Minimaxity and admissibility of the product limit estimator

In this article we examine the minimaxity and admissibility of the product limit (PL) estimator under the loss function {Mathematical expression}. To avoid some pathological and uninteresting cases, we restrict the parameter space to Θ={F: F(ymin) ≥ ∈}, where ε∈(0, 1) and y₁,... y,_n are the censoring times. Under this set up, we obtain several interesting results. When y₁=···=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.