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

The inverse of the nonparametric information operator is key for 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. The Neumann series is usually used for approximating the inverse. However, the Neumann series approximation is only known to converge in L2L2 norm, which is not sufficient for establishing statistical properties of the estimators yielded from the approximation. In this work, we show that L∞L∞ convergence of the Neumann series approximations to the inverse of the nonparametric information operator and to the efficient scores in missing data problems can be obtained under very simple conditions. This paves the way to a study of the asymptotic properties of the doubly robust estimators and the locally semiparametric efficient estimator in those difficult situations.