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October 1997 On second-order optimality of the observed Fisher information
Bruce G. Lindsay, Bing Li
Ann. Statist. 25(5): 2172-2199 (October 1997). DOI: 10.1214/aos/1069362393


The realized error of an estimate is determined not only by the efficiency of the estimator, but also by chance. For example, suppose that we have observed a bivariate normal vector whose expectation is known to be on a circle. Then, intuitively, the longer that vector happens to be, the more accurately its angle is likely to be estimated. Yet this chance, though its information is contained in the data, cannot be accounted for by the variance of the estimate. One way to capture it is by the direct estimation of the realized error. In this paper, we will demonstrate that the squared error of the maximum likelihood estimate, to the extent to which it can be estimated, can be most accurately estimated by the inverse of the observed Fisher information. In relation to this optimality, we will also study the properties of several other estimators, including the inverse of the expected Fisher information, the sandwich estimators, the jackknife and the bootstrap estimators. Unlike the observed Fisher information, these estimators are not optimal.


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Bruce G. Lindsay. Bing Li. "On second-order optimality of the observed Fisher information." Ann. Statist. 25 (5) 2172 - 2199, October 1997.


Published: October 1997
First available in Project Euclid: 20 November 2003

zbMATH: 0881.62023
MathSciNet: MR1474089
Digital Object Identifier: 10.1214/aos/1069362393

Primary: 62F10 , 62F12

Keywords: ancillarity , asymptotically linear estimator , Bhattacharyya scores , bootstrap , conditional inference , cumulant , estimation of loss , generalized inverse , jackknife , observed and expected Fisher information , sandwich estimators

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 5 • October 1997
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