# Stochastic Extreme Wind Speed Modeling and Bayes Estimation under the Inverse Rayleigh Distribution
## 3 Hints to the Classical Inference Methods of Estimation and Identification of the IRD Model  
While the present paper is focused upon new Bayes estimation methods, the topic of classical estimation has been already discussed in the literature [1; 2; 31; 4; 5; 6; 31], so in this brief section, only hints to the classical ML (Maximum Likelihood) estimation method of the IRD distribution are illustrated with some further discussion on the model identification. The extension to the RI estimation is straightforward and is briefly illustrated here for the ML method: by virtue of the well-known property of invariance of the ML estimation, if \(\phi\) * and \(F\) * are respectively the Maximum Likelihood estimates (MLE) of the quantities \(\phi\) and \(F\) in (10), then the MLE of the RI in (10) is expressed for any fixed value of time t and threshold \(z\) by the following:  
\[R^{*}(t,z)=1-e^{-\phi\tau[1-F^{*}(z)]} \tag{12}\]  
The MLE of the Poisson frequency is also well known in terms of the number of exceedances in a given interval (0,\(t\)) denoted by the random variable \(M\) (which counts the number of WS values in the given sample which exceed the value \(z\)) and is given by the following [31]:  
\[\phi^{*}=\frac{M}{t} \tag{13}\]  
Then, let us consider the MLE of the CDF \(F(z)\) appearing in (12) for an IRD model. First, we consider the MLE of parameter \(\alpha\) of the IRD model in Equations (1) and (2). The log-likelihood function \(\mathrm{L}(x|\alpha)\) of a dataset \(\mathbf{x}=\{x_{1},x_{2},\ldots,x_{n}\}\) of EWS values from an IRD model is simply deducible as follows:  
\[\mathrm{L}(x|\alpha)=N\log 2+N\log\alpha-\alpha\overset{N}{\underset{i=1}{ \sum}}\left[\left(\frac{1}{x_{i}}\right)^{3}\right]-3\overset{N}{\underset{i=1 }{\sum}}\log x_{i} \tag{14}\]  
Equating to zero the partial derivative of the above function with respect to parameter \(\alpha\) *, the MLE of such parameter is as follows:  
\[\alpha^{*}=\frac{N}{\underset{i=1}{\sum}}x_{k}^{-2} \tag{15}\]  
It is remarked that each RV 1/\(X_{k}\) follows a Raleigh distribution, so its square obeys an exponential law [1; 2; 31; 4; 5; 6; 31].  
It is known that a sum of independent exponentially distributed random variates gives a gamma distributed variable, giving the probability density function of the sum in the denominator of (15) as a gamma PDF with parameters: \(N\) (shape) and \(\alpha\) (scale).  
Finally, since--as already recalled--the MLE is invariant, the MLE of the CDF \(F(z)\) appearing in (12) is provided, looking at (2), by the following:  
\[F^{*}(z)=F(z|\alpha^{*})=e^{-(\frac{\alpha^{*}}{x^{2}})} \tag{16}\]  
Let us give a brief hint to the IRD model identification, which is very simple given the basic properties of the inverse Rayleigh model, i.e., the fact that it describes the distribution of \(X=1/\gamma\), with \(Y\) being a Rayleigh RV, for which identification is a well-established topic. Therefore, the goodness of fit tests for assessing the validity of an IRD model for a given set of EWS data \(\mathbf{x}=\{x_{1},x_{2},\ldots,x_{n}\}\) is simply performed (as exemplified in the Appendix B) assessing the validity of Weibull model with shape parameter \(\mathrm{b}=2\), i.e., a Rayleigh model, for the data \(\mathbf{y}=\{y_{1},y_{2},\ldots,y_{n}\}\), where \(y_{\mathrm{k}}=1/x_{\mathrm{k}}\).