## 3 Robust Parametric Algorithms and Their Application for Estimating Kinetic Wind Energy  
Let us consider the problem at the parametric level of prior uncertainty. For the sample \(\overrightarrow{Z}_{N}=(\zeta_{1},\ldots,\zeta_{N})\) with the distribution \(F(\zeta,\overrightarrow{\theta})\), we first derived the maximum likelihood estimate (MLE) \(\overrightarrow{\theta}_{N}^{*}\) in \(F(\zeta,\overrightarrow{\theta})\). By analogy with [5; 8], it can be shown that the consistent asymptotically unbiased effective estimates \(\overset{\rightarrow}{\theta}_{N}^{*}=\left(\theta_{1N}^{*},\ldots,\theta_{KN}^{*} \right)^{T}\) for the prior distribution \(G(\zeta,\overset{\rightarrow}{\theta})\) are determined from the system of estimation equations:  
\[\frac{1}{N}\overset{\rightarrow}{\overset{\rightarrow}{\overset{\rightarrow }{\overset{\rightarrow}{\theta}}}}U\left(\zeta,\overset{\rightarrow}{\overset{ \rightarrow}{\overset{\rightarrow}{\theta}}}_{N}\right)W\left(\zeta,\overset{ \rightarrow}{\overset{\rightarrow}{\overset{\rightarrow}{\theta}}}_{N} \right)=0,\;j=\overline{1,k}, \tag{3}\]  
\[U_{j}(\zeta,\overset{\rightarrow}{\overset{\rightarrow}{\overset{ \rightarrow}{\theta}}}_{N})=\frac{\partial}{\partial\theta_{j}}\ln g(\zeta, \overset{\rightarrow}{\overset{\rightarrow}{\theta}}_{N})\bigg{|}_{\Theta= \overset{\rightarrow}{\overset{\rightarrow}{\theta}}_{N}^{*}}, \tag{4}\]  
\[W(\zeta,\overset{\rightarrow}{\overset{\rightarrow}{\theta}}_{N})=\frac{(1- \epsilon)g(\zeta,\overset{\rightarrow}{\overset{\rightarrow}{\theta}}_{N})}{ f(\zeta,\overset{\rightarrow}{\overset{\rightarrow}{\theta}}_{N})}. \tag{5}\]  
The estimates \(\overset{\rightarrow}{\overset{\rightarrow}{\theta}}_{N}^{*}\) from the system of estimation Equation (3) are determined by iterations and are estimates of the weighted maximum likelihood method (WMLM). The efficiency of the estimates follows from the MLM, the unbiasedness of the estimates for \(G(\zeta,\overset{\rightarrow}{\overset{\rightarrow}{\theta}})\) was shown in [12], and their robustness follows from the bounded variances provided with the weight functions \(W(\zeta,\overset{\rightarrow}{\overset{\rightarrow}{\theta}}_{N}^{*})\) given by Equation (5). As an example, we considered the algorithms for robust estimates of the mean and variance \(\overset{\rightarrow}{\theta}=(\mu,s)\) of the prior normal distribution \(G(\zeta,\overset{\rightarrow}{\theta})\):  
\[g(\zeta,u)=\frac{1}{s\sqrt{2\pi}}\exp\left(-\frac{u^{2}}{2}\right),\;f(\zeta,u, \
u,\lambda)=(1-\epsilon)g(\zeta,u)+\epsilon g(\zeta,\
u,\lambda),\]  
\[U_{1}(\zeta,u)=u,\;U_{2}(\zeta,u)=\left[u^{2}-1\right],\;W(\zeta,\
u,\lambda)= \frac{(1-\epsilon)g(\zeta,\mu,s)}{f(\zeta,\
u,\lambda)},\;u=\frac{\zeta-\mu}{ s}. \tag{6}\]  
The plots of the corresponding distribution densities, weight functions, and estimate functions for the normal distribution are shown in Figure 1 for the indicated outlier fractions. Figure 2a shows an example of the parametric estimates of the vertical profiles of the total kinetic wind energy without (\(E_{0}\), the orange curve) and with an allowance for the contribution of the kinetic outlier energy (\(E\), the blue curve) retrieved from measurements with the commercial triaxial Doppler monostatic minisodar AV4000 (Atmospheric Systems Corporation, Santa Clara, CA, USA) [4]; its sounding range was 5-200 m with vertical resolution \(\Delta z=5\) m. The acoustic antenna was an array of 50 loudspeakers used to both transmit and receive acoustic signals at a frequency of 4900 Hz. This loudspeaker array was electrically steered to generate three independent beams: one vertical and two others at elevation angles of 76\({}^{\circ}\) in two mutually orthogonal planes. The minisodar had a pulse repetition period of 4 s and a pulse duration of 60 ms.