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

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, .. XN) to be the maximum 10 value of a random sample of size N drawn from f(x). Also, define the transformation Yi = g(Xi) where g(X) is any positive monotonically decreasing function of X. This would include, for example, Y = X but not Y = -X. Because the Yi are independent random variables bounded from below at some non-negative value , it follows from extreme value theory for minima that for sufficiently large N the random variable Y* = min (Y1, Y2, .. YN) will follow a Weibull distribution with cumulative distribution function: 15 * ( ) ( ) 1 exp{ [( ) / ] } 0, 0, 0 c F y pr Y y y c              where  = g(ω) and  ,  , 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* = 1 * X  has no less validity as a single expression for obtaining exceedance probabilities than the generalized extreme value distribution applied directly 20 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 to Weibull random variables for   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 25 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   0. Hydrol. Earth Syst. Sci. Discuss., doi:10.5194/hess-2017-198, 2017 Manuscript under review for journal Hydrol. Earth Syst. Sci. Discussion started: 17 May 2017 c © Author(s) 2017. CC-BY 3.0 License.

introduced the univariate generalized extreme value distribution (GEV) to analysis of environmental maxima or minima, subsequent to its original mathematical formulation by von Mises (1936). The GEV distribution for largest extremes is a single expression which incorporates the three asymptotic extreme value distributions of sample maxima derived by early workers in the field (Fisher and Tippett, 1928;Gnedenko, 1943). An historical overview is given 5 by Kotz and Nadarajah (2000). Because of its natural linkage to maxima, the GEV distribution has been applied for design purposes in numerous instances with respect to annual maxima such as flood discharges, wave heights, and wind speeds (Coles, 2001). Focus for maxima is of course on distribution upper tails but it is more reflective of reality if probability distributions for design purposes are defined within the same bounds as the physical variable concerned, noting that many environmental variables with hazard potential such as wind speed and river discharge are bounded below at zero. In this 10 regard the Type 3 and Gumbel asymptotic distributions of maxima both extend into the negative domain, albeit with very small probability in practical applications.
This brief communication makes an alternative extreme value argument leading to a class of distributions, all bounded below at zero, which could have potential application to annual maxima and with no less theoretical justification than the GEV. 15 There is no data-based argument made for any one distribution but the approach is illustrated for the particular case of reciprocals of 3-parameter Weibull random variables.

Alternative distribution class
Let f(x) be an unknown probability distribution defined over the range, 0 ≤ x ≤ ω, where 0 < ω ≤ ∞. Define X* = max (X1, X2, .. XN) to be the maximum value of a random sample of size N drawn from f(x). Also, define the transformation Yi = g(Xi) 20 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 Yi are independent random variables bounded from below at some non-negative value , it follows from extreme value theory for minima that for sufficiently large N the random variable Y* = min (Y1, Y2, .. YN) will follow a Weibull distribution with cumulative distribution function: where  = g(ω) and  ,  , and c are respectively location, scale, and shape parameters.
The distribution F(y) holds generally as a single extreme value expression for sufficiently large N, irrespective of which of the three possible asymptotic extreme value distributions of largest extremes holds for X*. Therefore, the limit Weibull 30 distribution for, say, Y* = 1 * X  has no less validity as a general expression for obtaining exceedance probabilities than the generalized extreme value distribution applied to X*.
If follows that a class of probability distributions for possible use with positive-valued annual maxima (taking a "year" as a random sample) can be defined from the application of the inverse function g -1 to random variables generated from a threeparameter Weibull distribution with   0 . All the distributions so obtained are defined over the range 0 ≤ x ≤ ω, which excludes all of the asymptotic extreme value distributions of maxima except for the special case of the Type 2 extreme value 5 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.
Each permissible inverse function transformation of Weibull random variables will define a different distribution in the class. The transformation approach is illustrated in the next section, with respect to the particular case of the distribution of reciprocals of random variables generated from a 3-parameter Weibull distribution with   0.

Illustration (Weibull reciprocal transformation)
For N sufficiently large Y* = 1 * X  will follow a Weibull distribution as defined by Eq. (1) and the inverse transformation of the Weibull random variable W is given by Z = W -1 . The resulting distribution of Z is referenced here for convenience as the H distribution, with cumulative distribution function and probability density function respectively given by:  Hydrol. Earth Syst. Sci. Discuss., doi:10.5194/hess-2017-198, 2017 Manuscript under review for journal Hydrol. Earth Syst. Sci. Discussion started: 17 May 2017 c Author(s) 2017. CC-BY 3.0 License.