Rate of convergence of k-step Newton estimators to efficient likelihood estimators

We make use of Cramér conditions together with the well-known local quadratic convergence of Newton's method to establish the asymptotic closeness of k-step Newton estimators to efficient likelihood estimators. In Verrill and Johnson [2007. Confidence bounds and hypothesis tests for normal distribution coefficients of variation. USDA Forest Products Laboratory Research Paper FPL-RP-638], we use this result to establish that estimators based on Newton steps from n-consistent estimators may be used in place of efficient solutions of the likelihood equations in likelihood ratio, Wald, and Rao tests. Taking a quadratic mean differentiability approach rather than our Cramér condition approach, Lehmann and Romano [2005. Testing Statistical Hypotheses, third ed. Springer, New York] have outlined proofs of similar results. However, their Newton step estimator results actually rely on unstated assumptions about Cramér conditions. Here we make our Cramér condition assumptions and their use explicit.