Gatha Cognition®
Perception, Learning and Reasoning

#### Attraction Structure and Speed of Convergence

Statistical Theory of Extremes

45-62

Tail equivalent , Slow variation , Quantile function , Speed of convergence , Attraction coefficients

In this chapter results are presented that make it possible to analyse if a sequence of random variables is attracted for maxima to one of the three limiting distributions, for i.i.d. case, how to obtain some normalizing coefficients and how quick the convergence is. This involves necessary and sufficient conditions on the tail of the underlying distribution to be in the respective domain of attraction for maxima; moreover, sufficient conditions and miscellaneous examples are also provided, including normal, geometric, Poisson and lognormal models.

In applications, observations are often not i.i.d., but under some conditions, for large samples, the limiting distributions of maxima of a sequence of non-i.i.d. observations are still the same as in the i.i.d. case.

The quantile function is very important in this study and also in the problems of the right tail estimation using the largest values of a sample.

To obtain the Fréchet distribution as a limit we must have the right-end point infinite and to obtain the Weibull distribution as a limit we must have a finite right-end point; the Gumbel distribution can be attained either with finite or infinite right-end point.

The subject of tail equivalence of distributions is relevant to evaluate what can be the asymptotic behaviour of maxima of some distribution by substituting it with another one which is easier to manipulate.

If two distributions are tail equivalent for maxima, then if one of the distributions is attracted to some max-stable distribution the other distribution is also attracted to the same limit and with the same coefficients.

The way a sequence of distributions of maxima converge to its limit is a very important question: either it converges quickly to the limit and this limit can be used as an approximation to the real distribution, or the approach is slow and the limit, from the statistical standpoint, has little relev