## 2 Problem Formulation
The unbiased and consistent estimates of \(\hat{E}_{0}=(\hat{E}_{M0},\hat{E}_{T0})\) may be calculated from the unbiased and consistent estimates of the distribution \(G(\zeta,\overrightarrow{\theta})\) that may be obtained using the robust algorithms for the distribution \(F(\zeta,\overrightarrow{\theta})\) based on the samples \(V_{xi}(z_{j},t_{k})\), \(V_{yi}(z_{j},t_{k})\), \(V_{zi}(z_{j},t_{k})\), \(i=1,\ldots,N\), \(t_{k}\in T\), from the distribution \(F(\zeta,\overrightarrow{\theta})\). Finally, the unbiased and consistent estimate of the outlier energy \((\hat{E}_{M0},\hat{E}_{T0})\) is equal to the difference \(\hat{E}_{\text{out}}=(\hat{E}-\hat{E}_{0})\). Hence, the main problem is reduced to obtaining asymptotically unbiased, consistent, and effective estimates of the parameters of the distribution \(G(\zeta,\overrightarrow{\theta})\) based on the samples from the distribution \(F(\zeta,\overrightarrow{\theta})\); that is, the robust estimates for the distribution \(F(\zeta,\overrightarrow{\theta})\) based on prior information on the forms of the distributions \(F(\zeta,\overrightarrow{\theta})\), \(G(\zeta,\overrightarrow{\theta})\), and \(H(\zeta)\) and their superpositions. As already indicated above, robust statistics based on different robustness criteria cannot guarantee unbiased, consistent, and effective estimates of parameters, especially for asymmetric distributions \(G(\zeta,\overrightarrow{\theta})\)[6; 10; 11; 19]. Considering the algorithms for robust estimation on different levels of prior information based on the weighted maximum likelihood method, we can formalize the problem. To simplify the derivation of the robust algorithms, we designated the samples \(V_{xi}(z_{j},t_{k})\), \(V_{yi}(z_{j},t_{k})\), and \(V_{zi}(z_{j},t_{k})\), \(i=1,\ldots,N\), by \(\overrightarrow{Z}_{N}=(\zeta_{1},\ldots,\zeta_{N})\). Let \(\overrightarrow{Z}_{N}=(\zeta_{1},\ldots,\zeta_{N})\) be the inhomogeneous sample of independent random variables with the distribution function \(F(\zeta,\overrightarrow{\theta})\in P\), where \(P\) is the class of distributions in the form of the mixture of the distributions satisfying the regularity conditions of the maximum likelihood method (MLM) [6; 10; 11]:  
\[F(\zeta,\overrightarrow{\theta})=(1-\epsilon)G(\zeta,\overrightarrow{\theta })+\epsilon H(\zeta), \tag{2}\]  
where \(G(\zeta,\overrightarrow{\theta})\in P_{0}\) is the a priori model of the distribution function, \(H(\zeta)\in P\) is the outlier distribution, \(\epsilon\geq 0\) is the outlier fraction, and \(\overrightarrow{\theta}=(\theta_{1},\ldots,\theta_{k})^{T}\) is the unknown parameter vector of the distribution. We designated by \(f(\zeta,\overrightarrow{\theta})\), \(g(\zeta,\overrightarrow{\theta})\), and \(h(\zeta)\) the corresponding distribution densities. Considering the problem of constructing the robust effective estimate \(\overrightarrow{\theta}_{N}^{*}=(\theta_{1N}^{*},\ldots,\theta_{kN}^{*})^{T}\) for the unknown parameter \(\overrightarrow{\theta}=(\theta_{1},\ldots,\theta_{k})^{T}\) of the prior distribution \(G(\zeta,\overrightarrow{\theta})\) from the inhomogeneous sample \(\overrightarrow{Z}_{N}=(\zeta_{1},\ldots,\zeta_{N})\) with the distribution \(F(\zeta,\theta)\in P_{0}\), the mathematical model (2) can be based on both parametric and nonparametric models of the distribution function and on their superpositions--semiparametric and semi-nonparametric models.