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For a version of the interval censoring model, case 2, in which the observation intervals are allowed to be arbitrarily small, we consider estimation of functionals that are differentiable along Hellinger differentiable paths. The asymptotic information lower bound for such functionals can be represented as the squared L2-norm of the canonical gradient in the observation space. This canonical gradient has an implicit expression as a solution of an integral equation that does not belong to one of the standard types. We study an extended version of the integral equation that can also be used for discrete distribution functions like the nonparametric maximum likelihood estimator NPMLE., and derive the asymptotic normality and efficiency of the NPMLE from properties of the solutions of the integral equations

Asymptotically optimal estimation of smooth functionals for interval censoring, case $2$