Bayesian Estimation of Nonparametric Failure Time Distribution for Arbitrary Censored Data
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It is recommended to begin the analysis of reliability data free of distributional assumptions Meeker, Escobar, and Pascual (2022). A common approach is to treat the data as a multinomial distribution and estimate the fraction failing at set time intervals. The well-known Kaplan-Meier nonparametric estimator can be viewed as a special case of this approach, see Kaplan and Meier (1958). If one can specify a likelihood function for the data, then one can use optimization methods to maximize the likelihood to estimate the faction failure at each time interval. This method works well when the data is exact, or censoring is only of one type, right, left, or interval, and the intervals do not overlap. When the data is more complex and contains both right and left censored observations or overlapping intervals, one must use an alternate method proposed by Peto (1973) and later generalized to allow for truncated data by Turnbull (1976). The Peto-Turnbull estimator provides the point estimates of each time step, but the confidence limits estimation is more complex. This paper describes a Bayesian method to determine the time to failure point estimates and credible limits for each time step. The proposed Bayesian estimators use two different reference priors, one group (Jeffreys) and an ordered K-group reference prior. This study compares these estimators to those of the Peto-Turnbull estimator and uses Monte Carlo simulation to evaluate coverage probabilities.