Uncertainty Estimates in the Fuzzy-Product-Limit Estimator

Abstract

The Fuzzy-Product-Limit Estimator (FPLE) is a method for estimating a survival curve and the mean survival time when very few data are available and a high proportion of the data are censored. Considering censored times as vague failure times, the censored values are represented by fuzzy membership functions that represent a belief of continued survival of the associated unit. Associated with any estimate is uncertainty. With the FPLE two distinct types of uncertainty exist in the estimate, the uncertainty due to the randomness in the recorded times and the vague uncertainty in the failure of the censored units. This paper addresses the problem of providing confidence bounds and estimates of uncertainty for the FPLE. Several methods for estimating the vague uncertainty in the estimator are suggested. Among them are the use of Efron's Bootstrap that obtains a confidence interval of the FPLE to quantify random uncertainty and produces an empirical distribution that is used to quantify properties of the vague uncertainty. Also, a method to obtain a graphical representation of the random and vague uncertainties is developed. The new methods provide confidence intervals that quantify statistical uncertainty as well-as the vague uncertainty in the estimates. Finally, results of simulations are provided to demonstrate the efficacy of the estimator and uncertainty in the estimates.Keywords: Bootstrap methods; fuzzy theory; product-limit estimator; confidence bounds. Int. J. Unc. Fuzz. Knowl. Based Syst. 2005.13:11-26. Downloaded from www.worldscientific.com by MONASH UNIVERSITY on 02/04/15. For personal use only. Uncertainty Estimates in the Fuzzy-Product-LimitEstimator 13The censored time t h the number of data points, the total operating time (TOT), the proportion of censored times, and the number of failures that occurred after time t t all provide information that is used to form the shape of the membership function for the censored time t t . In addition, the method allows the user to influence the membership functions through the parameters U and L. In general, the membership functions for censored and failed times are expressed as:

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“…One measure of the uncertainty in the fuzzy information is called the relative vague uncertainty [12].…”