Can anyone give me a bell-curve shaped leptokurtic distribution?

[Grisgra]

That is to say, a distribution with thicker tails and a higher peak than your standard normal. (The reverse, a platykurtic distribution, would have a squatter peak and thinner tails). Need it for a poker sim I'm writing but you guys are probably where I should start /images/graemlins/smile.gif.

IMPORTANT: I need to be able to set the standard deviation of the distribution without too much difficulty, and we're talking a known mean here, not one that needs to be estimated (so I think trying to use a t-distribution, which *is* leptokurtic, is out).


-------
Kind of a cross-post, I apologize to all moderator-types, but it's a pure stat question for which the application to poker theory is indirect, so it ended up in both places. Not sure in which I'll have the best chance of success.

[Borodog]

Won't anything of the form exp(-|x/a|^n), where 1<n<2 work?

[Grisgra]

[ QUOTE ]
Won't anything of the form exp(-|x/a|^n), where 1<n<2 work?

[/ QUOTE ]

But is that going to look bell-curve'ish?

[Copernicus]

Its been a long time but I thought leptokurtosis was only found in distributions that have no known mean. Or is that only Paretian distributions?

[Borodog]

[ QUOTE ]
[ QUOTE ]
Won't anything of the form exp(-|x/a|^n), where 1<n<2 work?

[/ QUOTE ]

But is that going to look bell-curve'ish?

[/ QUOTE ]

Edit: Whoops - misread the question.

Yes, try it. Don't forget the absolute values.

[Grisgra]

[ QUOTE ]
Its been a long time but I thought leptokurtosis was only found in distributions that have no known mean. Or is that only Paretian distributions?

[/ QUOTE ]

nah, you can have a known mean no prob. In fact given the kind of distribution I'm looking for, it would simply be the median/mode.

Useful link:

http://www.riskglossary.com/link/kurtosis.htm

[BruceZ]

[ QUOTE ]
That is to say, a distribution with thicker tails and a higher peak than your standard normal. (The reverse, a platykurtic distribution, would have a squatter peak and thinner tails). Need it for a poker sim I'm writing but you guys are probably where I should start /images/graemlins/smile.gif.

IMPORTANT: I need to be able to set the standard deviation of the distribution without too much difficulty, and we're talking a known mean here, not one that needs to be estimated (so I think trying to use a t-distribution, which *is* leptokurtic, is out).

[/ QUOTE ]

You can use the t-distribution with n > 2 degrees of freedom as this has variance n/(n-2) and mean 0. It is only for n <= 2 that the variance does not exist. For n = 1, the mean does not exist either, and this is a Cauchy distribution. In fact, for n degrees of freedom, the first n-1 moments all exist, and the higher moments diverge. As n -> infinity the pdf converges to the pdf of a standard normal distribution.

[Grisgra]

[ QUOTE ]
You can use the t-distribution with n > 2 degrees of freedom as this has variance n/(n-2) and mean 0. It is only for n <= 2 that the variance does not exist. For n = 1, the mean does not exist either, and this is a Cauchy distribution. In fact, for n degrees of freedom, the first n-1 moments all exist, and the higher moments diverge. As n -> infinity the pdf converges to the pdf of a standard normal distribution.

[/ QUOTE ]

Thanks -- I'll give this a try, see if it gives me what I'm hoping for. I wasn't aware that the variance of the t-distribution was n/n-2, I think I can use that to get what I need.

What if I wanted to generate a platokurtic distribution? Any ideas? I was wondering if I could get different kurtosis values by playing around with normal mixtures, but I don't know much about that there stuff except that conceptually I think it might work.

[BruceZ]

The t-distribution does not have a higher peak than the standard normal for any value of n, so it might not suit your needs, but it does have thicker tails. According to the table at Mathworld (http://mathworld.wolfram.com/Kurtosis.html), it is leptokurtic for n >= 5 since this value gives it positive kurtosis "excess", meaning that it has more kurtosis than the standard normal which has kurtosis excess of 0 and kurtosis proper of 3. Apparently having a higher peak than the standard normal is not a necessary condition for being leptokurtic.

The standard normal has a peak of 1/sqrt(2*pi) =~ 0.3989. The t-distribution has a peak of GAMMA[(n+1)/2]/GAMMA(n/2)/sqrt(n*pi), which approaches 1/sqrt(2*pi) as n -> infinity:

<font class="small">Code:</font><hr /><pre>
n peak of t-distribution

1 0.318309884
2 0.353553388
3 0.367552594
4 0.374999997
5 0.379606687
6 0.382732769
7 0.384991448
8 0.386699018
9 0.388034906
10 0.389108381
100 0.397946184
300 0.398609965
</pre><hr />