[R] SLLN for loggamma fails for 2 order moments

[G. Jay Kerns]

Dear Michael,

The SLLN hasn't failed.  :-)    In this particular example, it looks
like the sample means are simply converging at a rate which is slower
than desired.

Unfortunately, your particular loggamma doesn't have a finite fourth
moment; so you don't have the usual (worst case) convergence rate
proportional to sqrt(log (log(n)) / n)  by the law of the iterated
logarithm.  In fact, your loggamma has only finite moments less than
2.6... see below.

Again, the SLLN still holds.  But convergence is slow.  I have
double-checked and the source code for both mlgamma AND rlgamma in the
actuar package are correct.

There may be other people who can identify numerical issues that slow
down the convergence rate even more.

Best,
Jay

P.S.  the MGF of gamma is all you need for the moments of loggamma...

E(X^k) = (1 - k / 2.6)^(-25)



On Sat, Jun 14, 2008 at 4:54 AM, michaelhk82 <michaelhk82 at yahoo.dk> wrote:
>
> Hi.
> I was trying to calculate 2. order moment for a loggamma sample.
> Also, I checked it with the function mlgamma from the package actuar, which
> calculates moments based on loggamma parameters.
> Se below - I get big deviations. Numerical integration suggests that mlgamma
> gets it right. What fails
> in my attempt to use SLLN to estimate the 2. order moments?
>
> Code:
>
> sum(rlgamma(10^6,25,2.6)^2)/10^6 #Estimation by SLLN
> [1] 1.964011e+13
>> mlgamma(2,25,2.6)                      #2.order moment by mlgamma
> [1] 8.328225e+15
>
> The difference is more than a factor 100!
> I have tried to run the code several times but I never manage to exceed the
> value of mlgamma - that is, by SLLN I keep underestimating the 2. order
> moment.
> (and tried with larger samples as well, but my cpu wont accept sample larger
> than 10^7).
>
> Does anyone have an idea of what is happening?
>
> Michael
> --
> View this message in context: http://www.nabble.com/SLLN-for-loggamma-fails-for-2-order-moments-tp17837233p17837233.html
> Sent from the R help mailing list archive at Nabble.com.
>
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>



-- 


***************************************************
G. Jay Kerns, Ph.D.
Assistant Professor / Statistics Coordinator
Department of Mathematics & Statistics
Youngstown State University
Youngstown, OH 44555-0002 USA
Office: 1035 Cushwa Hall
Phone: (330) 941-3310 Office (voice mail)
-3302 Department
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E-mail: gkerns at ysu.edu
http://www.cc.ysu.edu/~gjkerns/