Minimaxity and admissibility of the product limit estimator

Abstract

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

% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGmbGaaiikai% aadAeacaGGSaGabmOrayaajaGaaiykaiabg2da9maapeaabaGaaiik% aiaadAeacaGGOaGaamiDaiaacMcaaSqabeqaniabgUIiYdGccqGHsi% slceWGgbGbaKaacaGGOaGaamiDaiaacMcacaGGPaWaaWbaaSqabeaa% caaIYaaaaOGaamOramaaCaaaleqabaaccmGae8xSdegaaOGaaiikai% aadshacaGGPaGaaiikaiaaigdacqGHsislcaWGgbGaaiikaiaadsha% caGGPaGaaiykamaaCaaaleqabaGaeqOSdigaaOGaamizaiaadEfaca% GGOaGaamiDaiaacMcaaaa!5992!\[L(F,\hat F) = \int {(F(t)} - \hat F(t))^2 F^\alpha (t)(1 - F(t))^\beta dW(t)\].

To avoid some pathological and uninteresting cases, we restrict the parameter space to Θ={F: F(ymin) ≥ ∈}, 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.