Journal of Statistics: Advances in Theory and ApplicationsAuthor's: AO YUAN and QIZHAI LI
Pages: [105] - [133]
Received Date: June 18, 2013
Submitted by:
The convolution theorem (Hájek [8]) characterizes the weak limit of any regular estimator as a convolution of two independent components. One is an optimal achievable part and another is a noise. Therefore, the optimal estimator is one without the noise part in its weak limit, which is a deeper characterization than the Cramer-Rao bound. However, this result is derived under the assumption that the specified model is the true one generating the data. In practice, any subjectively specified model is more or less deviated from the true one. The convolution representation (and the Cramer-Rao bound) should be modified to reflect this fact. Here, we study such modifications for the estimation of parameters under several cases: Euclidean parameter, Euclidean parameter with side information; Euclidean parameter with infinite-dimensional nuisance parameter; and the case of infinite-dimensional parameter. In each case, we decompose the weak limit of a regular estimator into three independent components, with one achievable optimal part, and two noise parts. When the specified model is indeed the true one, it reduces to existing convolution representation of two components.
convolution theorem, Cramer-Rao bound, efficient estimator, Euclidean parameter, infinite-dimensional parameter, nuisance parameter, optimal weak limit, side information.
