Extreme value and related models with applications in engineering and science

Extreme Value and Related Models with Applications in Engineering and Science Enrique Castillo University of Cantahria and University ofCastilla La Manchu Ali S Hadi The American Universiw in Cairo and Cornell University N Balakrishnan McMaster University Jose Maria Sarabia Uni\[...]... earthquake intensity or the minimum strength of the concrete used in building the dam It is well known t o engineers that design values of engineering works (e.g., dams, buildings, bridges, etc.) are obtained based on a compromise between safety and cost, that is, between guaranteeing that they survive when subject to extreme operating conditions and reasonable costs Estimating extreme capacities or operating... reprinted from the book Extreme Value T h e o q in Engineering, by E Castillo, Copyright @ Academic Press (1988), with permission from Elsevier 6 Chapter 1 Introduction and Motivation A building or engineering work will survive if it is designed to withstand the most severe earthquake occurring during its design period Thus, the maximum earthquake intensity plays a central role in design The probabilistic... corresponding t o various specified mean recurrence intervals, (b) design loads, and (c) seismic incidence in the form of areas of equal risk Wind speeds are estinlates of extreme winds that can occur at the place where the building or engineering work is to be located and have a large irlfluence on their design characteristics and final costs Design loads are also closely related to the largest loads acting... such as x, y, and z or x1,x2, ,x, A random variable is said to be discrete if it can assume only a finite or countably infinite number of distinct values Otherwise, it is said to be continuous Thus, a continuous random variable can take an uncountable set of real values The random variable X in Example 2.1 is discrete, whereas the random variable Y is continuous When we deal with a single random quantity,... under study However, in some other areas of applications, we are not interested in estimating the average but rather in estimating the maximum or the minimum (see Weibull (1951, 1952), Galambos (1987), Castillo (1994)) For example, in designing a dam, engineers, in addition to being interested in the average flood, which gives the total amount of water to be stored, are also interested in the maximum flood,... satisfactory in terms of safety and cost, because high probabilities of failure can be obtained on one hand, and large and unnecessary waste of money, on the other Consequently, the safety factor approach is not an optimal solution to the engineering design problem The knowledge of the distributions of the maxima and minima of the relevant phenomena is important in obtaining good solutions to engineering design... crack in the piece If the size and shape of the crack were known, the lifetime, measured in number of cycles to failure, could be deterministically obtained However, the presence of cracks in pieces is random in number, size, and shape, and, thus, resulting in a random character of fatigue strength Assume a longitudinal piece hypothetically subdivided into subpieces of the same length and being subjected... 1998, 2000), Galambos and Macri (2000), Kotz and Nadarajah (2000), and Nadarajah (2003) 1.3 Examples of Applications 1.3 Examples of Applications In this section examples of some of the fields where common engineeririg problerris involve extremes or other order statistics are given.' 1.3.1 Ocean Engineering In the area of ocean engineering, it is known that wave height is the main factor to be considered... assessment studies Stream discharge and flood flow have long been measured and used by engineers in the design of hydraulic structures (dams, canals, etc.), flood protection works, and in planning for floodplain use Riverine flooding and dams overtopping are very common problems of concern A flood frequency analysis is the basis for the engineering design of many projects and the economic analysis of flood-control... structures are in poor condition Then the pmf is The mean and variance of X are p = p = 0.2 and a2 = p ( l - p) = 0.16 I Bernoulli random variables arise frequently while handling extremes Engineers are often interested in events that cause failure such as exceedances of a random variable over a threshold value Definition 2.2 (Exceedances) Let X be a random variable and u a given threshold value The event ... solution to the engineering design problem The knowledge of the distributions of the maxima and minima of the relevant phenomena is important in obtaining good solutions to engineering design problems... reprinted from the book Extreme Value T h e o q in Engineering, by E Castillo, Copyright @ Academic Press (1988), with permission from Elsevier 6 Chapter Introduction and Motivation A building... measured and used by engineers in the design of hydraulic structures (dams, canals, etc.), flood protection works, and in planning for floodplain use Riverine flooding and dams overtopping are