Let for any df \(F\) \[ d_ F(s)=\int^ T_ 0(F(x)-F_ 0(x/s))^ 2w(x,s)dx,\;s>0, \] where \(F_ 0\) is a fixed distribution function. Let the minimum distance functional of scale \(\theta(F)\) satisfy \(\min_{s>0}d_ F(s)=d_ F(\theta(F))\). Let \(X_ 1,\dots, X_ n\) be iid distributed survival times with continuous density \(f\) and distribution function \[ F\in{\mathcal F}=\{F\mid F(x)=F_ 0(x/\theta),\;\theta\in(0,\infty)\}, \] where \(F_ 0\) is a fixed distribution on \((0,\infty)\) with continuous density \(f_ 0(x)\). Let \(Y_ 1,\dots,Y_ n\) be continuous iid censoring times independent of the survival times distributed according to the df \(G\).
The author concentrates on the classical Kaplan-Meier estimator \(\hat F_ n\) of \(F\) and examines how the level of censoring affects the performance of estimators of scale that minimize a weighted Cramér-von Mises distance between the Kaplan-Meier product-limit estimator of the survival function and an assumed model. The article shows that under certain conditions these estimators are asymptotically normal if the survival function is a member of the assumed scale family. Weights that minimize the asymptotic variance under these conditions are found. Asymptotic coverage probabilities of confidence intervals are found for percentiles constructed from exponential MDE’s of scale under departures from the exponential distribution, and what happens in the limiting cases of no censoring and complete censoring is examined. Using the asymptotic results and Monte Carlo experiments, several exponential MDE’s of scale are compared to the exponential maximum likelihood estimator of scale.