Paul H. Kvam

Degradation models and implied lifetime distributions

In experiments where failure times are sparse, degradation analysis is useful for the analysis of failure time distributions in reliability studies. This research investigates the link between a practitioners selected degradation model and the resulting lifetime model. Simple additive and multiplicative models with single random effects are featured. Results show that seemingly innocuous assumptions of the degradation path create surprising restrictions on the lifetime distribution. These constraints are described in terms of failure rate and distribution classes.

A Nonlinear Random-Coefficients Model for Degradation Testing

As an alternative to traditional life testing, degradation tests can be effective in assessing product reliability when measurements of degradation leading to failure can be observed. This article presents a degradation model for highly reliable light displays, such as plasma display panels and vacuum fluorescent displays (VFDs). Standard degradation models fail to capture the burn-in characteristics of VFDs, when emitted light actually increases up to a certain point in time before it decreases (or degrades) continuously. Random coefficients are used to model this phenomenon in a nonlinear way, which allows for a nonmonotonic degradation path. In many situations, the relative efficiency of the lifetime estimate is improved over the standard estimators based on transformed linear models.

Estimating Load-Sharing Properties in a Dynamic Reliability System

An estimator for the load-share parameters in an equal load-share model is derived based on observing k-component parallel systems of identical components that have a continuous distribution function F (˙) and failure rate r (˙). In an equal load-share model, after the first of k components fails, failure rates for the remaining components change from r (t) to γ1r (t), then to γ2r (t) after the next failure, and so on. On the basis of observations on n independent and identical systems, a semiparametric estimator of the component baseline cumulative hazard function R = –log(1 – F) is presented, and its asymptotic limit process is established to be a Gaussian process. The effect of estimation of the load-share parameters is considered in the derivation of the limiting process. Potential applications can be found in diverse areas, including materials testing, software reliability, and power plant safety assessment.

Nonparametric Statistics with Applications to Science and Engineering: Kvam/Nonparametric Statistics

Preface. 1. Introduction. 2. Probability Basics. 3. Statistics Basics. 4. Bayesian Statistics. 5. Order Statistics. 6. Goodness of Fit. 7. Rank Tests. 8. Designed Experiments. 9. Categorical Data. 10. Estimating Distribution Functions. 11. Density Estimation. 12. Beyond Linear Regression. 13. Curve Fitting Techniques. 14. Wavelets. 15. Bootstrap. 16. EM Algorithm. 17. Statistical Learning. 18. Nonparametric Bayes. A. MATLAB. B. WinBUGS. MATLAB Index. Author Index. Subject Index.

The Effect of Active Learning Methods on Student Retention in Engineering Statistics

Abstract An experiment was carried out to investigate the long-term effects of active learning methods on student retention in an introductory engineering statistics class. Two classes of students participated in the study—one class was taught using traditional lecture-based learning, and the other class stressed group projects and cooperative learning-based methods. Retention was measured by examining the students immediately after the course finished, and then again eight months later. The findings suggest that active learning can help to increase retention for students with average or below average scores. Graphical displays of the data, along with standard statistical analyses, help explain the observed difference in retention between students in the two different learning environments.

Nonparametric Maximum Likelihood Estimation Based on Ranked Set Samples

Abstract A ranked set sample consists entirely of independently distributed order statistics and can occur naturally in many experimental settings, including problems in reliability. When each ranked set from which an order statistic is drawn is of the same size, and when the statistic of each fixed order is sampled the same number of times, the ranked set sample is said to be balanced. Stokes and Sager have shown that the edf F n of a balanced ranked set sample from the cdf F is an unbiased estimator of F and is more precise than the edf of a simple random sample of the same size. The nonparametric maximum likelihood estimator (MLE) F of F is studied in this article. Its existence and uniqueness is demonstrated, and a general numerical procedure is presented and is shown to converge to F. If the ranked set sample is balanced, it is shown that the EM algorithm, with F n as a seed, converges to the unique solution (F) of the problems self-consistency equations; the consistency of every iterate of the EM a...

Ranked Set Sampling Based on Binary Water Quality Data With Covariates

A ranked set sample (RSS) is composed of independent order statistics, formed by collecting and ordering independent subsamples, then measuring only one item from each subsample. If the cost of sampling is dominated by data measurement rather than collection or ranking, the RSS technique is known to be superior to ordinary sampling. Experiments based on binary data are not designed to exploit the advantages of ranked set sampling because categorical data typical are as easily measured as ranked, making RSS methods impractical. However, in some environmental and biological studies, the success probability of a bivariate outcome is related to one or more covariates. If the covariate information is not easily quantified, but can be objectively ordered with respect to this success probability, the RSS method can be used to improve the analysis of binary data. This article considers the case in which the covariate information is modeled in terms of a mixing distribution for the success probability, and the expected success probability is of primary interest. The inference technique is demonstrated with water-quality data from the Rappahannock river in Virginia. In a general setting, the RSS estimator is shown to be superior, including cases in which error in judgment ranking is present.

Nonparametric Estimation of a Distribution Subject to a Stochastic Precedence Constraint

For any two random variables X and Y with distributions F and G defined on [0,∞), X is said to stochastically precede Y if P(X≤Y) ≥ 1/2. For independent X and Y, stochastic precedence (denoted by X≤spY) is equivalent to E[G(X–)] ≤ 1/2. The applicability of stochastic precedence in various statistical contexts, including reliability modeling, tests for distributional equality versus various alternatives, and the relative performance of comparable tolerance bounds, is discussed. The problem of estimating the underlying distribution(s) of experimental data under the assumption that they obey a stochastic precedence (sp) constraint is treated in detail. Two estimation approaches, one based on data shrinkage and the other involving data translation, are used to construct estimators that conform to the sp constraint, and each is shown to lead to a root n-consistent estimator of the underlying distribution. The asymptotic behavior of each of the estimators is fully characterized. Conditions are given under which each estimator is asymptotically equivalent to the corresponding empirical distribution function or, in the case of right censoring, the Kaplan–Meier estimator. In the complementary cases, evidence is presented, both analytically and via simulation, demonstrating that the new estimators tend to outperform the empirical distribution function when sample sizes are sufficiently large.

A change-point analysis for modeling incomplete burn-in for light displays

In testing display devices such as Plasma Display Panels (PDPs), the observed degradation in luminosity can exhibit an unstable period due to incomplete burn-in during the manufacturing process. We introduce a log-linear model with random coefficients and a change point to describe the nonlinear degradation path. The change point represents the time at which the burn-in period has finished and the degradation in the luminosity changes to a slower and more stable rate. The inference procedure for the lifetime distribution is based on maximum likelihood estimators and results indicate that reliability estimation can be improved substantially by using the change-point model to account for product burn-in effects. An example based on laboratory tests of PDPs helps to illustrate the procedure.

Reliability Modeling in Spatially Distributed Logistics Systems

This article proposes methods for modeling service reliability in a supply chain. The logistics system in a supply chain typically consists of thousands of retail stores along with multiple distribution centers (DC). Products are transported between DC & stores through multiple routes. The service reliability depends on DC location layouts, distances from DC to stores, time requirements for product replenishing at stores, DCs capability for supporting store demands, and the connectivity of transportation routes. Contingent events such as labor disputes, bad weather, road conditions, traffic situations, and even terrorist threats can have great impacts on a systems reliability. Given the large number of store locations & multiple combinations of routing schemes, this article applies an approximation technique for developing first-cut reliability analysis models. The approximation relies on multi-level spatial models to characterize patterns of store locations & demands. These models support several types of reliability evaluation of the logistics system under different probability scenarios & contingency situations. Examples with data taken from a large-scale logistics system of an automobile company illustrate the importance of studying supply-chain system reliability