Estimation of parameters of boundary value problems for linear ordinary differential equations with uncertain data

TL;DR

This paper develops minimax estimation methods for boundary value problems of linear differential equations with uncertain data, providing optimal linear estimates and error bounds under various incomplete data scenarios.

Contribution

It introduces a novel minimax estimation framework for BVPs with uncertain data, deriving explicit solutions via differential equations and extending to incomplete data cases.

Findings

Minimax estimates are expressed through solutions of special differential systems.

Optimal estimates minimize the maximum mean square error over uncertainty sets.

Explicit representations for estimation errors are obtained.

Abstract

In this paper we construct optimal, in certain sense, estimates of values of linear functionals on solutions to two-point boundary value problems (BVPs) for systems of linear first-order ordinary differential equations from observations which are linear transformations of the same solutions perturbed by additive random noises. It is assumed here that right-hand sides of equations and boundary data as well as statistical characteristics of random noises in observations are not known and belong to certain given sets in corresponding functional spaces. This leads to the necessity of introducing minimax statement of an estimation problem when optimal estimates are defined as linear, with respect to observations, estimates for which the maximum of mean square error of estimation taken over the above-mentioned sets attains minimal value. Such estimates are called minimax estimates. We establish that the minimax estimates are expressed via solutions of some systems of differential equations of special type. Similar estimation problems for solutions of BVPs for linear differential equations of order n with general boundary conditions are considered. We also elaborate minimax estimation methods under incomplete data of unknown right-hand sides of equations and boundary data and obtain representations for the corresponding minimax estimates. In all the cases estimation errors are determined.