Estimation under unknown correlation: Covariance intersection revisited

This note addresses the problem of obtaining a consistent estimate (or upper bound) of the covariance matrix when combining two quantities with unknown correlation. The combination is defined linearly with two gains. When the gains are chosen a priori, a family of consistent estimates is presented in the note. The member in this family having minimal trace is said to be "family-optimal." When the gains are to be optimized in order to achieve minimal trace of the family-optimal estimate of the covariance matrix, it is proved that the global optimal solution is actually given by the Covariance Intersection algorithm, which conducts the search only along a one-dimensional curve in the n-squared-dimensional space of combination gains.