Abstract
The paper is devoted to the guaranteeing estimation of parameters in the uncertain stochastic nonlinear regression. The loss function is the conditional mean square of the estimation error given the available observations. The distribution of regression parameters is partially unknown, and the uncertainty is described by a subset of probability distributions with a known compact domain. The essential feature is the usage of some additional constraints describing the conformity of the uncertain distribution to the realized observation sample. The paper contains various examples of the conformity indices. The estimation task is formulated as the minimax optimization problem, which, in turn, is solved in terms of saddle points. The paper presents the characterization of both the optimal estimator and the set of least favorable distributions. The saddle points are found via the solution to a dual finite-dimensional optimization problem, which is simpler than the initial minimax problem. The paper proposes a numerical mesh procedure of the solution to the dual optimization problem. The interconnection between the least favorable distributions under the conformity constraint, and their Pareto efficiency in the sense of a vector criterion is also indicated. The influence of various conformity constraints on the estimation performance is demonstrated by the illustrative numerical examples.
Highlights
The problems of the heterogeneous parameter estimation in the regression under the model uncertainty are considered intensively from the various points of view
The paper aims to present a solution to the minimax estimation problem under additional constraints, which are determined by a conformity index of the uncertain parameters to the available observations
We present an assertion that the least favorable distributions (LFD) in the minimax estimation problem is Pareto-efficient in the sense of the introduced vector criterion
Summary
The problems of the heterogeneous parameter estimation in the regression under the model uncertainty are considered intensively from the various points of view. The parameters are supposed to be random with unknown distribution, and the uncertainty set is formed by all the admissible distributions In both cases, the guaranteeing estimation presumes a solution to a two-person game problem: the first player is “a statistician”, and the performer of the second,. Let us explain this point by an example: the statistician knows that the source of the uncertainty is nature This means he/she “should bear in mind that nature, as a player, is not aiming for a maximal win (that is, does not want us to suffer a maximal loss), and in this sense, it is ‘impartial’ in the choice of strategies” [12]. The paper aims to present a solution to the minimax estimation problem under additional constraints, which are determined by a conformity index of the uncertain parameters to the available observations. EF { X } is a mathematical expectation of the random vector X with the distribution F; conv(S) is a convex hull of the set S
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