A definition of estimator efficiency ink-parameter case
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The paper introduces a new definition of efficiency in the multiparameter case (θ1,...,θk) when the variance-covariance matrix of the vector estimator (t 1, ...t k) exists. The definition is also applicable to the asymptotically unbiased estimators.
The basic idea is that, as we want in general to estimate some function g(θ1,...θk) of the parameters, efficiency of the vector estimator shall be defined as the smallest efficiency of the estimatorg(t 1, ...t k),g being regular. It is shown that this definition is asymptotically equivalent to the one obtained by any linear combination of the estimators, as it happens, naturally, for quantile estimation in the location-dispersion case. This efficiency is larger than Cramér efficiency which is, thus, not attained, apart from a very exceptional case.
Finally, a lower bound for the asymptotic variance is obtained.
Key wordsMultidimensional parameters and estimators functions of parameters and estimators asymptotic unbiasedness variance-covariance matrices efficiency lower bound of the variance
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