Confidence Limits for Weibull Regression With Censored Data

The response variable in an experiment follows a 2-parameter Weibull distribution having a scale parameter that varies inversely with a power of a deterministic, externally controlled, variable generically termed a stress. The shape parameter is invariant with stress. A numerical scheme is given for solving a pair of nonlinear simultaneous equations for the maximum likelihood (ML) estimates of the common shape parameter and the stress-life exponent. Interval and median unbiased point estimates for the shape parameter, stress-life exponent and a specified percentile at any stress, are expressed in terms of percentage points of the sampling distributions of pivotal functions of the ML estimates. A numerical example is given.     

Unbiased Maximum-Likelihood Estimation of a Weibull Percentile when the Shape Parameter is Known

The maximum-likelihood (ML) estimator for a percentile of a Weibull distribution with a known shape parameter is considered. Multiplicative correction factors are listed for rendering the ML estimator mean or median unbiased in the cases where the samples are type II censored with or without replacement. The correction factors depend upon the number of failures and the shape parameter but are independent of the sample size and the percentile being estimated.     

Multiple Comparison for Weibull Parameters

Two problems are considered: 1) testing the hypothesis that the shape parameters of k 2-parameter Weibull populations are equal, given a sample of n observations censored (Type II) at r failures, from each population; and 2) Under the assumption of equal shape parameters, the problem of testing the equality of the p-th percentiles. Test statistics (for these hypotheses), which are simple functions of the maximum likelihood estimates, follow distributions that depend only upon r,n,k,p and not upon the Weibull parameters. Critical values of the test statistics found by Monte Carlo sampling are given for selected values of r,n,k,p. An expression is found and evaluated numerically for the exact distribution of the ratio of the largest to smallest maximum likelihood estimates of the Weibull shape parameter in k samples of size n, Type II censored at r = 2. The asymptotic behavior of this distribution for large n is also found.     

Uniqueness of maximum likelihood estimators of the 2-parameterWeibull distribution

We present a simple statistic, calculated from either complete failure data or from right-censored data of type-I or -II. It is useful for understanding the behavior of the parameter maximum likelihood estimates (MLE) of a 2-parameter Weibull distribution. The statistic is based on the logarithms of the failure data and can be interpreted as a measure of variation in the data. This statistic provides: (a) simple lower bounds on the parameter MLE, and (b) a quick approximation for parameter estimates that can serve as starting points for iterative MLE routines; it can be used to show that the MLE for the 2-parameter Weibull distribution are unique     

Inference on Weibull Percentiles and Shape Parameter from Maximum Likelihood Estimates

Four functions of the maximum likelihood estimates of the Weibull shape parameter and any Weibull percentile are found. The sampling distributions are independent of the population parameters and depend only upon sample size and the degree of (Type II) censoring. These distributions, once determined by Monte Carlo methods, permit the testing of the following hypotheses: 1) that the Weibull shape parameter is equal to a specified value; 2) that a Weibull percentile is equal to a specified value; 3) that the shape parameters of two Weibull populations are equal; and 4) that a specified percentile of two Weibull populations are equal given that the shape parameters are. The OC curves of the various tests are shown to be readily computed. A by-product of the determination of the distribution of the four functions are the factors required for median unbiased estimation of 1) the Weibull shape parameter, 2) a Weibull percentile, 3) the ratio of shape parameters of two Weibull distributions, and 4) the ratio of a specified percentile of two Weibull distributions.     

Maximum Likelihood Estimation For The 2-Parameter Weibull Distribution Based On Interval-Data

MLE techniques are presented for estimating time-to-failure distributions from interval-data. Interval-data consist of adjacent inspection times that surround an unknown failure time. Censored interval-data bound the unknown failure time with only a lower time. The 2-parameter Weibull distribution is examined as the failure distribution. Parameter estimates from interval-data and from the midpoints of the intervals are compared for 6 shapes of the Weibull distribution. The results from Monte Carlo simulation runs are used to examine the s-bias and S-variability of the parameter estimates.     

Solving ML equations for 2-parameter Poisson-process models forungrouped software-failure data

Existence conditions are given for maximum likelihood (ML) parameter estimates for several families of 2-parameter software-reliability Poisson-process models. For each such model, the ML equations can be expressed in terms of one equation in one unknown. Bounds are given on solutions to these one equation problems to serve as initial intervals for search algorithms like bisection. Uniqueness of the solutions is established in some cases. Solutions are also tabulated for certain simple cases. Results are given for ungrouped failure data (exact times are available for all failures). ML estimation problems for such a situation are treated as limiting cases of problems based on failure times grouped into intervals of decreasing mesh     

A general linear-regression analysis applied to the 3-parameterWeibull distribution

The conventional techniques of linear regression analysis (linear least squares) applied to the 3-parameter Weibull distribution are extended (not modified), and new techniques are developed for the 3-parameter Weibull distribution. The three pragmatic estimation methods in this paper are simple, accurate, flexible, and powerful in dealing with difficult problems such as estimates of the 3 parameters becoming nonpositive. In addition, the inherent disadvantages of the 3-parameter Weibull distribution are revealed; the advantages of a new 3-parameter Weibull-like distribution over the original Weibull distribution are explored; and the potential of a 4-parameter Weibull-like distribution is briefly mentioned. This paper demonstrates how a general linear regression analysis or linear least-squares breaks away from the classical or modern nonlinear regression analysis or nonlinear least-squares. By adding a parameter to the simplest 2-parameter linear regression model (AB-model), two kinds of ABC models (elementary 3-parameter nonlinear regression models) are found, and then a 4-parameter AABC model is built as an example of multi-parameter nonlinear regression models. Although some other techniques are still necessary, additional applications of the ABC models are strongly implied     

Moment estimators for the 3-parameter Weibull distribution

Weibull moments are defined generally and then calculated for the 3-parameter Weibull distribution with non-negative location parameter. Sample estimates for these moments are given and used to estimate the parameters. The results of a simulation investigation of the properties of the parameter estimates are discussed briefly. A simple method of deciding whether the location parameter can be considered zero is described     

Weibull accelerated life testing with unreported early failures

Situations arise in life testing where early failures go unreported, e.g. a technician believes an early failure is “his fault” or “premature” and must not be recorded. Consequently, the reported data come from a truncated distribution and the number of unreported early failures is unknown. Inferences are developed for a Weibull accelerated life-testing model in which transformed scale and shape parameters are expressed as linear combinations of functions of the environment (stress). Coefficients of these combinations are estimated by maximum likelihood methods which allow point, interval, and confidence bound estimates to be computed for such quantities of interest for a given stress level as the shape parameter, the scale parameter, a selected quantile, the reliability at a particular time, and the number of unreported early failures. The methodology allows lifetimes to be reported as exact, right censored, or interval-valued, and to be subject optionally to testing protocols which establish thresholds below which lifetimes go unreported. A broad spectrum of applicability is anticipated by virtue of the substantial generality accommodated in both stress modeling and data type     

Inspection Intervals for Fail-Safe Structure

The Fail-Safe principle as applied to aircraft structural design implies that there is insufficient knowledge of the life capability of the design. Control of inspection intervals is not supported by risk calculations, yet only a sample of aircraft is inspected, at intervals whose duration is rapidly increased. This paper provides risk estimates based on a simple mathematical model. Catastrophic failure is treated in two stages modeled respectively by 2-parameter and 3-parameter Weibull distributions. Bayes inferences are made about the scale parameter using in-service survivor times. Only those cases are treated for which no failures have occurred. This results in a suggested form of inspection policy. A separate non-Bayes analysis confirms the Bayes risk estimate; thus the assumed improper prior is interesting. This prior, the only simple one which is tractable for the case of no failures, transforms, for the exponential distribution, to the uniform prior, in contrast to the hyperbolic one usually used. The analysis is simplistic but provides a ball-park estimate which would otherwise be unavailable. It can be used with caution as a check on inspection programs already derived by other means. It can also serve in tutorial demonstration of the statistical effects of the various parameters, to airworthiness managers. Possibly it might form the basis of a more sophisticated analysis.     

Estimating Weibull Parameters for a General Class of Devices from Limited Failure Data

Relatively simple approaches to estimating Weibull parameters for a general class of devices are developed through regression models. It is assumed that data are collected on a number of device types belonging to a general class. For each device type, the only information available is the number of devices being observed, the total time observed and the total number of failures. By assuming a constant shape parameter and a scale parameter that may vary with the characteristics of the device-type, the least squares method is used to provide estimates of the parameters of a two-parameter Weibull distribution for both replacement and nonreplacement data. An approach is also suggested for dealing with troublesome cases of zero failure occurrences. A numerical example is provided to illustrate the approach.     

Parameter estimation for the 3-parameter Gamma distribution usingthe continuation method

This paper shows a maximum-likelihood (ML) parameter estimation algorithm for the 3-parameter Gamma distribution. The algorithm, a combination of the continuation method and the extended Gamma distribution model, can find the local ML estimates of the parameters without a careful selection of the starting point in the iterative process. This algorithm is more efficient than previous algorithms, and can find the multiple local ML estimates     

Sequential tests for integrated-circuit failures

We propose a sequential probability ratio test (SPRT) based on a 2-parameter Weibull distribution for integrated-circuit (IC) failure analysis. The shape parameter of the Weibull distribution characterizes the decreasing, constant, or increasing failure rate regions in the bath tub model for IC. The algorithm (SD) detects the operating region of the IC based on the observed failure times. Unlike the fixed-length tests, the SD, due to its sequential nature, uses the minimum average number of devices for the test for fixed error tolerances in the detection procedure. We find that SD is, on average, 96% more statistically efficient than the fixed-length test. SD is highly robust to the variations in the model parameters, unlike other existing sequential tests. Since the accuracy of the tests and the test length are conflicting requirements, we also propose a truncated SD which allows a better control of this tradeoff. It has both the sequential nature of examining measurements and the fixed-length property of guaranteeing that the tolerances be met approximately with a specified number of available measurements     

Maximum likelihood estimation of exponential distribution parameters using interval data

This paper addresses the problem of estimating, by the method of maximum likelihood (ML), the location parameter (when present) and scale parameter of the exponential distribution (ED) from interval data. Interval data are defined as two data values that surround an unknown failure observation. Such observations occur naturally, during periodic inspections, for example, when only the time interval during which the failure occurred is known. The appropriate (conditional) log-likelihood functions are derived, as are expressions for the asymptotic variances and covariances of the ML parameter estimates. To illustrate the calculations involved, two numerical examples are discussed.     

Modified `practical Bayes-estimators' [reliability theory]

This paper presents a new formulation of `practical Bayes-estimators' (PBE) for the 2-parameter Weibull model when both parameters are unknown. Overcoming some limitations of the first formulation gave rise to this work, but the results are beyond this intent. These estimators are a tool to improve technical knowledge by using a few experimental data. In this case, the controversy about whether to use Bayes or classical methods is surmounted since estimators, like maximum likelihood, give estimates that often appear unlikely on the basis of technical knowledge of the engineers. A Monte Carlo study supports the following conclusions: if the shape parameter is greater than one, modified PBE maintain the good properties of practical Bayes estimators; otherwise the modified PBE are much better and do not suffer from the past limitation regarding the formulation of the prior interval on the shape parameter itself; and when there are very few data the modified PBE work as a filter that always improves (on average) the prior information if it is poor, or substantially confirms it if it is good. From this viewpoint, Bayes theorem allows statistics to help engineering and not vice versa     

Analysis of step-stress accelerated-life-test data: a new approach

A linear cumulative exposure model (LCEM) is used to analyze data from a step-stress accelerated-life-test, in particular, those with failure-free life (FFL). FFL is characterized by a location parameter in the distribution. For the 2-parameter Weibull distribution, the Nelson cumulative exposure model is a special case of LCEM. Under LCEM a general expression is derived for computing the maximum likelihood estimator (MLE) of stress-dependent distribution parameters under multiple censoring. The estimation procedure is simple and is illustrated by a set of experimental data using the 3-parameter Weibull distribution     

威布尔分布无故障数据的可靠性评估

对于威布尔分布无故障数据可靠性评估方法中形状参数已知和未知的两种方法,通过一个例子进行对比分析,指出当形状参数毫无所知时,所得到的基本可靠度置信下限估计最为保守。通过相似产品的信息和工程经验对形状参数作出一个较为精确的估计是可行的。     

Weibull extensions of the Freund and Marshall-Olkin bivariateexponential models

The physical motivation for the bivariate extensions of the exponential distribution due to Freund (1961) and Marshall-Olkin (1957) is common in engineering applications. The author extends their models to the case where the failure rate of one component changes upon the failure of the other and Poisson fatal shocks cause simultaneous failures of both components in order to derive bivariate extensions of the Weibull distribution. Some special cases, such as bivariate extensions of the linear hazard rate and minimum type distributions, are discussed     

Sensitivity of Bayes Estimates of Reciprocal MTBF and Reliability to an Incorrect Failure Model

A unit is placed on test for a fixed time, and the number of failures is observed. The stochastic process generating the failures is assumed to have s-independent, Erlang distributed times between failures. Bayes estimates of reciprocal MTBF (RMTBF) and reliability are given where the loss function is squared error and the prior distribution for RMTBF is gamma. We investigate what happens to the Bayes estimates when the shape parameter in the failure model is incorrectly specified (e.g., the failure model is assumed to be Poisson when it is not). This question is answered for parameters which are typical of a wide range of actual military equipment failure data. As the shape parameter in the failure model changes 1) there is only a small to moderate change in the estimates of RMTBF; 2) there is a small to moderate change in the estimate of reliability for small numbers of failures but a larger change for an unusually large number of failures; 3) there is little change in the s-efficiencies of the estimates as measured by s-expected squared error loss. For the range of parameters in this study, not much is lost in s-efficiency by restricting attention to the mathematically tractable Erlang failure model instead of using a more general gamma failure model.