Extreme value theory

[nanaba]

• Section 2.5 says 'heavy tail distributions whose higher moments can be infinite are of the Frechet type'. Now The Frechet type indicates a heavy tail and Pareto being the underlying distribution is one of them.
• But on the other hand, section 4.1 says that if the kth moment exists for all values of k, that indicates a light tail, f.e, the kth moment for a Pareto distribution only exists if k<a, which means a thicker tail.

Now I am a bit confused in whether or not having infinite moments or not indicates thicker/thinner tail. For me it sounds that the two paragraphs contradict each other.
Please advise on my confusion.

 

[Andrew Martin]

Hi Nanaba

The general result is that the fewer the number of moments that exist (ie are finite), then the heavier the tail of the distribution. The paragraphs are consistent with each other, both stating this result. Perhaps the confusion is around the statement that the higher moments are infinite? This is what is being referred to in the second paragraph when it is stated that not all moments exist (ie not all are finite).

So, for example, all the moments are finite for the exponential distribution, which we can think of as having a lighter tail. Not all moments are finite for the Pareto distribution, as per the second paragraph, and so we can think of this distribution as having a heavier tail.

This is all to do with how we calculate the moments, consider a distribution defined for x > 0:

\( E[X^k] = \int_{0}^{\infty} x^k f_X(x) dx \)

As we consider larger and larger values of k, x^k gets larger and larger as x gets larger in magnitude. If \(f_X(x)\) doesn't decrease down towards 0 quickly enough as x gets larger to offset this, then the integral won't converge and moments won't exist past a certain value of k.

If the pdf drops down towards 0 quickly enough, then all the moments exist, as is the case for the exponential distribution.

Hope this helps!

Andy

 

[nanaba]

Hi Andy,

thanks so much for this very clear depiction.