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MINIMAXITY AND ADMISSIBILITY OF THE PRODUCT

LIMIT ESTIMATOR

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E. G. PHADIA^{1} AND QIQING YU^{2}

^{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} *F*^{alpha}(*t*)(1-*F*(*t*))^{beta}
*dW*(*t*).

To avoid some pathological and uninteresting cases, we restrict the
parameter space to *Theta* = {*F*: *F*(*y*_{min}) __>__ *epsilon*}, where
*epsilon* \in (0, 1) and *y*_{1},...., *y*_{n} are the censoring times.
Under this set up, we obtain several interesting results. When
*y*_{1} = ··· = *y*_{n}, 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.

**Source**
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