# 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
The prior mean of V, as estimated from the simulated sample data, is equal to 1.6955, while the one obtained from the approximating prior gamma is equal to 1.6965, which confirms the adequacy of the approximation. Finally (as long as the priori assessment is concerned) the prior mean of R can be evaluated by the simulated sample, i.e., by the sample mean (average value), 0.8155. If, instead, the above approximating prior gamma is adopted for V, then \(1-R=e^{-V}\) has an NLGD prior PDF. Indeed, since the prior gamma PDF of V, as already computed, has parameters\(=218.6171\) and \(\delta_{0V}=0.0078\), then analogously to what was before illustrated, \(S-R\) has a prior PDF which can be approximated by a NGL PDF, as follows:  
\[p(r)=\frac{1}{\Gamma(\
u)\delta^{\
u}}(1-r)^{(\frac{1}{\delta}-1)}[-\log(1-r)]^ {\
u-1},\ 0\leq r\leq 1;\ \
u=v_{0V}\,\text{and}\ \delta=\delta_{0V} \tag{55}\]  
Also, in this case, the theoretical approximating prior NLGD PDF of R and the smoothed kernel PDF estimator of \(R\) are perfectly superimposed. Assuming the approximating NLGD prior PDF of \(R\), its mean is equal to  
\[E[R]=1-\frac{1}{\left(1+\delta\right)^{\
u}} \tag{56}\]  
with \(\
u=v_{0V}\) and \(\delta=\delta_{0V}\), which yields \(0.8155\), the same value computed by simulation. Also, as hinted at in a previous section, it can be roughly computed by means of the \"first order\" approximation: \(E[R]\approx 1-e^{(-E[V])}=1-e^{-1.6955}=0.8165\). Also, this value is close to the previous one and very close to what reported is in Table 1.  
It is well known, however, that the Bayes inference method, differently from the classical ones, produces not only point estimates of the parameters but also the probability distributions describing our uncertainty about their values. In this case, we can assess a whole prior PDF, which, for what concerns its numerical estimation, is shown in Figure 5 in the form of the smoothed kernel PDF estimator of R.  
Such a numerical prior PDF, i.e., the quantile estimates associated with the simulated sample, yields, for instance, the following prior 90% credibility interval: (0.7804 0.8483). \[P\ (0.7804<R<0.8483)=0.90\]  
Using the NLGD approximation, taking into account that U = (1-R) is a NLGD RV with parameters (\(\
u=v_{0V}\); \(\delta=\delta_{0V}\)), it is easy to see that the p-quantile of \(R\) is equal to 1, the (1-\(p\))-quantile of U:  
\[R_{p}=1-U_{(1-p)}=1-nlginv(1-p,\
u,\delta) \tag{57}\]  
where nlginv (\(q\), \(\
u\), and \(\delta\)) is the \(q\)-quantile of an NLGD RV with parameters \(\
u\) and \(\delta\). (For such a computation, the Matlab@function computing the quantile of a gamma RV, _gaminv_, was used, since Y = e\({}^{(-X)}\) is an NLGD RV when X is a gamma RV, so the \(p\)- quantile of Y is given by \(Y_{p}\) = e\({}^{(-X(1-p))}\)  
Figure 5: Prior PDF of R as smoothed kernel estimator for case C11. This yields the following prior 90% credibility interval (0.7796, 0.8489), which is practically the same interval above found numerically (0.7804, 0.8483), with precision up to the third decimal number. Of course, since the numerical computations are very reliable, the above gamma or NLGD approximation, as those dealt with in the following, are useful only because they allow easier and faster computations, a sensitivity or robustness analysis with respect to the assumed values of the parameters or the choice of the basic distributions (e.g., one could be interested in evaluating which are the changes in the range of results if the prior gamma PDF, here assumed for the parameters \(W\) and \(S\), is changed into, say, a lognormal or a uniform PDF. This is an issue discussed under the topic of \"robustness analysis\", briefly dealt with in the final part of the paper). Apart from this remark, in the following, more emphasis will be given to the numerical computations.