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.

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 |