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Reduction of bias and skewness with applications to second order accuracy. (English) Zbl 1238.62024

Summary: Suppose \({\widehat{\theta}}\) is an estimator of \(\theta \) in \({\mathbb{R}}\) that satisfies the central limit theorem. In general, inference on \(\theta \) is based on the central limit approximation. This has error \(O(n ^{ - 1/2})\), where \(n\) is the sample size. Many unsuccessful attempts have been made at finding transformations which reduce this error to \(O(n ^{ - 1})\). The variance stabilizing transformation fails to achieve this. We give alternative transformations that have bias \(O(n ^{ - 2})\), and skewness \(O(n ^{ - 3})\). Examples include the binomial, Poisson, chi-square and hypergeometric distributions.

MSC:

62F10 Point estimation
62F12 Asymptotic properties of parametric estimators
60F05 Central limit and other weak theorems
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