**2. Statement of Problem**

$$2.1. \text{ }Formulation$$

Let us consider the following observation model:

$$Y = A(X, \gamma) + B(X, \gamma)V. \tag{1}$$

Here:


The observation model is defined on the family of the probability triplets {(Ω, F, <sup>P</sup>*F*)}*F*∈F, where:


$$\mathbb{P}\_{\mathbb{F}}\{\gamma \in dq, \ X \in d\mathfrak{x}, \ V \in dv\} \stackrel{\triangle}{=} \mathbb{P}(d\mathbf{x}|q)\mathbb{P}(dq)\varrho\_{V}(v)dv. \tag{2}$$

Using the generalized Bayes rule [17], it is easy to verify that the function:

$$\mathcal{L}(y|q) \triangleq \int\_{\mathbb{R}^n} |\det(B(q, \mathbf{x}))|^{-1} \phi\_V(B^{-1}(q, \mathbf{x})(y - A(q, \mathbf{x}))) \Psi(d\mathbf{x}|q) \tag{3}$$

is the conditional pdf of the observation *Y* given *γ*: P*F*{*Y* ∈ *dy*|*γ* = *q*} = L(*y*|*q*)*dy*. Furthermore, the function:

$$\mathfrak{L}(y, F) \stackrel{\Delta}{=} \int\_{\mathcal{C}} \mathcal{L}(y|q) F(dq) \tag{4}$$

defines the pdf of the observation *Y* under the assumption that the distribution law of *γ* equals *F*:

$$\mathfrak{L}(y, F) = \frac{\mathbb{P}\_F\{Y \in dy\}}{dy} = \int\_{\mathcal{C}} \mathcal{L}(y|q) F(dq). \tag{5}$$

Below in the paper we refer to the function L(*y*, *F*) as *the sample conformity index based on the likelihood function.*

Our aim is to estimate the function *<sup>h</sup>*(*γ*, *<sup>X</sup>*) : C × <sup>R</sup>*<sup>m</sup>* <sup>→</sup> <sup>R</sup>*<sup>l</sup>* of (*γ*, *<sup>X</sup>*), and the admissible estimators are the functions *<sup>h</sup>*(*Y*) : <sup>R</sup>*<sup>k</sup>* <sup>→</sup> <sup>R</sup>*<sup>l</sup>* of the available observations.

The loss function is a conditional mean square of estimate error given the available observations:

$$J(\overline{h}, F | \underline{y}) \triangleq \mathbb{E}\_F \left\{ \|h(\gamma, X) - \overline{h}(Y)\|^2 | Y = \underline{y} \right\},\tag{6}$$

and the corresponding estimation criterion:

$$J^\*(\overline{h}|\mathcal{y}) \stackrel{\triangle}{=} \sup\_{F \in \mathbb{F}\_L} J(\overline{h}, F|\mathcal{y}) \tag{7}$$

characterizes the maximal loss for a fixed estimator *h* within the class F*<sup>L</sup>* of the uncertain distributions of *γ*, for which L(*y*, *F*) -*L*.

*The minimax estimation problem* for the vector *<sup>h</sup>* is to find an estimator *h*(·), such that:

$$
\widehat{h}(y) \in \underset{\overline{\mathfrak{h}} \in \mathbb{H}}{\text{Argmin}} \, J^\*(\overline{h}|y),
\tag{8}
$$

where H is a class of admissible estimators.
