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In this course we start by introducing order statistics and extremes. In particular, the density functions of any order statistic or any subset of them are given in closed form. Next, we indicate that no matter the parent distribution is, the limit distribution of the maximum belongs to the Levy-Jenkinson family, which includes only the families of Weibull, Gumbel and Frechet. The corresponding limit distributions for the case of minimum are given too. The cases of other order statistics are also analyzed and discussed. The use of probability papers to find the most adequate limit distribution is explained and justified. Exceedances are recommended instead of extremes, because of its better efficiency and the generalized Pareto distribution is analyzed. The case of dependence is shown to lead to the same limits distributions if the dependence is not too strong, and to other limit distribution if it is strong enough. One illustrative example of extreme waves is studied in some detail.
Me Gusta
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In this first block we introduce the course and provide a motivation for it, based on engineering design.
A course on extreme values
Enrique Castillo
Motivating a course on extreme values
We introduce the order statistics of a sample and the special order statistics minimum and maximum of the sample. We show how to obtain the joint distribution of any subset of order statistics, that includes the maximum, minimum and any particular order statistic or any subset of order statistics
Order Statistics
Limit distributions of extremes
We obtain the limit distributions for dependent random sequences and analyze when the limit coincides with the independence case. We also obtain the limit of not extreme order statistics and see the relation with non extreme cases
Limit distributions of dependent sequences
Limit distributions of order statistics
It is explained how probability papers are constructed and how can they be used to solve important problems in engineering. Exceedances are introduced and recommended as better than minima and maxima to identify the extreme values.
Probability Papers
Exceedances and return periods
In this block we present an interesting example in which the relevant variables to design a breakwater are modelled by a three-dimensional random variable including significant wave height, maximum wave height and period.
Examples of application to extreme values
Conclusions
Exercises
Bibliography
Here are all the slideshow from Extreme Values
Slideshows