Kurtosis and peakedness

[wrongsaidfred]

Hi David,

I know this may seem like an insignificant point, but I have read different things about peakedness and the student t distribution that do not seem to mesh.

We all know that there is excess kurtosis in a students t and that excess kurtosis approaches zero as the fd aproaches infinity.

I have read that the t distribution with 1 df is both flatter and has fatter tails that a normal distribution. I have also read that as the df increases, the students t becomes more peaked and the excess kurtosis approaches zero. What does not make sense is that in other readings, the kurtosis is almost defined by the "peakedness" of the distribution.

Long story short is that as df increases, the excess kurtosis decreases. How can it also be true that it becomes more peaked as this happens? It seems like more kurtosis=more peaked but this does not seem to be the case.

Thanks,
Mike

 

[David Harper CFA FRM]

Hi Mike,

It's a great observation.

First, I stopped (i think ~ 2 years ago) referring to kurtosis as a function of peakedness; while it may be true (i am not 100% convinced) that all distributions with kurtosis > 3, have higher peaks compared to normal, it is not always easy to do an apples to apples comparison with a standard normal pdf. And, i think it misses the point, anyway: kurtosis concerns us a function of the fourth moment and refers to higher density in the tails. (that this associates with a taller peak doesn't seem to be the point, to me).

Second, related, the student's t does have a higher peak, but it's never shown that way; e.g., http://en.wikipedia.org/wiki/Student's_t-distribution In wikipedia, the student's appears like you say, stubby moving up. But the "fallacy" is that they are comparing a standard normal with (unit) variance = 1.0 to a student's t with variance > 1.0; if you plotted the standard normal with the same variance, the studen't t would be taller and, as we know, with higher density tails. If we equalized variances, the studen't t would behave as you expect w.r.t. peakedness (see the difficulty? as we increase d.f., the excess kurtosis is tending to 0, but the variance is also changing).

so, i am happy to let kurtosis refer to tail density or tail heaviness (I don't like fat/thin because that leads to misinterpretations for some)

(fwiw, it is good to know that EACH of the sampling distributions [ student's t, chi^2, and F] converge to normal as df --> infinity, which implies their excess kurtosis tending toward zero)

Thanks, David

 

[wrongsaidfred]

Nice!

Thanks.

If memory serves correctly, didnt we come to the conclusion that an F distribution approaches a Normal distribution only when DF numerator=1 and DF denominator approaches infinity?

Thanks,
MIke

 

[David Harper CFA FRM]

Mike,

Per Gujarati property 2 (p 97) I have "2: Also, like the t and chi-square distributions, the F distribution approaches the normal distribution as k1 and k2, the d.f., become large (technically, infinite)."

(I'm not sure about df1 = 1 , df2 --> infinite, i thought maybe that's a steep downward slope not a bell curve ... not sure ....)

Thanks, David

 

[wrongsaidfred]

Thanks again. I guess what I was thinking is that F=Z^2 when numerator df=1 and denominator df is infinity, but that is a seperate idea.

While we are ont he idea of these relationships, I have a strange question. I know from Stock-Watson that if Z and X^2 are independent, Z/sqrt(X^2/m) has a students t distribution. Can anything be done algebracially to (for instance) find the value of a standard normal variable if we know the students t and X^2 that correspond to it?

If the answer is no, a simple no will suffice because I do not want to take up any more of your time. You have already answered a couple of tricky questions for me.

Thanks,
Mike