E. G. PHADIA1 AND QIQING YU2

1 Department of Mathematics, William Paterson College of New Jersey,
Wayne, NJ 07470, U.S.A.

2 Department of Applied Mathematics and Statistics, State University of New York
at Stony Brook, Stony Brook, NY 11794, U.S.A.

(Received October 26, 1989; revised May 31, 1990)

Abstract.    In this article we examine the minimaxity and admissibility of the product limit (PL) estimator under the loss function

L(F, ^F) = \int (F(t) - ^F(t))2 Falpha(t)(1-F(t))beta dW(t).

To avoid some pathological and uninteresting cases, we restrict the parameter space to Theta = {F: F(ymin) > epsilon}, where epsilon \in (0, 1) and y1,...., yn are the censoring times. Under this set up, we obtain several interesting results. When y1 = ··· = yn, we prove the following results: the PL estimator is admissible under the above loss function for alpha, beta \in {-1, 0}; if n = 1, alpha = beta = -1, the PL estimator is minimax iff dW({y}) = 0; and if n > 2, alpha, beta \in {-1, 0}, the PL estimator is not minimax for certain ranges of epsilon. For the general case of a random right censorship model it is shown that the PL estimator is neither admissible nor minimax. Some additional results are also indicated.

Key words and phrases:    Minimaxity, censored data, admissibility, nonparametric estimation, product limit estimator.

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