Richard L. Warr

Expanding The Statistical Flowgraph Model Framework To Use Any Transition Distribution

Historically, parametric statistical flowgraph models (SFGMs) have exclusively used distributions with moment generating functions (MGFs). This is a significant limitation because it does not allow the use of some common distributions. This paper extends SFGM methodology by using the mathematical construct of a complex Laplace transform in lieu of MGFs. This extension enables modeling all “smooth” densities in SFGMs. We demonstrate this method using an illustrative and a real data example; both the frequentist and Bayesian approaches are considered. This enhancement of parametric SFGMs notably extends their use and flexibility. R-code is available from the authors.

Bayesian nonparametric models for combining heterogeneous reliability data

Modern complex engineering systems often present the analyst with a mix of data types that can be used for reliability prediction: system test results, lifetime data from unit tests of components, and subsystem data, all of which may have predictive value for the system lifetime. We present a hierarchical nonparametric framework, using Dirichlet processes, in which time-to-event distributions may be estimated from sample data or derived based on physical failure mechanisms. By applying a Bayesian methodology, the framework can incorporate prior information, including expert opinion.

Should the Interquartile Range Divided by the Standard Deviation be Used to Assess Normality

We discourage the use of a diagnostic for normality: the interquartile range divided by the standard deviation. This statistic has been suggested in several introductory statistics books as a method to assess normality. Through simulation, we explore the rate at which this statistic converges to its asymptotic normal distribution, and the actual size of tests based on the asymptotic distribution at several sample sizes. We show that there are nonnormal distributions from which this method cannot detect a difference. Additionally, we show the power of this test for normality is quite poor when compared with the Shapiro–Wilk test.

Accelerated Test Methods for Reliability Prediction

Perusal of quality- and reliability-engineering literature indicates some confusion over the meaning of accelerated life testing (ALT), highly accelerated life testing (HALT), highly accelerated stress screening (HASS), and highly accelerated stress auditing (HASA). In addition, there is a significant conflict between testing as part of an iterative process of finding and removing defects and testing as a means of estimating or predicting product reliability. We review the basics of these testing methods and describe how they relate to statistical methods for estimation and prediction of reliability and reliability growth. We also outline potential synergies to help reconcile statistical and engineering approaches to accelerated testing, resulting in better product quality at lower cost.

A comprehensive method for solving finite-state semi-Markov processes

Semi–Markov processes (SMPs) provide a rich framework for many real–world problems. However, owing to difficulty in implementing practical solutions they are rarely used with their full capability. The theory of SMPs is quite mature but was mainly developed at a time when computational resources were not widely available. With the exception of some of the simplest cases, solutions to SMPs are inherently numerical, and SMPs have been underutilised by practitioners because of difficulty in implementing the theory in applications. This paper demonstrates the theory and computational methods needed to implement SMP models in practical settings. Methods are illustrated with an application modelling the movement of coronary patients in a hospital. Our aim is to allow practitioners to use richer SMP models without being burdened with the rigorous mathematical theory.

Visualizing discrepancies from nonlinear models and computer experiments

Plutonium-238 is an important specialized power source that radiates heat, which can be converted into electricity. This case study models the thermal output of samples of Pu-238, in which the underlying theoretical model of its decay summarizes a large portion of the observed behavior. A discrepancy function is used to account for missing structure seen in the observed data, but is not included in the physical model. The model combines the assumed physics model, discrepancy and experimental error with an expression of the form, f(x,θ) + δ(x) + ɛ. The combined model improves prediction of new observations in the future by accounting for shortcomings or omissions in the physical model and provides quantitative summaries of the relative contributions of the discrepancy and physics model. In this work, we illustrate how to visualize the discrepancy function when it is modeled using a Gaussian process. With the visualization, scientists can gain understanding about the differences between the observed data and the current scientific model and develop proposals of how to potentially improve their model. A secondary example illustrates how the visualization methods can help with understanding in higher dimensions.

Bridging the Gap between Quantitative and Qualitative Accelerated Life Tests

Test planners have long sought the ability to incorporate the results of highly accelerated life testing (HALT) into an early estimate of system reliability. While case studies attest to the effectiveness of HALT in producing reliable products, the capability to translate the tests limited failure data into a meaningful measure of reliability improvement remains elusive. Further, a review of quality and reliability literature indicates that confusion exists over what defines a HALT and how HALT differs from quantitative accelerated life testing methods. Despite many authors making a clear distinction between qualitative and quantitative accelerated life tests, an explanation as to why this delineation exists cannot be found. In this paper, we consider an exemplary HALT composed of a single stressor to show that the HALT philosophy precludes the estimation of a systems hazard rate function parameters because of the tests fix implementation strategy. Four common accelerated failure data analysis methods are highlighted to show their limitations with respect to estimating reliability from HALT data. Finally, a modified accelerated reliability growth test is proposed as a way forward for future research in HALT scenarios to characterize the risk of attaining a reliability requirement and improve parameter estimation. Copyright

Analyzing Deficient Response Summaries from Designed Experiments

ABSTRACT All too often statisticians do not have access to raw experimental data. These scenarios require additional methodology to properly account for the missing information. In this article, we demonstrate a technique for analyzing averages of lifetime data collected at various experimental conditions that provides inference for factor effects. To handle these summaries, we use some numerical techniques to calculate the probability density function of the average of independent and identically distributed lognormal random variables. We illustrate our method with an example from the literature and provide some R code that implements a Bayesian analysis. We also provide recommendations for more informative summary statistics than lifetime averages for lognormal data.

An introduction to statistical flowgraph models for engineering systems

Statistical flowgraph models have proven useful for analysis and modeling of complex systems viewed as multistate processes that lead to outcomes such as degraded operation or failure. This article provides an engineering-oriented introduction to statistical flowgraph models: system representation, setting up a flowgraph model, parameter estimation, solution of the model (using either a frequentist or Bayesian approach), and interpretation of model outputs. The method is illustrated with a model for piping reliability in a nuclear power plant, and compared with alternative solution methods.