# Stochastic Extreme Wind Speed Modeling and Bayes Estimation under the Inverse Rayleigh Distribution
## 4 A Bayes Method of Inference for the Risk Index Based upon the IRD Model
### The Case of a Gamma Prior PDF on the IRD Parameter \(\alpha\): Prior Analysis
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Case** & **E[\(\phi\)]** & **E[\(\alpha\)]** & **CV[\(\phi\)]** & **CV[\(\alpha\)]** \\ \hline A11 & 2.5 & 15 & 0.05 & 0.05 \\ A12 & 2.5 & 15 & 0.05 & 0.10 \\ A21 & 2.5 & 15 & 0.10 & 0.05 \\ A22 & 2.5 & 15 & 0.10 & 0.10 \\ B11 & 5.0 & 45 & 0.05 & 0.05 \\ B12 & 5.0 & 45 & 0.05 & 0.10 \\ B21 & 5.0 & 45 & 0.10 & 0.05 \\ B22 & 5.0 & 45 & 0.10 & 0.10 \\ C11 & 10 & 90 & 0.05 & 0.05 \\ C12 & 10 & 90 & 0.05 & 0.10 \\ C21 & 10 & 90 & 0.10 & 0.05 \\ C22 & 10 & 90 & 0.10 & 0.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Combinations of prior values with respect to mean value and coefficient of variation (CV) of the input random variable (RV) (\(\phi\), \(\alpha\)). Of course, the primary interest here is not the estimation of the above parameters per se, but they are the input RVs which allow the BE of the RI. Let us first evaluate the prior mean of RI implied by the above assessments on \(\phi\) and \(\alpha\). By simulation, numerical integration, or following the approximated method illustrated in the previous section, it is not difficult to evaluate such a prior mean and afterwards also the posterior mean based upon the following relations:  
\[\left\{ \begin{array}{l}R=1-e^{-V}\\ V=WS\end{array} \right. \tag{50}\]  
Figure 4: Two prior gamma PDFs for V are superimposed: the theoretical approximating prior gamma PDF of V and the one estimated from simulated data for case C11. Figure 3: The prior gamma PDF of a with mean E[a] = 90 and CV: CV[a] = 0.05 for case C11. First, notice that the RV \(S=1-e^{-(\frac{\alpha}{z^{2}})}\) has a prior PDF which can be easily assessed in terms of the NLGD PDF (Appendix C). Indeed, since the RV \(\alpha\) has a gamma PDF with parameters \(\
u_{0}=400\) and \(\delta_{0}\) = 0.2250, it is easy to verify that also the RV,  
\[\beta=\frac{\alpha}{z^{2}} \tag{51}\]  
has a prior gamma PDF, with parameters \(\
u\) and \(\delta\) expressed by \(\
u=\
u_{0}=400\) and \(\delta=\delta_{0}/z^{2}=0.2250/z^{2}=4.6488\)e-04. Therefore, by definition of the NLGD model (Appendix C), U = \(1-S=e^{-\beta}\) has an NLGD prior PDF, here denoted by \(p(u)\):  
\[p(u)=\frac{1}{\Gamma(\
u)\delta^{\
u}}u^{(\frac{1}{S}-1)}(-\log u)^{\
u-1},\ 0 \leq u\leq 1 \tag{52}\]  
This implies that \(S=1-U\) has a prior PDF which can be easily computed from the above NGL PDF, evaluated in \(u=1-s\), as follows:  
\[p(s)=\frac{1}{\Gamma(\
u)\delta^{\
u}}(1-s)^{(\frac{1}{S}-1)}[-\log(1-s)]^{ \
u-1},\ 0\leq s\leq 1 \tag{53}\]  
Therefore, utilizing the equations relevant to the NLGD model in Appendix C, prior mean and variance of \(S\) are, respectively, equal to  
\[\left\{ \begin{array}{l}\mu_{S0}=1-\frac{1}{(1+\delta)^{\
u}}\\ \sigma_{S0}^{2}=\frac{1}{(1+2\delta)^{\
u}}-\frac{1}{(1+\delta)^{2\
u}}\end{array} \right. \tag{54}\]  
The values of the mean and variance of \(S\) and \(W\) allow the computation of the mean and variance of \(V=WS\) by simple properties of the product of RV [31], with the following results:  
\[E[V]=1.6965;VAR[V]=0.0132\rightarrow\)CV[V] = 0.0676\]  
This, in turn, allows the computation of the parameter of the approximating prior gamma PDF of V, which are equal to \(\
u_{0V}=1/(0.0676)^{2}=218.62\) and \(\delta_{0V}=E[V]/218.6171=0.0078\). In Figure 4, two prior gamma PDFs are superimposed:  
1. The one corresponding to the theoretical approximating prior gamma PDF of V, and
2. The one estimated by means of the \"smoothed kernel PDF estimator\" [31], obtained after generating by Monte Carlo simulation (The Monte Carlo simulation of samples from RV W and S was obtained through the function \"_gammrad_\" of Matlab(r)used for generating samples of the gamma RV (\(\phi\), a), and then, the corresponding sample of S was obtained through the function \(S=S(\alpha)=1-\exp(-\alpha/z^{2})\)) 5000 values of W and S and so of \(V=WS\), according to their above prior PDF.