(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.