Approximate maximum entropy on the mean for instrumental variable regression

We want to estimate an unknown finite measure μXμX from a noisy observation of generalized moments of μXμX, defined as the integral of a continuous function ΦΦ with respect to μXμX. Assuming that only a quadratic approximation ΦmΦm is available, we define an approximate maximum entropy solution as a minimizer of a convex functional subject to a sequence of convex constraints. We establish asymptotic properties of the approximate solution under regularity assumptions on the convex functional, and we study an application of this result to instrumental variable estimation.