## 4 Robust Semiparametric Algorithms and Their Application for Estimating Kinetic Wind Energy  
Let us consider the problem at the semiparametric level of prior uncertainty. In this case, it is assumed that \(G(\zeta,\overset{\rightarrow}{\theta})\) is known to within a finite number of the unknown parameters \(\overset{\rightarrow}{\theta}\), and the information on the outliers \(\{\varepsilon,H(\zeta)\}\), that is, on the form of the distribution function \(H(\zeta)\) and, hence, \(F(\zeta,\overset{\rightarrow}{\theta})\) is unknown. For the sample \(\overset{\rightarrow}{Z}_{N}=(\zeta_{1},\ldots,\zeta_{N})\) from \(F(\zeta,\overset{\rightarrow}{\theta})\), we obtained the estimate \(\overset{\rightarrow}{\theta}_{N}\) by the maximum likelihood method. By analogy with [12], it is possible to show that the asymptotically effective (unbiased and consistent) estimates \(\overset{\rightarrow}{\theta}_{N}^{*}=(\theta_{1N}^{*},\ldots,\theta_{kN}^{*} )^{T}\) of the WMLM for the prior distribution \(G(\zeta,\overset{\rightarrow}{\theta})\) are determined from the system of the estimation equations:  
\[\frac{1}{N}\sum_{i=1}^{N}U(\zeta_{i},\overset{\rightarrow}{\theta}_{N}^{*})W _{N}(\zeta_{i},\overset{\rightarrow}{\theta}_{N}^{*})=0,\ j=\overline{1,k}, \tag{7}\]  
\[U_{j}(\zeta,\overset{\rightarrow}{\theta}_{N}^{*})=\frac{\partial}{\partial \theta_{j}}\ln g(\zeta,\overset{\rightarrow}{\theta}_{N}^{*})\Bigg{|}_{ \overset{\rightarrow}{\theta}_{N}^{*}}, \tag{8}\]  
Figure 2: Vertical profiles of the parametric estimates of the kinetic wind energy in the atmosphere obtained by postprocessing of the 10 min AV4000 minisodar measurement series on 10 September 2003, started at 12:00, local time: (**a**) the total kinetic energy without (\(E_{0}\), the orange curve) and with allowance for the contribution of the kinetic outlier energy (\(E\), blue curve), and (**b**) turbulence kinetic energy component without (\(E_{\rm OT}\), the orange curve) and with allowance for the contribution of the kinetic outlier energy (\(E_{\rm T}\), the blue curve).  
To determine the kinetic energy \(E_{0}\) of the stationary process, we needed to determine the mean \(\overline{X}(\overrightarrow{\theta})\) and variance \(\sigma^{2}(\overrightarrow{\theta})\) of the distribution \(G(\zeta,\overrightarrow{\theta})\):  
\[\overline{X}(\overrightarrow{\theta})=\int\limits_{-\infty}^{\infty}\zeta \cdot g(\zeta,\overrightarrow{\theta})d\zeta,\ \sigma^{2}(\overrightarrow{\theta})=\int\limits_{-\infty}^{\infty}\left( \zeta-\overline{X}(\overrightarrow{\theta})\right)^{2}\cdot g(\zeta, \overrightarrow{\theta})d\zeta. \tag{12}\]  
Defining from Equations (8)-(10) the asymptotically effective robust estimates of the parameters \(\overrightarrow{\theta}_{N}^{*}=\left(\vartheta_{1N}^{*},\ldots,\vartheta_{ kN}^{*}\right)^{T}\) and substituting them into Equation (8), we obtained for \(\overline{X}(\overrightarrow{\theta})\) and \(\sigma^{2}(\overrightarrow{\theta})\) the asymptotically effective robust estimates of the mean \(\overline{X}_{N}(\overrightarrow{\theta}_{N}^{*})\) and variance \(\sigma_{N}^{2}(\overrightarrow{\theta}_{N}^{*})\) and determined on their basis the kinetic energy \(E_{0}\) of the stationary process.