MINIMAXITY AND ADMISSIBILITY OF THE PRODUCT LIMIT ESTIMATOR

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

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

2Department 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, F̂) = ∫ (F (t) − F̂ (t))^{2}F^{α}(t)(1 − F (t))^{β}dW (t).

To avoid some pathological and uninteresting cases, we restrict the parameter space to Θ = {F: F (y_{min}) ≥ ϵ}, where ϵ ∈ (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 α, β ∈ { − 1, 0}; if n = 1, α = β = −1, the PL estimator is minimax iff dW ({y}) = 0; and if n ≥ 2, α, β ∈ { − 1, 0}, the PL estimator is not minimax for certain ranges of ϵ. 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.