On L convergence of Neumann series approximation in missing data problems.

The inverse of the nonparametric information operator is key to finding doubly robust estimators and the semiparametric efficient estimator in missing data problems. It is known that no closed-form expression for the inverse of the nonparametric information operator exists when missing data form nonmonotone patterns. Neumann series is usually applied to approximate the inverse. However, Neumann series approximation is only known to converge in L(2) norm, which is not sufficient for establishing statistical properties of the estimators yielded from the approximation. In this article, we show that L(∞) convergence of the Neumann series approximations to the inverse of the non-parametric information operator and to the efficient scores in missing data problems can be obtained under very simple conditions. This paves the way to the study of the asymptotic properties of the doubly robust estimators and the locally semiparametric efficient estimator in those difficult situations.