Zero variance self-normalized importance sampling via estimating equations

In ordinary importance sampling with a nonnegative integrand there exists an importance sampling strategy with zero variance. Practical sampling strategies are often based on approximating that optimal solution, potentially approaching zero variance. There is a positivisation extension of that method to handle integrands that take both positive and negative values. Self-normalized importance sampling uses a ratio estimate, for which the optimal sampler does not have zero variance and so zero variance cannot even be approached in practice. Strategies that separately estimate the numerator and denominator of that ratio can approach zero variance. This paper develops another zero variance solution for self-normalized importance sampling. The first step is to write the desired expectation as the zero of an estimating equation using Fieller's technique. Then we apply the positivisation strategy to the estimating equation. This paper give conditions for existence and uniqueness of the sample solution to the estimating equation. Then it give conditions for consistency and asymptotic normality and an expression for the asymptotic variance. The sample size multiplied by the variance of the asymptotic formula becomes arbitrarily close to zero for certain sampling strategies.