A class of probability distributions for application to non-negative annual maxima

Abstract. Many environmental variables of interest as potential hazards take on only positive values, such a wind speed or river discharge. While recognising that primary interest is for largest extremes, it is desirable that distributions of maxima for design purposes should be subject to similar bounds as the physical variable concerned. A modified univariate extreme value argument defines a set of distributions, all bounded below at zero, with potential for application to annual maxima. Let f(x) be a probability distribution over the range, 0 ≤ x ≤ ω, where 0 < ω ≤ ∞. Define X* = max (X₁, X₂, .. X_N) to be the maximum value of a random sample of size N drawn from f(x). Also, define the transformation Y_i = g(X_i) where g(X) is any positive monotonically decreasing function of X. This would include, for example, Y = X⁻¹ but not Y = −X. Because the Y_i are independent random variables bounded from below at some non-negative value ɛ, it follows from extreme value theory for minima that for sufficiently large N the random variable Y* = min (Y₁, Y₂, .. Y_N) will follow a Weibull distribution with umulative distribution function:

F(y) = pr(Y_* ≤ y) = 1 − exp {−[(y − ɛ) / ɑ]^c}        ɛ ≥ 0, ɑ > 0, c > 0

where ɛ = g(ω) and ɛ, ɑ, and c are respectively location, scale, and shape parameters. The distribution F(y) holds generally as an extreme value expression for sufficiently large N, irrespective of which of the three possible asymptotic extreme value distributions of sample maxima holds for X*. Therefore, the limit Weibull distribution for, say, Y* = X_*⁻¹ has no less validity as a single expression for obtaining exceedance probabilities than the generalized extreme value distribution applied directly to X*. If follows that a class of probability distributions for possible use with positive-valued annual maxima can be defined from the application of the inverse function g⁻¹ to Weibull random variables for ɛ ≥ 0. All distributions so obtained are defined over the range 0 ≤ x ≤ ω, which actually excludes all of the asymptotic extreme value distributions of maxima except for the special case of the Type 2 extreme value distribution with location parameter at zero. It is to be expected, however, that the asymptotic distributions will sometimes hold to a high level of approximation within the 0, ω interval. No specific distribution is advocated for annual maxima application because concern here is only with drawing attention to the existence of the distribution class. The transformation approach is illustrated with respect the distribution of reciprocals of random variables generated from a 3-parameter Weibull distribution with ɛ ≥ 0.