# Stochastic Extreme Wind Speed Modeling and Bayes Estimation under the Inverse Rayleigh Distribution
## 1 Introduction
Such a model is adopted as the parent distribution for assessment by means of a novel Bayes approach of a risk index model accounting for the most severe EWS amplitudes as disturbances for wind tower safety [18]. For the definition of such a risk index, this paper follows a probabilistic approach established in some basic literature on stochastic processes but probably not so widespread as it would be opportune in wind energy applications. Under such a stochastic approach--originated by classical studies on risk analysis as the authoritative book by [PERSON] [25]--safety is evaluated probabilistically in terms of a suitable stochastic counting process, measuring risk in terms of the occurrence of disturbance levels (i.e., EWS amplitude, in the case under study) higher than a prefixed threshold value. The risk index (RI) here adopted is defined with reference to a given time interval (0,\(t\)) as the probability that a given EWS amplitude is exceeded at least once in a time interval. As will be discussed with more details in Section 2 and Appendix A, if a threshold value \(z\) is assigned to EWS amplitudes, which is large enough, the occurrences of wind speed values which exceed such thresholds are rare events. Therefore, under general hypotheses, the process which counts the number of EWS values of exceeding \(z\) in the time interval (0,\(t\)) can be described, as deduced in advanced bookson extremal processes [21; 22; 23], by the well-known Poisson stochastic counting process. Denoting by \(\phi\) the parameter of such a Poisson process, i.e., the mean frequency of EWS values exceeding a given \"threshold\" value \(z\), then, assuming that the EWS values \(Z_{k}\) are statistically independent and identically distributed with the common, time-independent CDF \(F(z)\), which is equal to the CDF of (2) according to the supposed model, the above defined RI can be expressed, as it will be shown, by the following:  
\[R=R(t,z)=1-e^{-\varphi[1-F(z)]} \tag{4}\]  
With reference to such an RI, this paper also discusses the problem of its statistical estimation in the presence of a few field data with the purpose of efficient assessment of wind tower reliability and safety, not forgetting that such kinds of analyses also allow to obtain useful information for wind power generation statistics, as above outlined. In brief, the novelty of the proposed procedure discussed in this manuscript is twofold:  
1. The use of a new wind-speed model, the IRD one, to characterize extreme wind-speed distribution; such a model is proposed for the first time in the literature on the object, and its validity is illustrated with reference to real wind-speed data. 2. The use of a new suitable and practical Bayes approach is introduced for the estimation of the above RI with reference to the above IRD wind-speed model. The numerical part of the manuscript shows that such a Bayesian statistical inference approach for the estimation of the above RI from available data possesses a very good level of efficiency and precision, as proven by the very low Root Mean Square errors (RMSE). It is apparent that such results constitute novelties with respect to the current state-of-art on the subject. The application of the proposed wind-speed model to real field data is reported in Appendix B. Moreover, in order to better highlight the application of the proposed wind-speed model to real field data, in Section 4, a brief summary of the steps which are necessary to perform the application of the proposed model to real field data is reported. The rest of this paper is organized as follows. In Section 2, the basic steps leading to the RI of (4) are illustrated, together with the key features of the IRD model, characterizing the CDF \(F(z)\) of (4). In Section 3, a summary of the methods for IRD classical estimation is outlined; then, in Section 4, a suitable and practical Bayes approach is introduced for the estimation of the RI, with reference to the IRD model.