# 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 \(\alpha\)--Posterior Analysis
\begin{table}
\begin{tabular}{l|c c c c|c c c c} \hline \hline
**Parameters** & **n = 3** & **n = 10** & **n = 30** & **n = 50** & **n = 3** & **n = 10** & **n = 30** & **n = 50** \\ \hline  & \multicolumn{4}{c|}{**Case B11**} & \multicolumn{4}{c}{**Case B12**} \\ \hline _AV(R)_ & 0.3581 & 0.3580 & 0.3582 & 0.3581 & 0.3581 & 0.3574 & 0.3581 & 0.3576 \\ _AV(\(\alpha\))_ & 44.9549 & 44.9780 & 45.0124 & 44.9782 & 44.9782 & 44.9632 & 44.9782 & 45.0137 \\ _AV(\(\phi\))_ & 5.0033 & 5.0011 & 4.9994 & 5.0014 & 5.0014 & 5.0004 & 5.0014 & 4.9987 \\ _AV(RL)_ & 0.0435 & 0.3169 & 0.2118 & 0.3173 & 0.3173 & 0.0871 & 0.3173 & 0.3227 \\ _AV(RB)_ & 0.3551 & 0.3577 & 0.3569 & 0.3579 & 0.3579 & 0.3553 & 0.3579 & 0.3580 \\ _RMSEL_ & 0.3308 & 0.1388 & 0.1907 & 0.1400 & 0.1408 & 0.2877 & 0.1400 & 0.1396 \\ _RMSEB_ & 0.0198 & 0.0189 & 0.0195 & 0.0190 & 0.0160 & 0.0294 & 0.0190 & 0.0263 \\ _RARE_ & 20.5448 & 7.4852 & 10.5744 & 7.4852 & 11.6283 & 7.4852 & 5.3642 \\ _RMRE_ & 6.4392 & 5.6895 & 3.8621 & 5.6895 & 5.6895 & 2.8046 & 5.6895 & 3.7746 \\ _REFF_ & 16.7142 & 7.3439 & 9.9739 & 7.3539 & 8.8000 & 2.8046 & 7.3539 & 5.3030 \\ \hline  & \multicolumn{4}{c|}{**Case B21**} & \multicolumn{4}{c}{**Case B22**} \\ \hline _AV(R)_ & 0.3581 & 0.3580 & 0.3579 & 0.3581 & 0.3568 & 0.3576 & 0.3576 & 0.3572 \\ _AV(\(\alpha\))_ & 44.9549 & 44.9780 & 44.9658 & 44.9782 & 44.9622 & 45.0668 & 44.9961 & 45.0137 \\ _AV(\(\phi\))_ & 5.0033 & 5.0011 & 4.9987 & 5.0014 & 4.9982 & 5.0003 & 5.0088 & 4.9987 \\ _AV(RL)_ & 0.0435 & 0.3169 & 0.2144 & 0.3173 & 0.0426 & 0.0862 & 0.2126 & 0.3227 \\ _AV(RB)_ & 0.3551 & 0.3577 & 0.3568 & 0.3579 & 0.3453 & 0.3472 & 0.3519 & 0.3580 \\ _RMSEL_ & 0.3308 & 0.1388 & 0.1910 & 0.1400 & 0.3308 & 0.2899 & 0.1918 & 0.1459 \\ _RMSEB_ & 0.0198 & 0.0189 & 0.0191 & 0.0190 & 0.0411 & 0.0396 & 0.0378 & 0.0368 \\ _RARE_ & 20.5448 & 7.4852 & 10.7949 & 7.4852 & 9.9445 & 8.7859 & 5.4961 & 4.0660 \\ _RMRE_ & 6.4392 & 5.6895 & 5.7265 & 5.6895 & 2. For each of the above cases, \(N\) random samples of size \(n\) (with \(n\) varying as specified below) of EWS values \(X_{j}\) have been generated according to the assumed IRD PDF of stress \(X\) with parameter \(\alpha\) and with a Poisson parameter \(\phi\) for the process of exceedances. The parameters (\(\phi\) and \(\alpha\)) are in turn RVs generated according to their prior gamma distributions, as shown in a previous section (different priors were also investigated, with similar results to those shown in the following). For each sample size \(n\), a number \(N=10^{4}\) of replications has been performed; let us refer to each considered couple of values of \(n\) and \(N\) as a single simulation case study; different simulation case studies were considered by testing different sample sizes. Here, in particular, the results for small (\(n=3\), \(n=10\)) or moderate (\(n=30\), \(n=50\)) sample sizes are reported in the tables that follow, among the many more performed. The choice to show also the results for very small sample sizes as \(n=3\) takes its origin in view of the previously discussed features of Bayesian information updated also in the case of a limited amount of data has mainly a theoretical interest; nonetheless, it can also have a practical aspect when dealing with very high values of WS, so that their occurrence is very rare and only very few data are available. With reference to the generic parameter \(\zeta\) to be estimated (which in this case is basically the RI R, but one could be interested also in the BE of \(\phi\) and \(\alpha\)), it will be denoted as \(\zeta^{\circ}\), the estimate of the \"true\" value \(\zeta_{j}\) of \(\zeta\) relevant to the particular \(n\)-sized sample generated at the \(j\)th simulation cycle.