Classical Extreme Value Theory Research Articles

Extreme Values of Queue Lengths in M/G/1 and GI/M/1 Systems

We study the limiting behavior of maximum queue lengths in the M/G/1 and GI/M/1 service systems. When the systems are positive recurrent, the distributions of their maximum queue lengths, under standard linear normalizations, either do not converge or they converge to degenerate limits. Consequently, one cannot use classical extreme value theory to characterize their limiting behavior. We show, however, that by varying the system parameters in a certain way as the time interval grows, these maxima do indeed have three possible limit distributions. Two of them are classical extreme value distributions and the third one is a new distribution. The latter distribution is the best one for practical approximations.

The Effect of Seasonal Variation and Serial Correlation on the Extreme Value Distribution of Rainfall Data

Abstract For practical applications both the parent distribution of rainfall intensifies and the distribution of their annual maxima are of interest. The relation between these two distributions cannot be obtained from classical extreme value theory because of seasonal variation and serial correlation in the data. Mathematical results for the distribution of maxima in mdependent sequences (e.g., an mth order moving average process) are given to illustrate the effect of local dependence on the extreme value distribution. High-level exceedances occur in clusters when there is strong local dependence. The average number of exceedances in a cluster is an important parameter in the relation between the parent and extreme value distribution. For 5-min rainfall data from De Bilt, quantities of the annual maxima are overestimated by about 10 mm h−1 if the affect of serial correlation is ignored. This bias can easily be removed by taking local clustering of large rainfall intensities in a rainy spell into account....

Extremes and local dependence in stationary sequences

Extensions of classical extreme value theory to apply to stationary sequences generally make use of two types of dependence restriction: (a) a weak “mixing condition” restricting long range dependence (b) a local condition restricting the “clustering” of high level exceedances.

Extreme value theory for continuous parameter stationary processes

In this paper the central distributional results of classical extreme value theory are obtained, under appropriate dependence restrictions, for maxima of continuous parameter stochastic processes. In particular we prove the basic result (here called Gnedenko's Theorem) concerning the existence of just three types of non-degenerate limiting distributions in such cases, and give necessary and sufficient conditions for each to apply. The development relies, in part, on the corresponding known theory for stationary sequences. The general theory given does not require finiteness of the number of upcrossings of any levelx. However when the number per unit time is a.s. finite and has a finite meanμ(x), it is found that the classical criteria for domains of attraction apply whenμ(x) is used in lieu of the tail of the marginal distribution function. The theory is specialized to this case and applied to give the general known results for stationary normal processes for whichμ(x) may or may not be finite). A general Poisson convergence theorem is given for high level upcrossings, together with its implications for the asymptotic distributions ofr th largest local maxima.

Local limit theorems for the maxima of discrete random variables

Let , where the Xi, i = 1, 2, … are independent identically distributed random variables. Classical extreme value theory, described for example in the books of do Haan(6) and Galambos(3) gives conditions under which there exist constants an > 0 and bn such thatwhere G(x) is taken to be one of the extreme value distributions G1(x) = exp (− e−x), G2(x) = exp (− x−a) (x > 0, α > 0) and G3(x) = exp (−(− x)α) (x < 0, α > 0).

Note on the application of classical extreme value theory to flood data

Todorovic and Zelenhasic and others have advocated the use of exact distributions of order statistics from random sample size to describe flood maxima from fixed time intervals. It is suggested that the asymptotic extreme value distributions are not necessarily inappropriate in this situation and possess some useful advantages over the exact distributions.

On the joint distribution of the largest flood and its time of occurrence

In a partial duration series the total number of exceedances as well as the exceedance flows is formalized as a random sequence of random variables. This formalism along with some physically reasonable assumptions leads to a rigorous expression for the joint distribution function of the largest exceedance and its time of occurrence. The deviation provides one way to consider the temporal changes in the occurrence and the magnitudes of individual exceedances due to ‘seasonal’ influences. This approach is general enough to be applied to any time interval of interest; furthermore, the derived expressions are exact (nonasymptotic). It may thus represent an improvement over an application of Gumbel's classical extreme value theory to flood phenomena, which is based on the premise that a large number of exceedances are available over time intervals of interest and which provides only asymptotic expressions.

ON A SPECIAL DISTRIBUTION OF MAXIMUM VALUES1

Abstract The classical extreme-value theory does not give a good account of the distribution of maximum rainfall intensities in Belgium. Reasons are given for the use, in this case, of a probability function defined by a double exponential whose argument is a function represented by a curve with two asymptotes. The application of such a probability function, when the curve is a branch of a hyperbola, to the maximum rainfall, in 1 min., at Uccle, leads to a good fit.