Restoration of a Signal with the Bounded Second-Order Derivative by the Distributionally Robust Optimization

An estimation problem distributionally robust with respect to random noise is considered for a signal with the bounded second-order derivative on a finite number of observations. The objective functional is the probability that the L2-norm of the estimation error will exceed a pre-specified threshold. Its worst-case value on the set of all signals with the bounded second-order derivative and arbitrary distributions of the noise vector with fixed mean and covariance is to be minimized over the finite-dimensional class of spline estimators. The optimization problem is solved using the methods of convex programming by representing the objective functional in terms of the mean square bound following Markov's inequality and the tight bound in the form of the multivariate Selberg inequality. A numerical experiment is carried out to compare the obtained solutions to the problem of restoration of the trajectory of a target with bounded acceleration.