Chapter 1
Univariate Extreme Value
Distributions
1.1
Historical Survey
Probabilistic extreme value theory is a curious and fascinating blend of an enormous variety of applications involving natural phenomena such as rainfall, floods, wind gusts, air
pollution, and corrosion, and delicate advanced mathematical results on point processes and
regularly varying functions. This area of research thus attracted initially the interests of
theoretical probabilists as well as engineers and hydrologists, and only relatively recently of
the mainstream statisticians. For a number of years it was closely related to the activities
of E. J. Gumbel, a colorful personality, whose life and activities were affected by pre-World
War II upheavals.
The following pages are addressed not only or primarily to professionals in the field of
statistical distributions and statistical inference but to that much larger audience which is
interested in the topics without willing or being able to devote more than a limited amount
of time to considering them.
Probabilistic extreme value theory, first of all, deals with the stochastic behaviour of
the maximum and the minimum of i.i.d. random variables. The distributional properties
of extremes (maximum and minimum), extreme and intermediate order statistics, and exceedances over (below) high (low) thresholds are determined by the upper and lower tails of
the underlying distribution.
Conversely, the tail of the underlying distribution function or functional parameters
thereof may be evaluated by means of statistical procedures based on extreme and intermediate order statistics or exceedances over high thresholds. Focussing our attention on the
tails has the advantage that certain parametric statistical models, specifically tailored for
that part of the distribution, can be introduced.
Historically, work on extreme value problems may be traced back to as early as 1709
when Nicolas Bernouilli discussed the mean largest distance from the origin given n points
lying at random on a straight line of a fixed length t [see Gumbel (1958)].
Extreme value theory has originated mainly from the needs of astronomers in utilizing
or rejecting outlying observations. The early papers by Fuller (1914) and Griffith (1920) on
the subject were specialized both in fields of applications and in methods of mathematical
analysis (see below). A systematic development of the general theory may be regarded as
1
2
Extreme Value Distributions
having started with the paper by von Bortkiewicz (1922) that dealt with the distribution
of range in random samples from a normal distribution. The importance of the paper by
Bortkiewicz is inherent by the fact that the concept of distribution of largest value was
clearly introduced in it for the first time. (In his classical book, E. J. Gumbel (1958) devotes
a chapter to the memory of L. von Bortkiewicz.) In the very next year von Mises (1923)
evaluated the expected value of this distribution, and Dodd (1923) calculated its median,
also discussing some nonnormal parent distributions. Of more direct relevance is a paper
by Fréchet (1927) in which asymptotic distributions of largest values are considered. In the
following year Fisher and Tippett (1928) published results of an independent inquiry into the
same problem. While Fréchet (1927) had identified one possible limit distribution for the
largest order statistic, Fisher and Tippett (1928) showed that extreme limit distributions
can only be one of three types. Tippett (1925) had earlier studied the exact cumulative
distribution function and moments of the largest order statistic and of the sample range
from a normal population. Von Mises (1936) presented some simple and useful sufficient
conditions for the weak convergence of the largest order statistic to each of the three types
of limit distributions given earlier by Fisher and Tippett (1928). We shall discuss von Mises’
conditions in a subsequent section. In 1943, Gnedenko presented a rigorous foundation
for the extreme value theory and provided necessary and sufficient conditions for the weak
convergence of the extreme order statistics.
Mejzler (1949), Marcus and Pinsky (1969) (unaware of Mejzler’s result) and de Haan
(1970) (1971) refined the work of Gnedenko. An important but much neglected work of Juncosa (1949) extends Gnedenko’s results to the case of not necessarily identically distributed
independent random variables. Although of strong theoretical interest, Juncosa’s results do
not seem to have much practical utility. The fact that asymptotic distributions of a very
general nature can occur does not furnish much guidance for practical applications.
The theoretical developments of the 1920s and mid 1930s were followed in the late 1930s
and 1940s by a number of papers dealing with practical applications of extreme value statistics in distributions of human lifetimes, radioactive emissions [Gumbel (1937a,b), strength
of materials [Weibull (1939)], flood analysis [Gumbel (1941, 1944, 1945, 1949a), Rantz and
Riggs (1949)], seismic analysis [Nordquist (1945)], and rainfall analysis [Potter (1949)] to
mention a few examples. From the application point of view, Gumbel made several significant contributions to the extreme value analysis; most of them are detailed in his booklength account of statistics of extremes [Gumbel (1958)]. See the sections on Applications
for more details.
Gumbel was the first to call the attention of engineers and statisticians to possible applications of the formal “extreme-value” theory to certain distributions which had previously
been treated empirically. The first type of problem treated in this manner in the USA had
to do with meteorological phenomena — annual flood flows, precipitation maxima, etc. This
occurred in 1941.
In essence, all the statistical models proposed in the study of fracture take as a starting
point Griffith’s theory (already alluded to above), which states that the difference between
the calculated strengths of materials and those actually observed resides in the fact that
there exist flaws in the body which weaken it.
The first writer to realize the connection between specimen strength and distribution of
extreme values seems to be F. T. Peirce (1926) of the British Cotton Industry Association.
The application of essentially the same ideas to the study of the strength of materials was
carried out by the well-known Swedish physicist and engineer, W. Weibull (1939).
Univariate Extreme Value Distributions
3
The Russian physicists, Frenkel and Kontorova (1943), were the next to study these
problems. Another important neglected early publication related to extreme value analysis
of the distribution of feasible strengths of rubbers is due to S. Kase (1953).
A comprehensive bibliography of literature on extreme value distributions and their applications can easily be constructed to contain over 1,000 items at the time of this writing
(1999). While this extensive literature serves as a testimony to the great vitality and applicability of the extreme value distributions and processes, it also unfortunately reflects on the
lack of coordination between researchers and the inevitable duplication (or even triplication)
of results appearing in a wide range of diverse publications.
There are several excellent books that deal with the asymptotic theory of extremes and
their statistical applications. We cite a few known to us (without in any way dispraising
those that are not mentioned). David (1981) and Arnold, Balakrishnan, and Nagaraja (1992)
provide a compact account of the asymptotic theory of extremes; Galambos (1978, 1987),
Resnick (1987), and Leadbetter, Lindgren, and Rootzén (1983) present elaborate treatments
of this topic. Reiss (1989) discussed various convergence concepts and rates of convergence
associated with extremes (and also with order statistics). Castillo (1988) has successfully
updated Gumbel (1958) and presented many statistical applications of extreme value theory with emphasis on engineering. Harter (1978) prepared an authoritative bibliography
of extreme value theory which is still of substantial scientific value. Beirlant, Teugels and
Vynekier (1996) provide a lucid practical analysis of extreme values with emphasis on actuarial applications.
1.2
The Three Types of Extreme Value Distributions
Extreme value distributions are usually considered to comprise the following three families:
Type 1, (Gumbel-type distribution):
Pr[X ≤ x] = exp[−e(x−µ)/σ ] .
(1.1)
Type 2, (Fréchet-type distribution):




0,
( −ξ )
Pr[X ≤ x] =
x
−
µ


,

 exp −
σ
x < µ,
Type 3, (Weibull-type distribution):

( ξ )

µ
−
x


,
 exp −
σ
Pr[X ≤ x] =



0
x≤µ
(1.2)
x ≥ µ.
(1.3)
x>µ
where µ, σ(> 0) and ξ(> 0) are parameters.
The corresponding distributions of (−X) are also called extreme value distributions.
(Observe that Fréchet and Weibull distributions are related by a simple change of sign.)
Of these families of distributions, type 1 is the most commonly referred to in discussions
of extreme values. Indeed some authors call (1.1) the extreme value distributions. In view of