Bayesian estimation of mixed Weibull distributions

Estimation of mixed Weibull distribution by maximum likelihood estimation and other methods is frequently difficult due to unstable estimates arising from limited data. Bayesian techniques can stabilize these estimates through the priors, but there is no closed-form conjugate family for the Weibull distribution. This paper reduces the number of numeric integrations required for using Bayesian estimation on mixed Weibull situations from five to two, thus making it a more feasible approach to the typical user. It also examines the robustness of the Bayesian estimates under a variety of different prior distributions. It is found that Bayesian estimation can improve accuracy over the MLE for situations with low mixture ratios so long as the prior on the weak subpopulation's characteristic life has an expected value less than or equal to the true characteristic life.     

A comparison of maximum likelihood models for fatigue strength characterization in materials exhibiting a fatigue limit

In this study, various probabilistic models were considered to support fatigue strength design guidance in the ultra high-cycle regime (beyond 10⁸ cycles), with particular application to Ti-6Al-4V, a titanium alloy common to aerospace applications. The random fatigue limit model of Pascual and Meeker and two proposed simplified models (bilinear and hyperbolic) used maximum likelihood estimation techniques to fit probabilistic stress-life curves to experimental data. The bilinear and hyperbolic models provided a good fit to large-sample experimental data for dual-phase Ti-6Al-4V and were then applied to a small-sample data set for a beta annealed variant of this alloy, providing an initial probabilistic estimate of beta annealed Ti-6Al-4V fatigue strength in the gigacycle regime. The bilinear and hyperbolic models are recommended for use in estimating probabilistic fatigue strength parameters in support of very high-cycle design criteria for metals with clearly defined fatigue limits and fairly constant scatter in fatigue strength.     

Satellite and satellite subsystems reliability: Statistical data analysis and modeling

Reliability has long been recognized as a critical attribute for space systems. Unfortunately, limited on-orbit failure data and statistical analyses of satellite reliability exist in the literature. To fill this gap, we recently conducted a nonparametric analysis of satellite reliability for 1584 Earth-orbiting satellites launched between January 1990 and October 2008. In this paper, we extend our statistical analysis of satellite reliability and investigate satellite subsystems reliability. Because our dataset is censored, we make extensive use of the Kaplan–Meier estimator for calculating the reliability functions. We derive confidence intervals for the nonparametric reliability results for each subsystem and conduct parametric fits with Weibull distributions using the maximum likelihood estimation (MLE) approach. We finally conduct a comparative analysis of subsystems failure, identifying the “culprit subsystems” that drive satellite unreliability. The results here presented should prove particularly useful to the space industry for example in redesigning subsystem test and screening programs, or providing an empirical basis for redundancy allocation.     

Exploiting Structure of Maximum Likelihood Estimators for Extreme Value Threshold Selection

To model the tail of a distribution, one has to define the threshold above or below which an extreme value model produces a suitable fit. Parameter stability plots, whereby one plots maximum likelihood estimates of supposedly threshold-independent parameters against threshold, form one of the main tools for threshold selection by practitioners, principally due to their simplicity. However, one repeated criticism of these plots is their lack of interpretability, with pointwise confidence intervals being strongly dependent across the range of thresholds. In this article, we exploit the independent-increments structure of maximum likelihood estimators to produce complementary plots with greater interpretability, and suggest a simple likelihood-based procedure that allows for automated threshold selection. Supplementary materials for this article are available online.     

Maximum likelihood Estimation of Parameters in the Inverse Gaussian Distribution,With Unknown Origin

Maximum likelihood estimation is applied to the three-parameter Inverse Gaussian distribution, which includes an unknown shifted origin parameter. It is well known that for similar distributions in which the origin is unknown, such as the lognormal, gamma, and Weibull distributions, maximum likelihood estimation can break down. In these latter cases, the likelihood function is unbounded and this leads to inconsistent estimators or estimators not asymptotically normal. It is shown that in the case of the Inverse Gaussian distribution this difticulty does not arise. The likelihood remains bounded and maximum likelihood estimation yields a consistent estimator with the usual asymptotic normality properties. A simple iterative method is suggested for the estimation procedure. Numerical examples are given in which the estimates in the Inverse Gaussian model are compared with those of the lognormal and Weibull distributions.     

Statistical Methods for Estimating the Minimum Thickness Along a Pipeline

Pipeline integrity is important because leaks can result in serious economic or environmental losses. Inspection information from a sample of locations along the pipeline can be used to estimate corrosion levels. The traditional parametric model method for this problem is to estimate parameters of a specified corrosion distribution and then to use these parameters to estimate the minimum thickness in a pipeline. Inferences using this method are, however, highly sensitive to the distributional assumption. Extreme value modeling provides a more robust method of estimation if a sufficient amount of data is available. For example, the block-minima method produces a more robust method to estimate the minimum thickness in a pipeline. To use the block-minima method, however, one must carefully choose the size of the blocks to be used in the analysis. In this article, we use simulation to compare the properties of different models for estimating minimum pipeline thickness, investigate the effect of using different size blocks, and illustrate the methods using pipeline inspection data.     

Confidence Interval Estimation for the Weibull and Extreme Value Distributions

This paper reviews methods of constructing confidence intervals for parameters or other characteristics of the Weibull or extreme value distribution. The conditional method of obtaining confidence intervals is stressed, with emphasis on the flexibility of the method, and on the computations which are necessary to use it.     

Box–Cox Transformation in Big Data

The Box–Cox transformation is an important technique in linear regression when assumptions of a regression model are seriously violated. The technique has been widely accepted and extensively applied since it was first proposed. Based on the maximum likelihood approach, previous methods and algorithms for the Box–Cox transformation are mostly developed for small or moderate data. These methods and algorithms cannot be applied to big data because of the memory and storage capacity barriers. To overcome these difficulties, the present article proposes new methods and algorithms, where the basic idea is to construct and compute a set of summary statistics, which is termed as the Box–Cox information array in the article. According to the property of the maximum likelihood approach, the computation of the Box–Cox information array is the only issue to be considered in reading of data. Once the Box–Cox information array is obtained, the optimal power transformation as well as the corresponding estimates of model parameters can be quickly computed. Since the whole dataset is scanned only once, the proposed methods and algorithms can be extremely efficient and fast even when multiple models are considered. It is expected that the basic knowledge gained in this article will have a great impact on the development of statistical methods and algorithms for big data.     

Estimating Component Failure Rates From Combined Component and Systems Data: Exponentially Distributed Component Lifetimes

The situation in which test results are available on systems consisting of some of k independent but nonidentical components in series is studied. The underlying distribution of the lifetime of the component is assumed to be an exponential. The method of maximum likelihood is used to obtain estimates, a chi squared approximation is used to approximate the mean and variance of the maximum likelihood estimate, and a method for reducing its bias is presented.     

Probabilistic cumulative damage model to estimate fatigue life

This paper introduces a numerical model to estimate fatigue life under step‐stress conditions, using the Weibull and lognormal distributions. The maximum likelihood method was used to estimate the free parameters of the distributions. The model was fitted to an experimental data on fatigue life in the specimens of steel SAE 8620, by using evolutionary computation to optimize the likelihood function. Results are reported on the values of the parameters and their confidence interval. Also, a validation of the model is discussed using analysis of residuals.     

Reliability estimations of components from masked system life data

This paper introduces estimations of reliability values for the individual components in a series system using masked system life data. In particular, we compute the maximum likelihood and Bayes estimates of component reliabilities when the system components have constant failure rates. In obtaining Bayes estimates, it is assumed that the component reliabilities are independent random variables having piecewise linear prior distributions. The model is illustrated for a two-component series. A numerical simulation study is presented to show how one can utilize the present approach to compute estimations of component reliabilities for a practical problem. Further, we investigate the comparison between the maximum likelihood and Bayes estimates, based on the respective percentage errors.     

An approach for determining an appropriate assumed distribution of fatigue life under limited data

The case of limited data implies that some unknown uncertainties may be involved in fatigue reliability analysis. For the sake of statistical convenience, for consistency with the relevant physical arguments and, most importantly, to ensure the safety in design evaluation, an approach is developed to determine an appropriate distribution, from four possible assumed distributions—three-parameter Weibull, two-parameter Weibull, lognormal and extreme maximum-value distributions. The approach makes allowance for consistency with the fatigue physics and checking tail fit effects. An application to nine groups of fatigue life data of 16Mn steel (Chinese steel) welded plate specimens shows that the lognormal distribution and the extreme maximum-value distribution may be the appropriate distributions of the fatigue life under limited data.     

A study of two estimation approaches for parameters of Weibull distribution based on WPP

Least-squares estimation (LSE) based on Weibull probability plot (WPP) is the most basic method for estimating the Weibull parameters. The common procedure of this method is using the least-squares regression of Y on X, i.e. minimizing the sum of squares of the vertical residuals, to fit a straight line to the data points on WPP and then calculate the LS estimators. This method is known to be biased. In the existing literature the least-squares regression of X on Y, i.e. minimizing the sum of squares of the horizontal residuals, has been used by the Weibull researchers. This motivated us to carry out this comparison between the estimators of the two LS regression methods using intensive Monte Carlo simulations. Both complete and censored data are examined. Surprisingly, the result shows that LS Y on X performs better for small, complete samples, while the LS X on Y performs better in other cases in view of bias of the estimators. The two methods are also compared in terms of other model statistics. In general, when the shape parameter is less than one, LS Y on X provides a better model; otherwise, LS X on Y tends to be better.     

Missing Values in Response Surface Designs

The estimation of missing values for a general design is described and discussed. Formulae are provided for the estimation of missing values for two well-known, three factor, second order rotatable designs, with zero to six center points. A worked example illustrates the use of the formulae in the case of the cube plus octahedron plus one center point design.     

Quality Control Methods for Several Related Variables

The methods described in an earlier article devoted to control methods for two related variables are extended to the case of more than two related variables. The concept of matrix notation is introduced because of the resultant simplification in multivariate analysis and the original two-variable problem is restated in matrix form. The method of principal components is introduced both as a method of charscterizing a multivariate process and as a control tool associated with control procedures. These methods are illustrated with a numerical example from the field of ballistic missiles. Approximate multivariate techniques, designed to simplify the administration of these control programs, are also discussed.     

A Simple Method for Regression Analysis With Censored Data

Problems requiring regression analysis of censored data arise frequently in practice. For example, in accelerated testing one wishes to relate stress and average time to failure from data including unfailed units, i.e., censored observations.

Maximum likelihood is one method for obtaining the desired estimates; in this paper, we propose an alternative approach. An initial least squares fit is obtained treating the censored values as failures. Then, based upon this initial fit, the expected failure time for each censored observation is estimated. These estimates are then used, instead of the censoring times, to obtain a revised least squares fit and new expected failure times are estimated for the censored values. These are then used in a further least squares fit. The procedure is iterated until convergence is achieved. This method is simpler to implement and explain to non-statisticians than maximum likelihood and appears to have good statistical and convergence properties.

The method is illustrated by an example, and some simulation results are described. Variations and areas for further study also are discussed.     

Reliability analysis of wind turbine subassemblies based on the 3-P Weibull model via an ergodic artificial bee colony algorithm

The Weibull distribution is the most widely used model for the reliability evaluation of wind turbine subassemblies. Considering the important role of the location parameter in the three-parameter (3-P) Weibull model and its rare application in wind turbines, this study conducted a reliability analysis of wind turbine subassemblies based on field data that obeyed the 3-P Weibull distribution model via maximum likelihood estimation (MLE). An improved ergodic artificial bee colony algorithm (ErgoABC) was proposed by introducing the chaos search theory, global best solution, and Lévy flights strategy into the classical artificial bee colony (ABC) algorithm to determine the maximum likelihood estimates of the Weibull distribution parameters. This was validated against simulation calculations and proved to be efficient for high-dimensional function optimization and parameter estimation of the 3-P Weibull distribution. Finally, reliability analyses of the wind turbine subassemblies based on different types of field failure data were conducted using ErgoABC. The results show that the 3-P Weibull model can reasonably evaluate the lifetime distribution of critical wind turbine subassemblies, such as generator slip rings and main shafts, on which the location parameter has a significant effect.     

Accelerated Life Testing for a Class of Linear Hazard Rate Type Distributions

Accelerated life testing for distributions with hazard rate functions of the form r(t) = Ag(t) + Bh(t) are considered. Let V ₁, …, V _k be stress levels larger than V ₀—the stress level under normal conditions [V ₀ > 0]—and let a(v) be a nondecreasing function on (0, ∞). We discuss a generalization of the common accelerated models (the power rule model and the Arrhenius model) by assuming that the hazard rate under the stress level V, is of the form (a(V _t )) ^P (Ag(t) + Bh(t)). The maximum likelihood estimators of A, B and P for complete and censored samples are studied. The estimation procedure reduces to a solution of one equation with one unknown parameter. The estimation procedure under the assumption of aging is also described. The asymptotic variance-covariance matrix is given.     

An Approximation to Student's t

This paper describes two level fractional factorial designs in which some of the treatment combinations are duplicated. As is well known, the duplicated runs provide an unbiased estimate of error variance and more precise estimates of the effects. An example is given with the analysis procedure. Block designs for the corresponding fractional factorial designs are also given.     

Analysis of field data under two-dimensional warranty

This paper proposes a method of estimating lifetime distribution for products under two-dimensional warranty in which age and usage are used simultaneously to determine the eligibility of a warranty claim. To analyze two-dimensional warranty data, univariate and bivariate approaches which use a reliability model including both age and usage can be considered. Most of previous works are centered on the univariate approach which assumes a functional relationship between age and usage. We consider a bivariate approach which assumes that the two variables are statistically correlated in a bivariate distribution. Methods of obtaining maximum likelihood estimators are outlined, and specific formulas are obtained for the cases where marginal distributions are Weibull. An illustrative example is given and numerical studies are performed to compare the two approaches.