*4.2. The Least Favorable Distribution in the Light of the Pareto Efficiency*

The minimax estimation problem under the conformity constraints is tightly interconnected with the choice of the distribution *<sup>F</sup>* that is optimal in the sense of a vectorvalued criterion. On the one hand, the solution to the considered estimation problem is grounded on the evaluation of the distribution *<sup>F</sup>*, maximizing the dual criterion (12): **I**1(*F*|*y*) *J*∗(*F*|*y*) → max*F*. On the other hand, the distribution *F* should conform to the realized sample *Y* = *y*, and the maximization of the conformity index leads to the following optimization problem: **I**2(*F*|*y*) L(*y*, *F*) → max*F*.

Obviously, the criteria **I**<sup>1</sup> and **I**<sup>2</sup> are conflicting; hence the proper choice of *F* requires the application of the vector optimization techniques.

Let:


**Lemma 3.** *Any least favorable distribution <sup>F</sup><sup>M</sup>* <sup>∈</sup> <sup>F</sup>*<sup>M</sup> is Pareto-efficient with respect to the vector-valued criterion* (**I**1,**I**2)*.*

The proof of Lemma 3 follows directly from the Germeyer theorem [16].

Consideration of the constrained minimax estimation problem in light of the optimization by the vector criterion is somehow close to the one investigated in [31], where the

estimation quality is characterized by the <sup>2</sup> norm of the error, and the Shannon entropy is characterized as a measure of the statistical uncertainty of the estimated vector.
