Nov 17

3:30 pm

## Estimation in Restricted Parameter Spaces: Some History and Some Recent Developments

### Constance van Eeden

Seminar

University of British Columbia

Pacific Northwest Statistics Meeting

In this talk I will present results on restricted-parameter-space estimation problems. An example of such a problem is the case where X~N(theta,1) and theta is to be estimated when one knows that theta >= 0. That is, one is looking for an estimator theta-hat of theta satisfying theta-hat >= 0. Finding "good" estimators for such situations is a difficult problem. In the example above, e.g., the maximum likelihood estimator (MLE), Z=max(0,X), is inadmissible for squared-error loss. The estimator Z is known to be minimax, as is the MLE, X, in the unrestricted case. And the example is one of the relatively few cases where an admissible minimax estimator is known. The generalized Bayes estimator for the uniform prior on the parameter space [0,inf) is such an admissible minimax estimator. It does not dominate the MLE, i.e. it does not have a uniformly, in theta, lower risk function than the MLE. Estimators that do dominate the MLE have recently been found. None have been shown to be admissible, but, of course, all are minimax. In my talk I will trace the development of restricted-parameter-space problems from the early 50's through more recent times. The problems studied are admissibility, dominators, minimaxity and the presence of nuissance parameters.