統計学輪講（第４２回）

日時	2002年 1月29日(火)    15時〜16時40分
場所	経済学部第４教室
講演者　久保川達也(経済学部)
演題　	Empirical and Generalized Bayes Ridge Regression Estimators with 
        Minimaxity and Stability

概要： 
  I will talk about the classical problem of estimating the regression 
parameters in a multiple linear regression model when the multicollinearity 
is present.  The least squares estimator (LSE) is instable, and one 
candidate of stabilized procedure is the ridge regression estimator with 
parameter k, which is not minimax, however.  Also it includes arbitrariness 
of k, so that k may be estimated from the data.  However it is known that 
such adaptive ridge regression estimators do not satisfy the conditions for 
the minimaxity under the squared loss in the multicollinearity cases 
(Casella(1980)).  In this talk, I will employ a weighted squared loss 
suggested by Strawderman (1978) instead of the usual squared loss, and derive 
conditions for adaptive ridge regression estimators to be better than the LSE, 
namely, minimax.  Especially, the empirical Bayes estimator with estimating 
the parameter k by the root of the marginal likelihood equation is shown to 
satisfy the minimaxity and stability in the multicolliearity cases and to 
have very nice risk-performances even for the usual squared loss.  
The usefulness of the empirical Bayes estimator will be also illustrated 
through an example. As another candidate with stability, I will present 
the generalized Bayes estimator against a natural prior and give conditions 
for the minimaxity.  Hence admissible, minimax and stabilized estimators 
can be provided.