# 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
The basic statistics--estimated at the end of each simulation case study--which describe the efficiency of the proposed estimates are the following:  
\(\blacklozenge\)_RMSEB_: root mean square error of the Bayes estimator;  
\(\blacklozenge\)_RMSEL_: root mean square error of the ML estimator; and  
\(\blacklozenge\)_REFF_ = _RMSEL/RMSEB_: relative efficiency of the Bayes estimator with respect to the ML estimator. The _REFF_ index is the most widely adopted measure of the \"relative efficiency\" [26; 27] of the Bayes estimator with respect to the ML estimator: the more it exceeds unity, the more efficient is the Bayes estimate with respect to the ML estimate. The above-defined quantities are based on the concept of \"RMSE\". Given an estimator, \(\zeta^{\circ}\), of the parameter \(\zeta\), its RMSE is--as well known--defined as  
\[RMSE=\sqrt{\text{E}[(\zeta^{\circ}-\zeta)^{2}]} \tag{61}\]  
with both \(\zeta^{\circ}\) and \(\zeta\) as random variables in the Bayesian approach, while the latter is an unknown, deterministic constant in the classical statistical approach. The RMSE (which is not analytically computable) is evaluated here at the end of each simulation case study by means of the ordinary large-sample estimator [31] as follows:  
\[RMSE=\left[\frac{1}{N}\sum_{j=1}^{N}\left(\zeta^{\circ}_{\ j}-\ \zeta_{j}\right)^{2} \right]^{1/2} \tag{62}\]  
While the above measures concern the efficiency of the Bayes estimates, other significant \"performance indexes\" were also evaluated, as measure of their precision:  
\(\blacklozenge\) AREB: Average relative error of the Bayes estimator  
\(\blacklozenge\) AREL: Average relative error of the ML estimator  
\(\blacklozenge\) MREB: Maximum relative error of the Bayes estimator  
\(\blacklozenge\) MREL: Maximum relative error of the ML estimator  
The relative error of an estimator, \(\zeta^{\circ}\), of the parameter \(\zeta\) can be defined as (E[\(\zeta^{\circ}\)] - E[\(\zeta\)])/E[\(\zeta\)]. Therefore, as an overall \"sample\" measure of the relative bias, the \"Average Relative Error\", AREB, is used here, in which the quantities \(\mathrm{E}[\zeta^{\circ}]\) and \(\mathrm{E}[\zeta]\) are evaluated through their estimated mean (average) values at the end of each simulation case study:  
\[\mathrm{ARE}\ =\ \left|\frac{\mathrm{M}[\zeta^{\circ}]-\mathrm{M}[\zeta]}{ \mathrm{M}[\zeta^{\circ}]}\right| \tag{63}\]  
with \(\mathrm{M}[\zeta^{\circ}]\) and \(\mathrm{M}[\zeta]\) respectively the average values of the \(\zeta_{j}{}^{\circ}\) and \(\zeta_{j}\) values over the \(N\) performed simulations:  
\[\left\{ \begin{array}{l}\mathrm{M}[\zeta^{\circ}]=(1/N)\sum\limits_{k=1}^{N} \zeta^{\circ}{}_{k}\\ \mathrm{M}[\zeta]=(1/N)\sum\limits_{k=1}^{N}\zeta_{k}\end{array} \right. \tag{64}\]  
With the same notations, the Maximum Relative Error, MRE, of an estimator over \(N\) trials is given by the following:  
\[\mathrm{MRE}\ =\ \max\left|\frac{\zeta_{j}^{\circ}-\zeta_{j}}{\zeta_{j}} \right|,\ \ \mathrm{j}=1,\ldots,\mathrm{N} \tag{65}\]  
The values of RMSE, ARE, and MRE can be calculated for each simulation case study for both the ML estimator and the proposed Bayesian approach, thus obtaining the values of quantities RMSEB, RMSEL, \(\mathrm{\AAREB}\), \(\mathrm{\AAREB}\), \(\mathrm{\AAREB}\), \(\mathrm{\AAREB}\), \(\mathrm{\AAREB}\), and \(\mathrm{\AAREB}\) (defined above). In other words, \(\mathrm{\AAREB}\) denotes the RMSE value for a BE, \(\mathrm{\AAREB}\) denotes that for an MLE; \(\mathrm{\AAREB}\) denotes the ARE value for a BE; \(\mathrm{\AAREB}\) denotes that for an MLE; and so on. Note that, for an easier comparison, both \(\mathrm{\AARE}\) and MRE are evaluated in terms of absolute values.