Content:
Non-parametric estimation of a density function is used as a vehicle to illustrate the significance of including non-data information in the formulation of a statistical estimation problem. This, in turn, raises questions about consistency and other asymptotic properties of estimators that include such non-data information (shape, support, moment bounds, etc.). In this paper, consistency is derived as a consequence of a quantitative version of a strong law of large numbers for random upper semicontinuous functions.
The major objective of this paper is to lay the groundwork to include in the formulation of statistical estimation problems information beyond that provided by the observed samples. Of course, this is by no means the first article dealing with this issue! To begin with, every parametric estimation problem includes in its formulation significant non-data information. Even, in the formulation of non-parametric problems, there a large number of papers that deal with various ways to include non-parametric information. For example, simply assuming that the distribution of the random phenomena can be described by a density function implicitly includes non-data information in the formulation of the estimation problem. But, there is a large literature that goes much beyond that. As a few examples, one could refer to the work of Wellner and Groeneboom and some others on how to include shape information, Thompson and Tapia and Wahba on to include smoothness information about the density function, and so on. And then there is the extensive litterature that deals with Bayesian statistics.
What's different here is that we introduce a general framework that can applies to a wide variety, if not all, situations when there is non-data information availale, and that moreover, leads us to numerical procedures that can take advantage of such additional information; and this applies to the parametric as well as the non-parametric case. Essentially, this approach allows us to include in the formulation of the estimation problem any information one might have about the stochastic phenomena.
September 2004