## 5 Robust Semi-Nonparametric Algorithms and Their Application for Estimation of Kinetic Wind Energy
Let \(\beta\) be the unknown parameter of \(G(\zeta,\theta)\in\widetilde{P}_{0\theta}\) in the form:  
\[\beta=\int\rho(x,\theta)dG(x,\theta)\text{ or }\int\rho(x,\theta)dG(x,\theta)- \beta=0 \tag{19}\]  
To derive the robust estimate \(\beta_{N}\) of the parameter \(\beta\), consider the estimation equation in the form:  
\[\frac{1}{N}\sum_{i=1}^{N}\widetilde{\omega}_{N}(\zeta_{i},\theta_{N})=0, \tag{20}\]  
where the estimation function \(\widetilde{\varphi}_{N}(\zeta,\theta)\) has the form:  
\[\widetilde{\omega}_{N}(\zeta,\theta)=(\rho(\zeta,\theta)-\beta)\widetilde{W} _{N}(\zeta,\theta) \tag{21}\]  
\(\widetilde{W}_{N}(\zeta,\theta)=\frac{\widetilde{g}_{N}(\zeta,\theta)}{f_{N}( \zeta,\theta)}\) is the weight function, \(f_{N}(\zeta)\) is the nonparametric estimate of the Parzen-Rosenblatt density given by Equation (10), and \(\widetilde{g}_{N}(\zeta,\theta)\) is the modified nonparametric estimate of the Parzen-Rosenblatt density. Formulas (20) and (21) define the robust semi-nonparametric estimate \(\beta_{N}\) of the parameter \(\beta\). A mathematical analysis of these estimates is beyond the scope of the present work. Note only that they are asymptotically unbiased and efficient for the distribution \(G(\zeta,\theta)\in\widetilde{P}_{0\theta}\), provided that the sample \(\overset{\rightarrow}{Z}_{N}=(\zeta_{1},\ldots,\zeta_{N})\) is from \(F(\zeta,\overset{\rightarrow}{\theta})\). For example, let \(G(\zeta,\theta)\in\widetilde{P}_{0\theta}\) be the class of symmetric functions of the shear parameter \(\theta\). Let us designate by \((\mu,s)\) the average and variance of \(G(\zeta,\theta)\). In this case, the estimation equations for the robust nonparametric estimates \((\mu_{N},s_{N})\) and normal kernel function (7) take the following forms:  
\[\sum_{i=1}^{N}\sum_{j=1}^{N}\big{(}\mu_{N}-t_{ij}\big{)}\frac{\widetilde{g}_{N }\bigg{(}-\frac{\big{(}\mu_{N}-t_{ij}\big{)}^{2}}{2\sigma_{N}^{2}}\bigg{)}}{f_{ N}(\zeta_{i})}=0,\]  
\[\widetilde{g}_{N}(\zeta)=\frac{1}{N}\!\sum_{i=1}^{N}\frac{1}{h_{N}}k(\frac{ \zeta-\zeta_{i}}{h_{N}})\left[1-\left\|\frac{1}{N}\!\sum_{i=1}^{N}\psi_{I}( \zeta_{i})\right\|\Lambda_{N}^{-1}\left\|\frac{1}{N}\!\sum_{i=1}^{N}\psi_{j}( \zeta_{i})\right\|^{T}\right]. \tag{14}\]0  
where \(t_{ij}=\frac{\zeta_{i}+\zeta_{j}}{2}\) are the Walsh half-sums and \(\widetilde{g}_{N}(\zeta,\theta)\) is defined by Formula (10). The estimates are determined by iterations. The estimates of the kinetic energies of the stationary process without outliers, \(E_{0}\), and with the outliers, \(E_{\text{out}}\), were obtained using these formulas. Table 1 presents the semi-nonparametric estimates of the altitude dependence of the kinetic wind energy without outliers, \(E_{0}\), the kinetic wind energy with an allowance for the contribution of the outliers, \(E\), and their difference \(E-E_{0}=E_{\rm out}\) equal to the kinetic energy of the outliers obtained by the postprocessing of the minisodar measurements. Consider first the semi-nonparametric estimates \(E_{0}\) of the kinetic energy without outliers. Their nearly monotonic increase with altitude could be easily traced from 20 to 200 m. The large \(E_{0}\) value at \(z=10\) m is apparently caused by the effect of the underlying surface. The rate of \(E_{0}\) growth increased with altitude starting from \(z=150\) m. Our statistical analysis of the altitude behavior of the total kinetic energy \(E\) with an allowance for the contribution of the kinetic energy of outliers revealed the following. The \(E\) values differed only slightly from the corresponding \(E_{0}\) values at altitudes up to 50 m, that is, in this altitude range, the contribution of the kinetic outlier energy to the total kinetic energy was low. The situation radically changed above 50 m. An essential nonmonotonic increase in \(E\) values was observed with a further increase in the altitude, accompanied by considerable deviations from the monotonic dependence.