# 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  
### A Premise. Bayes Estimation of the Frequency of Exceedances  
Bayes method of inference or estimation--as illustrated in its general theory in [26; 27] and for some EWS applications in [12; 18; 20]--is proposed here for the RI under the IRD model. This kind of inference implies the assessment of prior information on the model parameters. In the cases under study, there are two parameters to be estimated:  
1. The mean frequency \(\phi\) of the process N(\(t\)) in (5), i.e., the expected number of exceedances of the EWS per unit time according (6). Briefly, \(\phi\) is denoted as frequency of exceedances. 2. The parameter a of the IRD model of (2) and (3) or any other parameter related to these ones. The above phrase \"any other parameter related to the parameter a of the IRD model\" means, for instance, the CDF of the above model, denoted as \(F(x|\alpha)\) in Section 1. Indeed, the mentioned relation represents a one-to-one correspondence between the parameter \(\alpha\) vs. the IRD CDF \(F(x)\) for any given EWS value \(x\). In the proposed approach, the prior information can be assessed equivalently both in terms of the \"basic\" parameters (\(\alpha\)) or of the corresponding CDF of their respective model. It is remarked that assessing the prior information in terms of the basic parameters (\(\alpha\)) is also equivalent to assessing the prior information in terms of the of the quantiles of the model, which are related to \(\alpha\). Whatever the choice, depending on the available information or past experience, for assessing the above information, the ultimate purpose of the inference is to evaluate the Bayes estimates of the above defined RI, in which time \(\mathrm{t}\) and threshold \(z\) are to be considered known constants:  
\[R=R(t,z)=1-e^{-\varphi t[1-F(z)]} \tag{17}\]  
It may be convenient in order to simplify some of the following formulations and considering that the time horizon \(\mathrm{t}\) and threshold \(z\) are to be held fixed to express more concisely the RI as follows:  
\[R=R(t,z)=1-e^{-WS} \tag{18}\]  
where  
\[W=W(t)=\varphi t\text{ and }S=S(z)=1-F(z) \tag{19}\]  
are respectively the \"expected number of exceedances\" (ENE) in (0,\(t\)), and the complementary CDF (CCDF) (or \"survival function\") of the EWS amplitudes evaluated at \(z\), i.e., \(S\) represents the probability that the threshold value \(z\) is exceeded by any single EWS values. Since \(S(z)=1-F(z|\alpha)\) for the IRD model, the RI to be estimated can be considered as functions of two RVs, \(R=p(\phi,\,\alpha)\). Moreover, the two input RVs (\(\phi\) and \(\alpha\)) for the RI estimation will be considered as statistically independent of each other. In general terms, denoting by T = (X,Y) the 2-dimensional vector of the two input RV, the problem of the Bayes estimation of the function \(R=p(\)X,Y) = \(p(\)T\()\) is briefly outlined in the following, leaving the details to the well-established literature on the subject [26; 27]. The Bayes estimator (BE) of \(R\) requires a joint prior PDF of the RV (X,Y), denoted by \(p(x\),\(y)\), expressing prior belief or information about the above parameters.