# 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
Since the RV \(\alpha\) has a posterior gamma PDF with parameters \(\
u_{1}\) = 403 and \(\delta_{1}\) = 0.2246, the RV \(\beta=\alpha/z^{2}\) has a posterior gamma PDF with parameters \(\
u\) and \(\delta\) expressed by the following:  
\[\
u=\
u_{0}=403;\,\delta=\delta_{0}/z^{2}=0.2250/z^{2}=4.6405\text{e-04}\]  
Therefore, U = 1 - S = e\({}^{-\beta}\) has an NLGD posterior PDF, already formulated in (52); S = 1 - U has a posterior PDF which can be computed from the above NLGD PDF, evaluated in u = 1 - s; and posterior mean is as follows:  
\[\mu_{S0}=1-\frac{1}{\left(1+\delta\right)^{\
u}}=0.1705\]  
Then, the computation of the mean and variance of V = WS are performed with the following results:  
\[\text{E}[\text{V}]=1.6965;\,\text{VAR}[\text{V}]=0.0132{\rightarrow}\text{ CV}[\text{V}]=0.0676\]  
This, in turn, allows the computation of the parameter of the approximating prior gamma PDF of V, and as in the previous subsection, the BE of R is as follows:  
\[\text{R}^{\circ}=0.8116\]  
which is of course (given also the very little sample size) much closer to the true value (0.8103) of R than the MLE (0.6672). It is also worth mentioning that, as in the previous subsection, the \"rough\"computation of such BE by means of the \"first-order\" approximation \(\text{E}[\text{R}|\text{D}]\approx 1-\text{e}^{-\text{E}[\text{V}|\text{D}]}=1- \text{e}^{-1.6955}\) = 0.8165. Finally, the numerical computations, both when performed by simulation or numerical computations, give essentially the same result: \(\text{E}[\text{R}|\text{D}]\approx 1\cdot\text{e}^{-\text{E}[\text{V}|\text{D}]}=1 \cdot\text{e}^{-1.6955}=0.8120\). Of course, the merits of the BE are not appreciable by a single simulation, here shown only to highlight the mathematics, but require an overall assessment, which is the object of the next sections. Numerical Application for Performances of the Bayes Approach in the Case of a Gamma Prior PDF on the IRD Parameter \(\alpha\)  
This and the next section follow the framework illustrated in the previous section, in which only the features of an estimation procedure for a given sample have been reported. Since in practice the Bayes estimate and their relevant features (e.g., their mean square errors) are computable only numerically, in order to assess the overall performance of the proposed Bayes estimation, a large series of Monte Carlo simulations have been carried out, relevant to different sample sizes and extending the same approach. In detail, the Bayesian methodology has been evaluated numerically, as illustrated here, by means of a large set of simulated experiments aiming at assessing the estimation efficiency of the Bayes approach. This is illustrated here in the case of a gamma prior PDF on the IRD parameter \(\alpha\), while the alternative assumption of a beta prior PDF on the IRD parameter \(\text{S}(\alpha)\) will be briefly illustrated in the next section. Both in this and in the next section, the assumption of a gamma prior PDF on the Poisson parameter (\(\alpha\), \(\phi\)) is maintained, as customary [26; 27]. The estimation efficiency of the Bayes estimates is assessed both \"per se\" (i.e., in \"absolute\" terms) and in terms of \"relative efficiency\" [26; 27; 28] of the Bayes estimates vs. the \"classical\" Maximum Likelihood (ML) estimators, more widely used in the field of wind speed PDF assessment [13; 14; 15; 16; 17]. After a sound investigation, different sets of typical values have been selected for the mean and the standard deviation (or the CV) of the main random parameters (\(\alpha\) and \(\phi\)). Different CV prior values were selected with the same values of prior means, since--as intuitive--the estimation performances are rather sensitive to variations in CV prior values while variations in the prior means are less effective in terms of estimation efficiency. As above said, 12 sets of simulation cases are illustrated here, among the many more performed, and shown in Tables 3-5.