ASYMPTOTIC PROPERTIES OF BOOTSTRAPPED

LIKELIHOOD RATIO STATISTICS

FOR TIME CENSORED DATA

Shuen-Lin Jeng$^1$, S. N. Lahiri$^2$ and William Q. Meeker$^2$

$^1$Tunghai University and $^2$Iowa State University

Abstract: Much research has been done on the asymptotic distributions of likelihood ratio statistics for complete data. In this paper we consider the situation in which the data are time censored and the distribution of the likelihood ratio statistic is a mixture of continuous and discrete distributions. We show that the distribution of a signed square root likelihood ratio statistic can be approximated by its bootstrap distribution up to second order accuracy. Similar results are shown to hold for likelihood ratio statistics with or without a Bartlett correction. The main tool used is a continuous Edgeworth expansion for the likelihood-based statistics, which may be of some independent interest. Further, we use a simulation study to investigate the adequacy of the approximation provided by the theoretical result by comparing the finite-sample coverage probability of several competing confidence interval (CI) procedures based on the two parameter Weibull model. Our simulation results show that, in finite samples, the methods based on the bootstrap signed square root likelihood ratio statistic outperform the bootstrap-$t$ and $BC_{a}$ methods in constructing one-sided confidence bounds (CBs) when the data are Type I censored.

Key words and phrases: Bartlett correction, confidence interval, life data, likelihood ratio, one-sided confidence bound, parametric bootstrap, signed square root likelihood ratio, Type I censoring.