Minimax Estimation of a Mean Vector

K E Y W O R D S AND PHRASES Minimax decision rule; squared error loss; Dirichlet process; compact sets; Bayes rules; isotonic regression. l . INTRODUCTION In an often cited paper, BOHLMANN (1976) has considered linear minimax estimators for the mean of a univariate distribution in a nonparametr ic setting under squared error loss. To be precise, BOHLMANN assumes that Fo(x ) is a family of C D F ' s indexed by the one-dimensional parameter 0 and that U(O) is a CDF. BOHLMANN also assumes that after observing the random variable X, distributed as Fo(x), the actuary chooses a linear estimator of the form d(X) = 7X+6 to estimate the mean of X. Nature chooses (a) a family of distributions Fo(x) and (b) a C D F U(O) for O. Because (a) and (b) together determine a joint distribution for X and O, a natural loss function is given by L[(F, U), (7, 6)] = E[TX+6-p(O)] 2 where = I (7x+6-u(°))~ ro(aX) U(dO) = 7 2 v + ( I 7 ) 2 w + [ ( 1 7 ) m 6 ] 2 v = E [ c r 2 ( O ) ] , w = Var[/.z(O)], and m = E [ p ( O ) ] . This work was partially supported by Office of Naval Research Contract N00014-83-K-0249. ASTIN BULLETIN, Vol. 20, No. 2