Parameterized Joint Densities with Gaussian Mixture Marginals and their Potential Use in Nonlinear Robust Estimation

Abstract — This paper addresses the challenges of the fusion of two random vectors with imprecisely known stochastic dependency. This problem mainly occurs in decentralized estimation, e.g. of a distributed phenomenon, where the stochastic dependencies between the individual states are not stored. To cope with such problems we propose to exploit parameterized joint densities with both Gaussian marginals and Gaussian mixture marginals. Under structural assumptions these parameterized joint densities contain all information about the stochastic dependencies between their marginal densities in terms of a generalized correlation parameter vector ξ. The parameterized joint densities are applied to the prediction step and the measurement step under imprecisely known correlation leading to a whole family of possible estimation results. The resulting density functions are characterized by the generalized correlation parameter vector ξ. Once this structure and the bounds of these parameters are known, it is possible to find bounding densities containing all possible density functions, i.e., conservative estimation results. I