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ESTIMATING A PARAMETER WHEN IT IS KNOWN THAT THE PARAMETER EXCEEDS A GIVEN VALUE
Author(s) -
Gordon Ian R.,
Hall Peter
Publication year - 2009
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/j.1467-842x.2009.00557.x
Subject(s) - estimator , mathematics , constraint (computer aided design) , variance (accounting) , simple (philosophy) , degree (music) , upper and lower bounds , value (mathematics) , statistics , econometrics , mathematical analysis , philosophy , physics , geometry , accounting , epistemology , acoustics , business
Summary In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘ θ > θ 1 ’ or ‘ θ < θ 1 ’, where θ 1 is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ 1 is to replace an estimator, , of θ by the maximum of and θ 1 . However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ 1 , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ 1 , strictly exceed θ 1 except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ 1 , and more generally produces estimators that respect the constraint in a smooth, realistic fashion.