William J. Zimmer

Comparison of estimation methods for weibull parameters: Complete and censored samples

This paper compares several methods for estimating the parameters of the two-parameter Weibull distribution with complete, multiply time censored, and type II censored samples. An extensive simulation study compares the performance of these estimators.

Maximum likelihood estimation of the Burr XII parameters with censored and uncensored data

Abstract This paper presents the methodology for obtaining point and interval estimates of the parameters of the Burr Type XII distribution with multiple-censored and singly-censored data (Type I censoring or Type II censoring) as well as complete data, using the maximum likelihood method. The basis is the likelihood expression for multiple-censored data. The other types of censoring and complete data are treated as special cases. A FORTRAN language program was used. Some illustrative examples are included.

Tables of Sample Sizes in the Analysis of Variance

Tables are provided which give the minimum sample sizes per treatment (or level) for all combinations of alpha = 0.5, 0.3, 0.25, 0.2, 0.1, 0.05, 0.01 and beta = 0.3, 0.2, 0.1, 0.05, for relative discrimination (Delta/Sigma) = 1.0(0.5)3.0, and for the nu..

A Nonparametric Approach to Accelerated Life Testing

Abstract A nonparametric model of acceleration is introduced. Based on this model, a procedure is suggested for estimating the nonaccelerated life distribution from accelerated observations. Unlike previous nonparametric estimation procedures, our method does not require nonaccelerated observations. A relationship between the accelerated and the nonaccelerated distributions is assumed but there is variety in that choice. A comparison of our method with the power rule method under the assumption of exponential lifetimes reveals that in some instances our method is asymptotically equivalent to the maximum likelihood method for estimating the nonaccelerated mean lifetime. Simulation for small sample sizes completes the comparison.

Monotone Log-Odds Rate Distributions in Reliability Analysis

Abstract Monotone failure rate models [Barlow Richard, E., Marshall, A. W., Proschan, Frank. (1963). Properties of probability distributions with monotone failure rate. Annals of Mathematical Statistics 34:375–389, and Barlow Richard, E., Proschan, Frank. (1965). Mathematical Theory of Reliability. New York: John Wiley & Sons, Barlow Richard, E., Proschan, Frank. (1966a). Tolerance and confidence limits for classes of distributions based on failure rate. Annals of Mathematical Statistics 37(6):1593–1601, Barlow Richard, E., Proschan, Frank. (1966b). Inequalities for linear combinations of order statistics from restricted families. Annals of Mathematical Statistics 37(6):1574–1592, Barlow Richard, E., Proschan, Frank. (1975). Statistical Theory of Reliability and Life Testing. New York: Holt, Rinehart and Winston, Inc.] have become one of the most important models of failure time for reliability practitioners to consider and use. The above authors also developed models and bounds for monotone increasing failure rates (IFR) and for monotone decreasing failure rates (DFR). The IFR models and bounds appear to be especially useful for describing and bounding the hazard of aging. This article extends a new model for time to failure based onthe log odds rate [Zimmer William, J., Wang Yao, Pathak, P. K. (1998). Log-odds rate and monotone log-odds rate distributions. Journal of Quality Technology 30(4):376–385.] which is comparable to the model based on the failure rate. It is shown that in the case of increasing log odds rate (ILOR) in terms of log time (ln t), the model is less stringent than the IFR model for aging. The characterization of distributions of failure time by log odds rate is also derived. It is shown that the logistic distribution has the property of constant log odds rate over time and that the log logistic distribution has the property of constant log odds rate with respect to ln t. Some other properties of ILOR distributions are presented and bounds based on the relationship to the log logistic distribution are provided for distributions which are ILOR with respect to ln t. Motivational examples are provided. The ILOR bounds are compared to the more stringent bounds based on the IFR model. Bounds on system reliability are also provided for certain systems.

Bayes estimation of hazard and acceleration in accelerated testing

In accelerated life testing, the time transformation function theta (t) is often unknown, even if that function is assumed to be linear. If theta (t) is known, data in the accelerated condition can be adjusted to provide information about the failure time distribution in the use condition. If theta (t) is unknown, the usual estimation procedures require data from the use condition as well as data from the acceleration condition. In this work it is assumed that the uncertainty about theta can be modeled by a prior distribution, chosen from the truncated Pareto family of distributions, and that the uncertainty in lambda , the failure rate, can be modeled by a prior distribution from the gamma family. Under these assumptions, the posterior distributions and their first two moments are provided for both lambda and theta . Thus, this complete Bayes approach to accelerated life testing with the assumed model allows the adjustment of data taken in the accelerated condition to provide the user with the important estimates in the use condition. The results are illustrated by examples. >

Bayes estimation of the linear hazard-rate model

In life testing, the failure-time distributions are often specified by choosing an appropriate hazard-rate function. The class of life-time distribution characterized by a linear hazard-rate includes the one-parameter exponential and Rayleigh distributions. Usually the parameters of the linear hazard-rate model are estimated by the method of least squares. This work is concerned with Bayes estimation of the two-parameters from a type-2 censored sample. Monte Carlo simulation is used to compare the Bayes risk of the regression estimator with the minimum Bayes risk. Discrete mixtures of decreasing failure rate distributions are known to have decreasing failure rates. The authors prove that the result holds for continuous mixtures as well. >

Comparisons of methods of estimation for a pareto distribution of the first kind

This paper compares methods of estimation for the parameters of a Pareto distribution of the first kind to determine which method provides the better estimates when the observations are censored, The unweighted least squares (LS) and the maximum likelihood estimates (MLE) are presented for both censored and uncensored data. The MLEs are obtained using two methods, In the first, called the ML method, it is shown that log-likelihood is maximized when the scale parameter is the minimum sample value. In the second method, called the modified ML (MML) method, the estimates are found by utilizing the maximum likelihood value of the shape parameter in terms of the scale parameter and the equation for the mean of the first order statistic as a function of both parameters. Since censored data often occur in applications, we study two types of censoring for their effects on the methods of estimation: Type II censoring and multiple random censoring. In this study we consider different sample sizes and several values of the true shape and scale parameters. Comparisons are made in terms of bias and the mean squared error of the estimates. We propose that the LS method be generally preferred over the ML and MML methods for estimating the Pareto parameter γ for all sample sizes, all values of the parameter and for both complete and censored samples. In many cases, however, the ML estimates are comparable in their efficiency, so that either estimator can effectively be used. For estimating the parameter α, the LS method is also generally preferred for smaller values of the parameter (α ≤4). For the larger values of the parameter, and for censored samples, the MML method appears superior to the other methods with a slight advantage over the LS method. For larger values of the parameter α, for censored samples and all methods, underestimation can be a problem.

A Nonparametric Bayes Empirical Bayes Procedures for Estimating the Percent Nonconforming in Accepted Lots

An estimator is presented for estimating the percentage of nonconforming product in lots accepted under a zero-defect acceptance sampling plan. Extensions to the case of a c-defect acceptance sampling criterion are also considered. Both point and probab..

Log-Odds Rate and Monotone Log-Odds Rate Distributions

Since the 1960s, reliability models for time to failure based on monotone failure rate models have become important models of failure time for reliability practitioners. Bounds for monotone increasing failure rates (IFR) have been developed and are esp..