I have read that a drawback with Edgeworth series expansion is that "... they can be inaccurate, especially in the tails, due to mainly two reasons: (1) They are obtained under a Taylor series around the mean. (2) They guarantee (asymptotically) an absolute error, not a relative one. This is an issue when one wants to approximate very small quantities, for which the absolute error might be small, but the relative error important."
So my resulting question is if there are any attractive alternative ways of approximating stochastic variables with some corresponding method that still is useful in the tails of the distribution, and does not (for example) result in negative probabilities (which is mentioned as another drawback with the approach).
I am honestly no expert on this, but I have a feeling that this paper may be of interest: http://www.mit.edu/~vchern/papers/EdgeworthRearranged-posted.pdf
Dear Lars,
I wouldn't recommend any normal-like function to model distributions
which may have hard tails. These types of distributions should be modelled
using Extreme Value Distributions (EVD). If you have yearly maximum data the
recommended EVD is the Generalised Extreme Value distribution, for maximum
daily data the Generalised Pareto Distribution works better. A good first
reading on the subject of Extreme Value Distributions is:
Coles, S. 2001. An Introduction to Statistical Modeling of Extreme Values.
Springer, London.
Hope it helps,
Augusto
Another criticism of the Edgeworth (and Gram-Charlier) series is that they do not always (usually?) converge--they are similar to asymptotic expansions).
Instead of expanding the density, you could expand a function that transforms it to normality. The Cornish-Fisher expansions (for such a transformation and its inverse) achieve this. These are based on the Edgeworth expansion. They might also fail to converge but still give approximations.
- Badges
- Science topic
- Similar topics
- Mathematics
- Statistics
- Probability
- Probability Theory