of independent normal deviates follow the chi-
squared distribution. What about the sums of squares of independent
Student-t deviates -- does this distribution have a name? I can
simulate the distribution, of course.
Research has found that the Student-t distribution fits daily stock
market log returns much better than the normal distribution, because
daily returns have excess kurtosis. I am studying the distribution of
stock market variance and volatility over N-day samples.
Johnson,Kotz & Balakrishnan - Continuous univariate distributions Vol2 - have some results for differences of two Student's ts, which is of course equivalent to the sum, but there aren't any simple explicit expressions. And of couse you want more than 2. If your degees of freedom are large enough for low order moments (up to 4) to exist then you might be able to work by find the exact 2nd and 4th cumulants for the sums you want, the finding a scaled Student t to match these cumulants.
David Jones |
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| Graham Jones |
| Posted: Fri May 02, 2008 11:33 am |
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"Beliavsky" <beliavsky@aol.com> wrote in message
news:cdbd63a3-27f8-4195-acb0-e18651c9ec18@w7g2000hsa.googlegroups.com...
Quote: The sums of the squares of independent normal deviates follow the chi-
squared distribution. What about the sums of squares of independent
Student-t deviates -- does this distribution have a name? I can
simulate the distribution, of course.
I think it is an F distribution, but I'm going by memory and don't have a
reference. Note that a t-distribution is a normal divided by an independent
rv Y where Y^2 ~ chi-square/dof, so you might use that to prove something.
(David: he wants *squares* of t-dist rvs)
Graham |
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| Beliavsky |
| Posted: Fri May 02, 2008 11:50 am |
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On May 2, 2:48 pm, "Graham Jones" <x...@x.x> wrote:
Quote: "Herman Rubin" <hru...@odds.stat.purdue.edu> wrote in message
news:fvfktv$21bs@odds.stat.purdue.edu...
In article <SoydnZ-0cZDB34bVnZ2dneKdnZydn...@bt.com>,
Graham Jones <x...@x.x> wrote:
I think it is an F distribution, but I'm going by memory and don't have a
reference. Note that a t-distribution is a normal divided by an
independent
rv Y where Y^2 ~ chi-square/dof, so you might use that to prove something.
I presume what is wanted is the distribution of a sum
of squares of independent t variables. What you state
might be good for the sum of squares of t variables
where the numerators are independent normal with the
same variance and the denominators are identical, but
otherwise not. The tails of the distribution are large
because of the denominators, not the numerators.
Ah yes. You've jogged my memory. The F distribution came up when I was
working with a multivariate t-distribution (defined as a multivariate normal
'divided' by a rv like Y) and assuming mean and covariance C of the
multivariate normal to be known. Then I wanted the distribution of the
Mahanalobis distance based on C, so I effectively had independent normal
rvs, not independent t rvs.
Having said that, I doubt that 'daily stock market log returns' are
independent, and the study of the 'distribution of stock market variance and
volatility over N-day samples' suggests my comment might still be relevant..
Thanks to you and the other people who responded. You are right about
the log returns not being independent, because empirically their
absolute values have positive serial correlation, which can be modeled
via GARCH with Student-t noise.
I can simulate from such a process to study the distribution of
realized stock market variance. |
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| Herman Rubin |
| Posted: Fri May 02, 2008 1:01 pm |
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In article <SoydnZ-0cZDB34bVnZ2dneKdnZydnZ2d@bt.com>,
Graham Jones <x@x.x> wrote:
Quote: "Beliavsky" <beliavsky@aol.com> wrote in message
news:cdbd63a3-27f8-4195-acb0-e18651c9ec18@w7g2000hsa.googlegroups.com...
The sums of the squares of independent normal deviates follow the chi-
squared distribution. What about the sums of squares of independent
Student-t deviates -- does this distribution have a name? I can
simulate the distribution, of course.
Quote: I think it is an F distribution, but I'm going by memory and don't have a
reference. Note that a t-distribution is a normal divided by an independent
rv Y where Y^2 ~ chi-square/dof, so you might use that to prove something.
I presume what is wanted is the distribution of a sum
of squares of independent t variables. What you state
might be good for the sum of squares of t variables
where the numerators are independent normal with the
same variance and the denominators are identical, but
otherwise not. The tails of the distribution are large
because of the denominators, not the numerators.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558 |
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| Graham Jones |
| Posted: Fri May 02, 2008 1:48 pm |
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"Herman Rubin" <hrubin@odds.stat.purdue.edu> wrote in message
news:fvfktv$21bs@odds.stat.purdue.edu...
Quote: In article <SoydnZ-0cZDB34bVnZ2dneKdnZydnZ2d@bt.com>,
Graham Jones <x@x.x> wrote:
I think it is an F distribution, but I'm going by memory and don't have a
reference. Note that a t-distribution is a normal divided by an
independent
rv Y where Y^2 ~ chi-square/dof, so you might use that to prove something.
I presume what is wanted is the distribution of a sum
of squares of independent t variables. What you state
might be good for the sum of squares of t variables
where the numerators are independent normal with the
same variance and the denominators are identical, but
otherwise not. The tails of the distribution are large
because of the denominators, not the numerators.
Ah yes. You've jogged my memory. The F distribution came up when I was
working with a multivariate t-distribution (defined as a multivariate normal
'divided' by a rv like Y) and assuming mean and covariance C of the
multivariate normal to be known. Then I wanted the distribution of the
Mahanalobis distance based on C, so I effectively had independent normal
rvs, not independent t rvs.
Having said that, I doubt that 'daily stock market log returns' are
independent, and the study of the 'distribution of stock market variance and
volatility over N-day samples' suggests my comment might still be relevant.
Graham |
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