Book Open Access
Statistical Theory of Extremes
- Academia das Ciências de Lisboa (Lisbon Academy of Sciences), Lisbon, Portugal.
About this book
José Tiago da Fonseca Oliveira (1928-1992) wrote about 'Statistical Theory of Extremes'. After a slow development since the 1930’s, this theory developed very quickly after, namely by the work of Gumbel (1958). The aim of the present book is to integrate, in one text, the state of knowledge of essential methodology for univariate and multivariate extremes and some stochastic processes, based on the fundamental probabilistic results.
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Contents
- Preface
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pp. I-III Open access
Preface
José Tiago de Fonseca Oliveira
The Statistical Extremes Theory, after a slow development for half a century, has been developing very quickly in the last ten years; it may be said to be an exploding area. Although the probabilistic aspects as well as the stochastic processes of extremes have recently been dealt with, statistical decision methods have not actually been described and integrated in a book, even though the area has
10.21523/gcb1.1701 1,151 views 2017
- Part-1: Probabilistic Patterns of Univariate Statistical Extremes
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pp. 1-10 Open access
Introduction
José Tiago de Fonseca Oliveira
Statistical Theory of Extremes deals with analysis and predict the aspects of natural phenomena using univariate or multivariate sampled data. Natural phenomena, as floods, large fire claims, heavy rains, gusts of winds, and large waves analysed using suitable or available samples of random sequences or of stochastic processes. Interplay between order statistics and exceedances. Exact distribution of order statistics. Asymptotic behaviour of the sample quantiles, i.e. of central order statistic, as an approximation tool for efficient estimation of “extreme” quantiles of the design of structures, evaluation of extreme conditions or forecasting catastrophes. The book deals with behaviour of maxima with conversion to minima results immediate.
10.21523/gcb1.1702 1,208 views 2017
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pp. 11-44 Open access
Asymptotics of Univariate Extremes
José Tiago de Fonseca Oliveira
Here it is stated the basis for the development of the whole book: finite behaviour of sample maxima and minima and behaviour of extremes in large samples; asymptotic distributions of extremes. The laws of large numbers (LLN) for extremes and samples; the additive and the multiplicative laws of large numbers (ALLN and MLLN); right-tail and left-tail behaviour of a distribution function and the Extremal Limit Theorem; domains of attraction for maxima and for minima; max-stability for the class of von Mises-Jenkinson limiting distributions; joint behaviour of sample minimum and sample maximum and the asymptotic distributions of the m-th extremes.
10.21523/gcb1.1703 1,326 views 2017
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pp. 45-62 Open access
Attraction Structure and Speed of Convergence
José Tiago de Fonseca Oliveira
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.
10.21523/gcb1.1704 2,436 views 2017
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pp. 63-64 Open access
Annex-1: On The “Duality” between Extremes and Sums
José Tiago de Fonseca Oliveira
For simplicity we will deal with some “duality” between sums (or averages) and maxima, the translation to minima being obvious from the relation \(\mathrm{ \mathrm{ \begin{array}{c} \mathrm{ n} \\ \mathrm{ { min} }\\\mathrm{1 } \end{array} }\{ X_{i} \} =-\mathrm{ \mathrm{ \begin{array}{c} \mathrm{ n} \\ \mathrm{ { max} }\\\mathrm{1 } \end{array} }\{ -X_{i} \}}}\). The “duality” is expressed by the
10.21523/gcb1.1705 1,605 views 2017
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pp. 65-66 Open access
Annex-2: On the Khintchine’s Convergence of Types Theorem
José Tiago de Fonseca Oliveira
As this theorem has been referred to and will be important to the proof of asymptotic univariate results — and to subsequent generalizations — we will give a proof of it. Let \(\mathrm{ \{ Vk \left( x \right) \} }\) be any sequence of distribution functions and suppose that for two sets \(\mathrm{ \{ ( \lambda _{k}, \delta _{k}) , \delta _{k}>0 \} }\) and \(\mathrm{ \{ \left( \lambda _{k}^{*}, \de
10.21523/gcb1.1706 1,519 views 2017
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pp. 67-69 Open access
Annex-3: On the δ-Method
José Tiago de Fonseca Oliveira
Essentially the \(\delta\)-method is based on the fact that if \(\mathrm{ \{ X_{k} \} }\) and \(\mathrm{ \{ Y_{k} \} }\) are sequences of random variables such that either \(\mathrm{ X_{k}-Y_{k}{\mathrm{\ }\stackrel{\mathrm{P}}{\rightarrow}}0 }\) or \(\mathrm{ X_{k}/Y_{k}{\mathrm{\ }\stackrel{\mathrm{P}}{\rightarrow}}1 }\) and \(\mathrm{ \{ X_{k} \} }\) has a limiting proper and non-degenerate dis
10.21523/gcb1.1707 1,509 views 2017
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pp. 70-77 Open access
Exercises
José Tiago de Fonseca Oliveira
1.1 Show that \(\mathrm { ( \begin{array}{c}\mathrm{ k} \\ \mathrm{ min} \\ \mathrm{ 1 } \end{array} X_i){\mathrm{\ }\stackrel{\mathrm{P}}{\rightarrow}}\underline{w} }\). 1.2 Study the Cauchy distribution for the LLN. 1.3 Prove the conditions for the validity of MLLN. 1.4 Consider a sample of \(\mathrm{ n }\) observations \(\mathrm{ \{ X_{i} \} }\) and a possible future sample of \(\mathrm{ m }\)
10.21523/gcb1.1708 1,635 views 2017
- Part-2: Statistics for Univariate Extremes
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pp. 78-92 Open access
A Quick Exploration of Extreme Data
José Tiago de Fonseca Oliveira
The chapter is devoted to quick exploration techniques on how statistically decide for one of the three models, Fréchet, Gumbel or Weibull, assuming a distribution that is one of these three forms. First, it is considered a model from previous knowledge. Afterwards, the technique of the probability paper for Gumbel distribution is explored for quick statistical choice between Weibull, Gumbel and Fréchet distribution. A simple statistical choice between the extreme models is presented (trilemma). Moreover, short illustrating examples of quick exploration of data are considered.
10.21523/gcb1.1709 1,652 views 2017
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pp. 93-120 Open access
Statistical Analysis of the Gumbel Model
José Tiago de Fonseca Oliveira
It is assumed that the samples of maxima come from a Gumbel distribution. Gumbel characteristics are studied: moment generating and characteristic functions, mean value, variance, skewness, kurtosis, quantiles and mode. Return periods and return levels are considered. Statistical inference for location/scale parameters: Maximum Likelihood Estimators (MLE), Moments Estimators (ME), Best Linear Unbiased Estimators (BLUE), Best Linear Invariant Estimators (BLIE), confidence interval, tests of hypotheses, point and interval prediction, tolerance intervals and overpassing probability of some threshold. A worked example on maximal yearly precipitations is presented.
10.21523/gcb1.1710 2,163 views 2017
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pp. 121-129 Open access
Statistics for Fréchet Distribution
José Tiago de Fonseca Oliveira
The Fréchet distribution for maxima is studied. Convenient transformations, leading to standard exponential, Gumbel and Weibull of minima, are given. From ML equations for dispersion and shape parameters, asymptotic normality is presented. Close to this, the asymptotic confidence region for the pair is obtained and similar results are presented for the quantile. The characteristics of the Fréchet distribution, moments and moment generating function are mentioned. Statistical decision for the 3-parameters case, estimation, testing, and point prediction are considered. In particular, MLE for location, dispersion and shape are given, solving ML equations, with first approximations using a quick tool from Gumbel. The overpassing probability for a level and a worked example close this section.
10.21523/gcb1.1711 1,514 views 2017
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pp. 130-157 Open access
Statistical Decision for Weibull Distribution for Minima
José Tiago de Fonseca Oliveira
The Weibull distribution for minima is explained. Weibull random variable (for minima) has a finite location parameter. The Weibull distribution deals with modeling of failure and helps to determine ‘time-to-failure’. Characteristics are presented from definition of distribution function. Statistical decision for the 3-parameters case, estimation, testing and point prediction are discussed. Prediction difficulties are depending on the interplay of the location and shape parameter of the distribution.
10.21523/gcb1.1712 1,573 views 2017
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pp. 158-173 Open access
Analitic Statistical Choice for Univariate Extremes
José Tiago de Fonseca Oliveira
Method of statistical choice between Weibull, Gumbel and Fréchet distribution is described. Locally optimal approach and variational approach are discussed to illuminate the problem from two points of view. Analysis begins with samples of reduced values and extended for generalization. The variational approach is coincident with the locally optimal approach. However, there does not exist an optimal procedure for statistical choice based on \(Q_n\) .
10.21523/gcb1.1713 1,489 views 2017
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pp. 174-200 Open access
Statistical Analysis With Partial Information
José Tiago de Fonseca Oliveira
The chapter deals with some sparse results in which the samples are not complete. The analysis concerned with drawing inferences about quantities associated with right tail of the real F(.) when n is large, using asymptotic extreme value theory. The corresponding quantiles for the lower tail can be dealt with by the usual transformation from maxima to minima. Largest observations and subsamples maxima are compared. Subsample maxima method shows definite superiority. Estimation using two or three quantiles, estimations of the parameters using block partitions of the samples and estimations using exceedances over thresholds are also discussed.
10.21523/gcb1.1714 1,736 views 2017
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pp. 201-219 Open access
Annex-4
José Tiago de Fonseca Oliveira
The estimation of the tail of a distribution, using extreme order statistics, sometimes referred as a possibility, has been approached in different ways. We will describe the essential features of these, with some comments and a comparison between them as well as some remarks about some open questions; we have drawn heavily on the review paper by Tiago de Oliveira (1991). As there is complete dual
10.21523/gcb1.1715 1,556 views 2017
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pp. 220-228 Open access
Exercises
José Tiago de Fonseca Oliveira
2.1 If \(\mathrm{ Z=max ( X-a,Y-b ) }\) with \(\mathrm{ ( X,Y ) }\) independent with the reduced Gumbel distribution as well as \(\mathrm{ z }\), show that \(\mathrm{ Prob \{ X-a \leq Z \leq z \} =e^{-b}~ \Lambda ( z ) }\). 2.2 Consider one of the sets of data given in the book (or elsewhere); verify using the probability paper if they can be fitted to the Gumbel, Weibull or Fréchet distributions;
10.21523/gcb1.1716 2,272 views 2017
- Part-3: Multivariate Extremes
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pp. 229-268 Open access
Bivariate Extremes: Probabilistic Results
José Tiago de Fonseca Oliveira
An asymptotic distribution of bivariate extremes is studied to obtain the asymptotic probabilistic behaviour and to provide bivariate models of (asymptotic) extremes. The distribution of bivariate extremes and the limiting distribution of maxima are discussed with complementary basic, correlation and regression results. The classical correlation coefficient is linearly invariant. Orthogonal polynomials with respect to \(\mathrm{ \Lambda(X) }\) is used to improve the regression lines.
10.21523/gcb1.1717 1,614 views 2017
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pp. 269-300 Open access
Bivariate Extremes: Models and Statistical Decision
José Tiago de Fonseca Oliveira
Large samples, \(\mathrm{ F^{n}( x,y ) }\) is replaced by an asymptotic approximation for statistical decision of extremes. The discussions decide the fitness and accuracy of those approximations. The differentiable models, non-differentiable models of statistical decision, intrinsic estimation of the dependence functions, ready-to-wear model and estimations of the association between the margins with some examples are discussed in this chapter. Differentiable models, the logistic one gives a good fit but shows the logistic and the natural models can be very close and difficult to separate for moderate samples.
10.21523/gcb1.1718 1,604 views 2017
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pp. 301-310 Open access
Multivariate Extremes
José Tiago de Fonseca Oliveira
The approach to multivariate extreme distributions is similar to that outlined previously for bivariate extreme distributions. The prosbabilistic theory of multivariate extremes and generalizations of bivariate models are discussed with some notes on linear regression. The models are important as the ways to predict future, possibly dangerous events like floods, high tides, gusts of winds, etc.
10.21523/gcb1.1719 1,693 views 2017
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pp. 311-312 Open access
Annex-5: On the Quadrants Method
José Tiago de Fonseca Oliveira
Consider a sample of \(\mathrm n\) i.i.d. random pairs \(\mathrm {\{(x_{1}, y_{1}),\dots, (x_n, y_n) \}}\) whose distribution function is \(\mathrm {F ( x,y \vert \theta ) }\), where \(\mathrm { \theta }\) is a (unique) dependence parameter and the margins \(\mathrm { F ( x, +\infty \vert \theta) }\) and \(\mathrm { F ( +\infty, \vert \theta) }\) are parameter-free. Let us denote by \(\mathrm { \x
10.21523/gcb1.1720 1,551 views 2017
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pp. 313-318 Open access
Exercises
José Tiago de Fonseca Oliveira
3.1 Obtain the inequality \(\mathrm{ \frac{max ( 1,e^{w} ) }{1+e^{w}} \leq k ( w ) \leq 1 }\) or \(\mathrm{max \left( u,1-u \right) \leq \Lambda \left( u \right) \leq 1 }\) from the Boole-Frechet inequality and a passage to the limit. 3.2 Show that the sets \(\mathrm{ \{ k \left( w \right) \} }\) or \(\mathrm{ \{ A \left( u \right) \} }\) are convex, closed and symmetrical (corresponding to exchan
10.21523/gcb1.1721 1,676 views 2017
- Part-4: Stochastic Sequences and Processes of Extremes
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pp. 319-332 Open access
Extremes in Random Sequences and Stochastic Processes
José Tiago de Fonseca Oliveira
The chapter analyses the development in time of some physical system subject to random influences, by means of a discrete or continuously registering instrument. The behaviour of maxima in random sequences and stochastic processes having as a target to obtain, under convenient conditions, the same limiting distributions is discussed in this chapter. Extremes in stochastic sequences, global extremes of Gaussian stationary processes, global extremes in Wiener-Lévy processes and local extremes in Gaussian stationary processes are given with some results. Some scattered examples of extreme value properties for other kinds of stochastic processes are also discussed.
10.21523/gcb1.1722 2,081 views 2017
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pp. 333-353 Open access
Extremal Sequences and Processes: Basics and Statistics
José Tiago de Fonseca Oliveira
This chapter reveals a definition of extremal (maxima) random sequences and stochastic processes. Extremal sequences are the usual sequences of maxima. It can be obtained as stochastic processes of independent extremes with extremal margins. Extremal sequences and processes (distribution and properties), limiting characterizations, second order properties of extremal processes, jumps of an extremal process, logarithmization of time and the ergodic theorem for extremal processes, parameterization of extremal processes and statistics for extremal sequences and processes are discussed with explanatory examples.
10.21523/gcb1.1723 1,746 views 2017
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pp. 354-379 Open access
Other Processes of Extremes: Basics and Statistics
José Tiago de Fonseca Oliveira
Properties of the extreme-Markovian-stationary (EMS) sequences analogous are introduced in this chapter. Statistical decision for EMS sequences and processes and Extreme-Markovian-evolutionary (EME) sequences with some remarks on statistical decision are given. Sliding extreme (SE) sequences and statistical decision for SE sequences are also given with applications for earthquake analysis and modelling.
10.21523/gcb1.17024 1,572 views 2017
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pp. 380-381 Open access
Annex-6: On The Choice of Stochastic Processes Models
José Tiago de Fonseca Oliveira
Up to now we have discussed some quick estimation, not necessarily optimal, if and when some model was assumed. It may be useful to take a look at the choice of the models, even if we do not use sound procedures, but bearing in mind the theory of regression. Let us recapitulate some second-order features of sequences of extremes, \(\mathrm{ X_{n}= \lambda + \delta ~Z_{n} }\).We have, where \(\math
10.21523/gcb1.17025 1,562 views 2017
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pp. 382-384 Open access
Exercises
José Tiago de Fonseca Oliveira
4.1 Consider the EMS sequence \(\mathrm{ \{ Z_{k} \} }\) . Obtain the expression of \(\mathrm{ Prob \{ min \left( Z_{1},\dots,Z_{k} \right) \geq z \} =Q_{k} \left( z,z \vert \theta \right) }\) where \(\mathrm{ Q_{k} \left( x,y \vert \theta \right) =Prob \{ \mathrm{ \begin{array}{c} { } \mathrm{k-1 } \\ \mathrm{ min } \\ \mathrm{1} \end{array} } \left( Z_{1},\dots,Z_{k-1} \right) \geq x,Z_{k} \geq
10.21523/gcb1.17026 1,098 views 2017
- Part-5: Complements
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pp. 385-389 Open access
GEA-General Extremes Analysis: A guide in natural language
José Tiago de Fonseca Oliveira
This chapter reviews the step-by-step writing of a program for the computer in the appropriate computer language. The model explained in four steps: Statistical Choice of Models (SCM), Statistical Analysis of Models (SAM), Statistical Decision for Multivariate Models (SDM) and Stochastic Processes Analysis (SPA). The exact distribution of extremes is approximated by one of the asymptotic distributions.
10.21523/gcb1.1727 1,606 views 2017
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pp. 390-399 Open access
Some Case Studies
José Tiago de Fonseca Oliveira
This chapter deals with the statistical analysis of data considered in the parts: univariate extremes data and multivariate extremes data related to stochastic processes and sequence of extremes connected to multivariate extremes data. The case studies of maximum wind speed data in Lisbon, flood discharges of the North Saskachevan River at Edmonton and flood discharges of the Fox River at Berlin and Wrightstown are analysed.
10.21523/gcb1.1728 1,589 views 2017
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pp. 400-401 Open access
Exercises
José Tiago de Fonseca Oliveira
5.1 Study the data given in Sneyers (1977) for early maximal precipitations in mm at Uccle for durations of 20 min, 2h and 6h (samples of 17 observations) Table 1 . ##table|95## 5.2 Study some of the data contained in the references to Part 5 (and in particular the lag-correlation) as well as data contained in Parts 2, 3 and 4. 5.3 For Sneyers (1977) data (Chapter 5) compare the estimators obtaine
10.21523/gcb1.1729 1,221 views 2017
Bibliographic information
- Publisher
- Gatha Cognition
- Published
- 6 Jan 2017
- ISBN
- 978-81-932622-3-8
- Language
- English
- Managing Editor
- Professor. José Carlos Brandão Tiago de Oliveira
1) UNESCO Chair in Intangible Heritage and Traditional Know-how: Linking Heritage, University of Évora, Portugal ; and 2) Center of Philosophy of Science, Faculty of Sciences, University of Lisbon, Lisbon, Portugal - Chapters
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