Abstract
A bound of Cramer-Rao type is provided for an estimator of a real-valued parameter θ in the presence of an abstract nuisance parameter η, such as an unknown distributional shape, on the basis of N i.i.d. observations. The bound consists of the reciprocal of the effective Fisher information in the sample, plus a term involving the integrated mean squared error of an estimator of a multiple of the so-called conditional score function for θ, for the case where θ is known. This implies that an estimator of θ can only perform well over a class of shapes η if it is possible to estimate the conditional score function for θ accurately over this class. For the special case where fully adaptive estimation may be possible, this result was given in a companion paper (Klaassen and van Zwet (1985)).
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Bibliography
Begun, J. M., Hall, W. J., Huang, W. M. and Wellner, J. A. (1983). Information and asymptotic efficiency in parametric-nonparametric models. Ann. Statist. 11, 432-452.
Klaassen, C. A. J. (1987). Consistent estimation of the influence function of locally asymptotically linear estimators. Ann. Statist., to appear.
Klaassen, C. A. J. and van Zwet, W. R. (1985). On estimating a parameter and its score function. Proc. Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer, L. Le Cam and R. A. Olshen, editors, Vol. II, 827-839.
van der Vaart, A. W. (1986). Estimating a real parameter in a class of semiparametric models. Submitted to Ann. Statist.
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Klaassen, C.A.J., van der Vaart, A.W., van Zwet, W.R. (2012). On Estimating a Parameter and Its Score Function, II. In: van de Geer, S., Wegkamp, M. (eds) Selected Works of Willem van Zwet. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1314-1_8
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DOI: https://doi.org/10.1007/978-1-4614-1314-1_8
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