# Stochastic Extreme Wind Speed Modeling and Bayes Estimation under the Inverse Rayleigh Distribution
## Appendix B The Inverse Rayleigh with an application to Extreme Wind Speed Modeling  
As introduced in Section 1, a positive RV X is said to possess an \"inverse Rayleigh\" distribution with positive scale parameter a if its CDF \(F(x)\) and its (PDF) \(f(x)\), i.e., the derivative of \(F(x)\), are expressible by the following equations:  
\[F(x|\alpha)=e^{-(\frac{\alpha}{x})},\ x>0 \tag{14}\]  
\[f(x|\alpha)=2\alpha x^{-3}e^{-(\frac{\alpha}{x})},\ x>0 \tag{15}\]  
The name \"inverse Rayleigh\" distribution takes its origin from the property that, if X is an IRD RV, then Y = 1/X has a Rayleigh distribution, which is a particular case of a Weibull distribution with shape parameter \(\beta\) = 2, being the Weibull CDF expressed by the following equation, with \(\lambda\) and \(\beta\) positive parameters:  
\[F(x)=1-e^{-\lambda x^{\beta}} \tag{16}\]  
[PERSON] (1972) presented some properties of the maximum likelihood estimator, for IRD; furthermore, confidence intervals and tests of hypotheses were developed. [PERSON] (2003) developed some estimation and prediction results for the IRD. [PERSON] and [PERSON] (2005) obtained Bayesian estimators of the parameter of the IRD under four loss functions. Bayesian of one and two sample predictions were also developed including point predictions and prediction intervals.  
More IRD parameters are illustrated as follows. The p-quantile \(X_{p}\), i.e., the unique solution of \(F(X_{p})=p\ (0<p<1)\), is easily computed as follows:  
\[X_{p}=\left[-\frac{\alpha}{\log p}\right]^{\frac{1}{2}} \tag{17}\]  
This relationship implies that the following equation can be used for obtaining a simulated value of an IRD RV \(X\) from an RV U which has a uniform distribution in (0,1).  
\[X=\left[-\frac{\alpha}{\log\text{U}}\right]^{\frac{1}{2}} \tag{18}\]  
The median value m, i.e., the solution of \(F(m)\) = 0.5, is computed as follows:  
\[m=\left(\frac{\alpha}{\psi}\right)^{\frac{1}{2}},\ \text{with}\ \psi=\log(2)=0.6931 \tag{19}\]so that any quantile \(X_{p}\) is proportional to the median through the relations:  
\[X_{p}=m\left[\frac{\psi}{K_{p}}\right]^{\frac{1}{2}};\text{ with }K_{\text{P}}=-\log(p)>0 \tag{11}\]  
Finally, the \(k\)th moment about the origin of the IRD model is expressed if \(k<2\) by the following:  
\[\text{E}\left[X^{k}\right]=\alpha^{m}\Gamma(1-m),\ m=\frac{k}{2} \tag{12}\]  
so that the mean value is given by the following:  
\[\text{E}[X]=\alpha\Gamma(0.5)=\alpha\sqrt{\pi} \tag{13}\]  
while the model variance does not exist.  
The application to EWS data is exemplified in this study by data analysis related to EWS data in [19], summarized in Table 11, where the 52 weekly maxima (\(z_{1},\ldots\ z_{n}\), with n = 52) of wind speed data in (m/s) are reported (These data represent the 52 maximum WS values registered in each of the 52 weeks of year 2014 in Boulder, CO, USA at latitude 39\({}^{\circ}\)54\({}^{\prime}\) north and longitude 105\({}^{\circ}\)14\({}^{\prime}\) west; they are part of large datasets reported in [19]).  
The goodness of fit tests for assessing the validity if an IRD model for the above data were all successfully satisfied. A simple way to get evidence of such a fitting is drawing a \"Weibull probability plot\" with shape parameter \(\beta=2\) for the data on the RV Y = 1/Z as shown in Figure 11 (The Weibull probability plot in Figure 11 was obtained by the Matlab@function \"_wblplot (Z)_\", with \(Z\) being the data sample vector of Table 11).  
For the (classical) estimation of parameter \(\alpha\) with the purpose of finding the best IRD fitting PDF, it is readily seen that the quantile estimation methods [24], the method of moments [31], and the maximum likelihood one yield essentially the same result. For brevity, here, the first one is reported. Since the median is equal to 33.9 m/s, the parameter \(\alpha\) can be estimated as follows:  
\[\text{a}\ ^{*}=33.9\ [-\log(0.5)]=28.22\]  
The graph of the corresponding IRD fitting PDF is shown in Figure 12 superimposed to the histogram.