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Abstract
We make use of Cramer 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 [square root]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 Crame´r 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 Cramer conditions. Here we make our Cramer condition assumptions and their use explicit.
Keywords
Cramer conditions,
quadratic mean differentiability,
likelihood ratio,
Wald tests,
Rao tests,
asymptotics,
confidence intervals,
mathematical statistics,
equations,
quadratic differentials,
statistical hypothesis testing,
asymptotic efficiences,
statistics,
efficient likelihood estimators,
Cramer conditions,
Newton estimators,
coefficients of variation,
statistical analysis,
asymptotic relative efficiency
Citation
Verrill, Steve. 2007. Rate of convergence of k-step Newton estimators to efficient likelihood estimators. Statistics & probability letters. Vol. 77 (2007): Pages 1371-1376.
