Small-sample asymptotic distributions of M-estimators of location

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Abstract
Asymptotic formulae for the distribution of M-estimators, i.e. maximum likelihood type estimators, of location, including the arithmetic mean, are derived which numerical studies show to give relative errors for densities and tail areas of the order of magnitude of 1% down to sample sizes 3 and 4 even in the extreme tails. The paper is the continuation of earlier work by the second author and is also closely related to Daniels's work on the saddlepoint approximation. The method consists in expanding the derivative of the logarithm of the unstandardized density of the estimator in powers of 1/n at each point, using recentring by means of conjugate distributions. This method yields a unified point of view for the comparison of other asymptotic methods, namely saddlepoint method, Edgeworth expansion and large deviations approach, which are also compared numerically.

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