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Second-order asymptotic optimality and positive solutions of Schrödinger’s equation. (English. Russian original) Zbl 0608.62035
Theory Probab. Appl. 30, 333-363 (1986); translation from Teor. Veroyatn. Primen. 30, No. 2, 309-338 (1985).
It is known that the best equivariant estimator X of a translation parameter for a Gaussian family $${\mathcal N}(\theta,E)$$ is admissible in $${\mathbb{R}}^ s$$ only for $$s=1,2$$. For $$s>2$$ Stein showed that estimators $$\theta$$ (X) exist such that $(1)\quad E_{\theta}| \theta (X)- \theta |^ 2=E_{\theta}| X-\theta |^ 2-q(\theta),\quad q(\theta)>0,\quad \theta \in {\mathbb{R}}^ s.$ The question of what functions q($$\theta)$$ are usable on the right hand side of (1) is of interest. In the present paper this question is considered in an asymptotic setup in which the covariance matrix E is replaced by $$\gamma$$ E with $$\gamma\to 0$$. Relation (1) is replaced by $(2)\quad E_{\theta}| \theta (X)-\theta |^ 2=E_{\theta}| X-\theta |^ 2-\gamma^ 2q(\theta)+o(q^ 2),\quad \theta \in \Theta.$ Such an approach has many advantages, but perhaps its main one is the establishment of a close relationship between the questions under consideration and the theory of elliptic differential equations. Previously it was shown by the author that the existence of a positive solution to an equation of the type $$\Delta \omega +(1/4)q\omega =0$$ is necessary and sufficient for a Hölder-continuous function q($$\theta)$$ to be usable. In addition, with each such solution $$\omega$$ there can be associated an estimator $$\theta_{\gamma,\omega}$$ satisfying (2) in such a way that the resultant class of estimators $$\{\theta_{\gamma,\omega}\}$$ is asymptotically complete up to $$o(\gamma^ 2)$$. Hence, the question of the ”best” functions q($$\theta)$$ reduces to investigating the optimality of estimators $$\theta_{\gamma,\omega}.$$
In connection with the classical optimality criteria, their asymptotic analogues such as second-order asymptotic admissibility and q- nonimprovability are introduced. It is shown that the admissibility conditions for the estimator $$\theta_{\gamma,\omega}$$ reduce to non- existence of nontrivial positive $$L_{\omega}$$-superharmonic functions for the operator $L_{\omega}u=\Delta u+\nabla \log \omega \cdot \nabla u$ in the domain $$\Theta$$. The latter is known to be equivalent to the recurrency of the diffusion process $$X_{\omega}(t)$$ described by the operator $$L_{\omega}$$ with absorption on the boundary $$\partial \Theta$$. There is a direct connection between the recurrency of $$X_{\omega}(t)$$ and the existence of a unique solution of the exterior boundary value problem for the operator $$L_{\omega}.$$
Some of the admissibility conditions obtained in the present paper are conceptually close to the non-asymptotic admissibility tests known in statistical literature. In addition a number of new tests for asymptotic admissibility of the estimators $$\theta_{\gamma,\omega}$$ are given by the author. Moreover, a number of new rather simple tests for q- improvability are presented here. Finally, the application of the method of differential equations to the estimation problem under consideration is illustrated by the construction of families of q-nonimprovable second- order admissible estimators.
Reviewer: J.Melamed

##### MSC:
 62F10 Point estimation 62C15 Admissibility in statistical decision theory 35J10 Schrödinger operator, Schrödinger equation 62F12 Asymptotic properties of parametric estimators
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