In a somewhat general context, and in a variety of special cases, we calculate the order in probability of the difference between consistent roots of rival estimating equations, with application to point estimators in parametric and nonparametric models, interval estimators, and test statistics. The emphasis is on comparison of statistics having the same first-order asymptotic distribution, our results indicating the degree to which their stochastic expansions correspond. Differences in the statistical performances of various commonly-used iterative procedures are detected. We discuss implications of our results for higher-order efficiency comparisons, and for matching the first-order efficiency of an implicitly defined target statistic in finitely many iterative steps, commenced by an estimator that is consistent but not T^1^/^2 -consistent, justifying as suitable initial estimators ones obtained via a search of the objective function, and nonparametric methods. Some of our results are applied to the linear-in-variables simultaneous equations system.