TABLE OF CONTENTS Three types of asymptotic distributions have been developed for minimum and maximum values based on different initial distributions [1; 2]. Among these three types of asymptotic distributions, the Type I asymptotic distribution for both the maxima and the minima and the Type III asymptotic distribution for minimum values are of interest in reliability engineering. The Type I asymptotic distribution is refered to as the "Type I extreme value distribution," or simply "the extreme value distribution (EVD)." The Type III asymptotic distribution for the minima is the well-known Weibull distribution, which has been discussed in Chapter 6. In this chapter only the Type I EVD is discussed.
The EVD pdf for the minima is given by [3, p. 113]
| (9.1) |  |
The EVD pdf for the maxima is given by
| (9.2) |  |
- ? < T < ?, - ? < ? max < ?, ? max > 0.
The parameters ? max and ? min are the location parameters and may assume any value. The parameters ? max and ? min are the scale parameters. These pdf's are shown in Fig. 9.1. The EVD has no shape parameter; consequently, it always has the same shape. From Eqs. (9.1) and (9.2) and Fig. 9.1, it may be seen that the probability density functions for the minimum and maximum values are mirror images of each other. The EVD for the minima is skewed to the left...
