Abstract: We consider nonparametric likelihoods for the mean of an unknown distribution using estimating equations for moments of order one and greater. Although empirical likelihood is the same regardless of the number of estimating equations used, use of two or more such estimating equations with the empirical exponential family gives a likelihood that agrees with empirical likelihood to third order. We show that the empirical exponential family using an arbitrary number of moments is a least favorable family. Simulations indicate that empirical exponential family using estimating equations for moments of order one and greater is very close to empirical likelihood and that a Wald statistic constructed using the empirical exponential family gives good coverage.
Key words and phrases: Empirical exponential family, empirical likelihood, estimating equation, least favorable family, nonparametric likelihood, nuisance parameter.