Third-order optimum properties of estimator-sequences |
| |
Authors: | Dr Th Pfaff |
| |
Institution: | 1. Regionales Hochschulrechenzentrum Kaiserslautern, Postfach 3049, D-6750, Kaiserslautern
|
| |
Abstract: | SequencesT (n) ,n∈N, are considered, whereT (n) estimates a vector parameter ?∈R p from an i.i.d. sample of sizen, and such sequences are compared on the basis of their risks ∫L(n 1/2(T n (x)?θ))P θ n (dx) relative to loss functionsL:R p →R. A characterization is given for sequencesT *(n) ,n∈N, which generate an essentially complete class in the following sense: For any sequenceT (n) ,n∈N, there exist functions Φ n ,n∈N, such that forn→∞ we have $$\begin{gathered} \smallint L (n^{1/2} (T^{*(n)} + n^{ - 1} \Phi _n (T^{*(n)} ) - \theta )) dP_\theta ^n \leqslant \hfill \\ \leqslant \smallint L (n^{1/2} (T^{(n)} - \theta )) dP_\theta ^n + o (n^{ - 1} ), \hfill \\ \end{gathered} $$ for every ? and everyL satisfying certain conditions. If the estimator-sequences are compared by their risks ∫W(T (n) d P θ n ,θ) with respect to loss functionsW:R p ×Θ→R then a similar result on asymptotically complete classes is valid. The results are obtained under the assumption thatT *(n) ,n∈N, andT (n) ,n∈N, admit stochastic expansions which are sufficiently regular, that the loss functionsL andW are sufficiently smooth and bounded by polynomials, and that the estimator-sequences have asymptotically bounded moments; the latter condition is not needed for bounded functionsL andW. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|