© 1984 by Biometrika Trust
Estimation of a structural parameter in the presence of a large number of nuisance parameters
Department of Mathematical Engineering and Instrumentation of Physics, University of Tokyo Tokyo, Japan
When the number of nuisance parameters increases in proportion to the sample size, the Cramer-Rao bound does not necessarily give an attainable lower bound for the asymptotic variance of an estimator of the structural parameter. The present paper presents a new lower bound under a criterion called information uniformity. The bound is expressed as the inverse of the sum of the partial information and a certain nonnegative term, which is derived by differential-geometrical considerations. The optimal estimating function meeting this lower bound, when it exists, is also obtained in a decomposed form. The first term is the modified score function, and the second term is, roughly speaking, given by the normal component of the mixture covariant derivative of some random variable. Furthermore, special versions of these results are given in concise form, and these are then applied to elucidate the efficiency of some examples.
Key Words: Asymptotic theory • Bound for asymptotic variance • Differential geometry • Estimating function • Exponential and mixture connexion • Mixture curvature • Nuisance parameter • Structural parameter