
\section{Introduction}

This book provides an introduction to the
calculation of extreme value statistics for measured or simulated data.
''Extreme'' here means ''the largest'', interpreted in a way that
follows from the context. As opposed to books on asymptotic extreme value statistics, the focus is also on methods specifically developed to work for real-life data. A consequence of this, is that the book contains much less theoretical issues about the asymptotic properties of extreme value statistics than is usual. However, the most important elements from the asymptotic extreme value statistics will be discussed, since they are still widely used in practical applications.  


Although two of the asymptotic methods described in this book have been used
extensively over several decades for prediction of the extreme value
statistics of many natural phenomena, the prerequisites for their application are often not satisfied, and in some cases, not even approximately. Under such circumstances, there would appear to be a problem. It is this situation that will be highlighted in this chapter.



\section{A Brief Summary of Status, Problems and Challenges}


Statistical distributions of the extreme values of large samples of data were derived almost one hundred years ago by \cite{fiti28}, cf. also \cite{frec27,gned43,deha70}.  The main prerequisite for the existence of the derived results were that the data could be considered as outcomes of independent and identically distributed random variables. As it turned out, in non-degenerate cases there are only three possible types of limiting extreme value distributions with increasing amounts of data. It means that these results are asymptotic, as the technical term goes. On the positive side, the fact that we know explicitly what the possible distributions look like, even if only in the limit of large samples increasing indefinitely, is very satisfactory. And there are criteria that can tell us which type of distribution applies if the underlying distribution of the data is known \citep{leliro83}. However, on the negative side, it is not possible to know to what extent one of the three types of limiting distributions actually applies to a real-life case with only a limited amount of data, even though there may be reasons to expect that the true extreme value distribution should not deviate too much from one of the limiting forms. Unfortunately, there are no useful convergence results that are precise enough to really help us decide quantitatively on this issue. Still, the common practice has been to assume an appropriate limiting form as the extreme value distribution to use. This can easily be understood from the simple fact that the limiting distributions are known explicitly, while the exact extreme value distributions inherent in the data, are largely unknown. The procedure to identify the appropriate limiting distribution is to optimize the fit of the extreme values derived from the observed data to the asymptotic forms. Typically, the extreme values from the data are taken as the maxima of specified blocks of data, e.g. annual maxima.


The three asymptotic types of extreme value distributions are essentially characterized by the value of one parameter, $\gamma$ say, called the shape parameter\index{shape parameter}. As will be seen later, the most important case for us in this book is when $\gamma = 0$. This is called a Type I, or Gumbel, distribution. For positive values of $\gamma$, Type II, or Fr\'{e}chet, distributions are obtained, while for negative values of $\gamma$, the distributions are of Type III, or Weibull (for maxima). As it turns out, all three distribution types may be expressed in terms of one parametric form called the generalized extreme value (GEV) distribution. A standard recommendation is then to use the GEV parametric form for the sake of optimized fitting of the obtained extreme value sample. There is, however, one serious flaw with this procedure. The extreme value sample being extracted from limited amounts of data, are hardly a sample from an asymptotic distribution. Hence, one cannot expect that the estimated parameters will point to the correct asymptotic distribution. This is an issue of importance for extrapolation to out-of-sample long return period levels. For example, a practical task may be to say something about a 100 year return period level on the basis of 25 years of measured data. Then the correct asymptotic distribution is of paramount importance  because the different types of extreme value distribution may lead to quite different extrapolation results. An additional issue is, of course, that with limited amounts of data follows considerable uncertainty on the estimated quantities. It may, in fact, happen that the estimated value of $\gamma$ is slightly negative, pointing to a Type III distribution, but with the confidence interval accounted for, also $\gamma = 0$, or even $\gamma > 0$ are possible candidates for the value of $\gamma$. Hence, all three types of extreme value distributions seems to be possible alternatives in such a case. Since these asymptotic distributions have very different behaviour when extrapolated to high quantile values, the previous comments on the importance of this aspect, would often necessitate a more careful analysis of the situation to decide which asymptotic distribution to apply. 

The peaks-over-threshold (POT) method for extreme value analysis will be discussed to some extent in this book. This method is also based on asymptotics. The data extracted for its use, are the exceedances above high thresholds. Asymptotically, these data are assumed to follow a generalized Pareto (GP) distribution, which is then the equivalent of the GEV for the block maxima method. The POT method also has three classes of distributions, again characterized by the $\gamma$ parameter. For example, the singular case $\gamma = 0$ corresponds to the exponential distribution. It is a rather popular method, mainly because it uses more of the data for inference. Unfortunately, it has certain deficiencies, which will be highlighted in this book. 


There is an important and interesting observation to be made at this initial stage of our exposition of extreme value statistics. As already been stated, for all negative values of the shape parameter $\gamma$, the Type III class of extreme value distributions apply, while for all positive values of $\gamma$, it is the Type II class of extreme value distributions that is obtained. This would seem to indicate that there are two huge classes of extreme value distributions that would tend to make the singular case $\gamma = 0$ a rather special and maybe uninteresting case. The fact of the matter is quite the opposite. For almost all environmental processes that will be dealt with in this book, it is the Gumbel distribution that has prevailed as the correct asymptotic extreme value distribution. There has over the years been some suggestions to the other types as well, but these have almost all been finally rejected in the face of overwhelming evidence for Type~I distributions. Of course, it is impossible to fully answer the fundamental question: To what extent do our statistical models apply to real-life data? But so far, it seems that these statistical methods work rather well on such data, but being overconfident in these methods is perhaps an unwarranted position to take. 


One important reason that the singular asymptotic Gumbel case is so important in practice, is that from the perspective of a sub-asymptotic world, the picture of the size of the extreme value distribution classes looks very different. When only limited amounts of data are available, the asymptotic limiting distributions, strictly speaking, do not apply, except in very special cases. Hence, we are in a sub-asymptotic situation. As will be seen in large parts of this book, there is a huge class of extreme value distributions that apply to a range of different problems, which all end up at the Gumbel distribution asymptotically. So, the apparent singularity of the Gumbel case is an artefact of the asymptotic limiting process, and does not reflect the situation in the sub-asymptotic regime. 


In an effort to resolve the inconsistency between real-life data and asymptotic distributions, a new method has been developed that is based on the concept of the average conditional exceedance rate (ACER). The method proceeds by establishing a cascade of empirical, non-parametric distribution functions that converge to the extreme value distribution inherent in the data. The advantages of the method is that no assumptions about independent and stationary data have to be made. For example, seasonal variations of the data do not require special modelling. The method also has a unique diagnostic feature in how it displays the effect of dependence between the data on the extreme value distribution. This may be of significance for the choice of which data can be included in the analysis. The ACER method will be discussed in detail in this book.


Whatever method of extreme value statistics is chosen for the analysis of the available data, the goal is almost always to predict extreme values with return periods larger, and often much larger, than the period of data collection. This inevitably requires extrapolation techniques to be used. The seemingly stochastic mechanism generating the sampled data is often sufficiently well understood to support the assumption of the validity of extrapolation. Unfortunately, this may not always be the situation. Ideally, in such cases, the predicted extreme values obtained by extrapolation should then be accompanied by a cautionary note. However, this is rarely done, simply because more credible alternatives for the prediction process are not available.


The extrapolation procedure is, in general, based on obtaining estimates of the parameters that determine the extreme value distribution type adopted for the data at hand. If an asymptotic approach is used, the GEV distribution is often preferred for parameter estimation in the case of the block maxima method, or the GP distribution for the POT method. Since these are parametrized forms covering all three types of asymptotic extreme value distributions, it is often recommended to use these forms, allowing the data to determine which type of extreme value distribution to use. As already mentioned, such a procedure may not always be a good idea. Also for the ACER method, a parametrized family of functions is proposed for the purpose of extrapolation, which is tailored to reflect the sub-asymptotic character of the data.     


The parameter estimates calculated for the examples in this book, are based on either the method of moments or the maximum likelihood method in the case of the GEV or the GP distributions. For the ACER method, the optimized fitting is obtained by using a Levenberg-Marquardt method on an objective function expressed as a weighted mean square deviation measure between the empirical and the proposed parametric ACER functions on the log level. Uncertainty quantification is also a very important aspect of any statistical inference. In this book, the use of bootstrapping will serve to illustrate this issue, since it has some attractive properties.
           