## - Extreme Value Distribution

The probability density function (pdf) of the extreme value distribution with the location parameter µ and the scale parameter σ is given by

f(x|µ, σ) = 1 σ exp (−z − exp(−z)) , (26.1)

where z = x−µ σ . The cumulative distribution function (cdf) is given by

F (x|µ, σ) = exp{− exp[−z]}, −∞ < x <∞, σ > 0. (26.2)

The inverse distribution function is given by

F−1(p|µ, σ) = µ− σ ln(− ln(p)), 0 < p < 1. (26.3)

The plots of the pdf in Figure 26.1 show that this distribution asymmetric and rightskewed. The family of distributions with the pdf in (26.1) is referred to as the type I largest extreme value distribution (LEV), as this is the sampling distribution of the largest observation in a sample from a continuous distribution. We shall refer to this distribution as LEV(µ, σ). We also note that if

Y ∼Weibull(b, c) then, − ln(Y ) ∼ LEV(µ = − ln b, σ = 1/c). (26.4)

This one-to-one relation allows us to find prediction intervals, tolerance limits, and confidence intervals for survival probability from those for a Weibull distribution.