INFERENCE FROM ACCELERATED DEGRADATION AND FAILURE DATA BASED ON
GAUSSIAN PROCESS MODELS
W. J. Padgett and Meredith A. Tomlinson
Department of Statistics, University of South Carolina, Columbia, SC 29208,
and Department of Mathematics, Winthrop University, Rock Hill, SC 29733
ABSTRACT
An important problem in reliability and survival analysis is that of modeling degradation together with any observed failures in a life test. Here, based on a continuous cumulative damage approach with a Guassian process describing degradation, a general accelerated test model is presented in which failure times and degradation measures can be combined for inference about system lifetime. Some specific models when the drift of the Gaussian process depends on the acceleration variable are discussed in detail. Illustrative examples using simulated data as well as degradation data observed in carbon-film resistors are presented.
KEY WORDS:
Inverse Gaussian distribution, Accelerated life test, Degradation process,
Fisher information, Power law, Arrhenius model, Censoring.
LOWER CONFIDENCE BOUNDS FOR PERCENTILES OF
WEIBULL AND BIRNBAUM-SAUNDERS DISTRIBUTIONS
W. J. Padgett and Meredith A. Tomlinson
Department of Statistics, University of South Carolina, Columbia, SC 29208
ABSTRACT
Approximate lower confidence bounds on percentiles of the Weibull and the Birnbaum-Saunders distributions are investigated. Asymptotic lower confidence bounds based on Bonferroni's inequality and the Fisher information are discussed, and parametric bootstrap methods to provide better bounds are considered. Since the standard percentile bootstrap method typically does not perform well for confidence bounds on quantiles, several other bootstrap procedures are studied via extensive computer simulations. Results of the simulations indicate that the bootstrap methods generally give sharper lower bounds than the Bonferroni bounds but with coverages still near the nominal confidence level. Two illustrative examples are also presented, one for tensile strength of carbon micro-composite specimens and the other for cycles-to-failure data.
KEY WORDS:
Quantiles, Bonferroni bounds, Strength distribution, Fatigue-life distribution, Parametric bootstrap.
A CUMULATIVE DAMAGE MODEL FOR STRENGTH OF MATERIALS
WHEN INITIAL DAMAGE IS A GAMMA PROCESS
W. J. Padgett and M. A. Tomlinson
Department of Statistics, University of South Carolina, Columbia, SC 29208
ABSTRACT
Based on a discrete cumulative damage approach with a gamma process describing "initial damage" for a carbon fibrous composite specimen, a new statistical model for the strength of such composites is presented. The model is an accelerated test form of an inverse Gaussian distribution and is shown to fit carbon micro-composite strength data better than previous models. Asymptotic lower confidence bounds for small percentiles of the strength distribution are obtained based on Bonferonni's inequality and the Fisher information. Simulation results indicate that the asymptotic Bonferonni lower bounds are quite conservative, and bootstrap methods for improving the bounds are considered. An illustrative example for actual observed tensile strengths of carbon micro-composite specimens is also presented.
KEY WORDS:
Strength distribution, Inverse Gaussian distribution, Accelerated test model, Cumulative damage, Gamma process, Confidence bounds, Strength percentiles.
RELIABILITY MODELS MOTIVATED BY FAILURE OF BRITTLE FIBERS AND COMPOSITES
J. D. Lynch and W. J. Padgett
Department of Statistics, University of South Carolina, Columbia, SC 29208
ABSTRACT
Some relationships between the failure rate properties of interacting components and the failure rate of systems of such components, or load-sharing systems, are summarized. These are based on a result giving a mixture representation of the system strength for a load-sharing system under arbitrary load-sharing rules. Then based on a continuous cumulative damage process, a family of statistical models for the strength of such systems is described. Asymptotic inference procedures are presented, and two models in the family are given as illustrative examples for actual observed tensile strengths of carbon fibers and of micro-composite specimens.
KEY WORDS:
Load-sharing system, Failure rate, Strength distribution, IFR, IFRA, Inverse Gaussian distribution, First passage, Cumulative damage, Fibrous composite, carbon fibers.
A PENALIZED LOCAL D-OPTIMALITY APPROACH TO DESIGN
FOR ACCELERATED TEST MODELS
Arzu Onar
Department of Management Science,
University of Miami,
Coral Gables, FL 33124-8237
W. J. Padgett
Department of Statistics,
University of South Carolina,
Columbia, SC 29208
ABSTRACT
The problem of choosing optimal levels of the acceleration variable for accelerated testing is an important issue in reliability analysis. Most recommendations have focused on minimizing the variance of an estimator of a particular characteristic, such as a percentile, for a specific parametric model. In this paper, a general approach based on "locally penalized" D-optimality (LPD-optimality) is proposed, which simultaneously minimizes the variances of the model parameter estimators. Application of the method is illustrated for inverse Gaussian-accelerated test models fitted to carbon fiber tensile strength data, where the fiber length is the "acceleration variable".
KEY WORDS:
Strength distributions; inverse Gaussian distribution; Fisher information; Size effect; Penalty function.
ACCELERATED TEST MODELS WITH THE BIRNBAUM-SAUNDERS DISTRIBUTION
W. J. Owen and W. J. Padgett
Department of Mathematics and Computer Science, University of Richmond, Richmond, VA 23173 and Department of Statistics, University of South Carolina, Columbia, SC 29208
ABSTRACT
This chapter will focus on the Birnbaum-Saunders distribution and its applications in reliability and life testing, and this distribution will be described later in this section. The Birnbaum-Saunders fatigue life distribution has applications in the general area of accelerated life testing, and two approaches given here yield various three-parameter generalizations or extensions to the original form of their model. The first approach uses the (inverse) power law model to develop an accelerated form of the distribution using standard methods. The second approach uses different assumptions of cumulative damage arguments (with applications to strengths of complex systems) to yield various strength distributions that are similar in form. Estimation and asymptotic theory for the models are discussed with some applications to various data sets from engineering laboratories.
KEY WORDS:
Fatigue-life distribution, Inverse power law, Cumulative damage, Reliability estimation.
SOME BAYESIAN PROCEDURES WITH THE ACCELERATED INVERSE GAUSSIAN MODEL
Arzu Onar and W. J. Padgett
Department of Mathematics and Statistics, University of Surrey, UK, and Department of Statistics, University of South Carolina, Columbia, SC 29208
ABSTRACT
Some Bayesian procedures are developed for a general family of accelerated inverse Gaussian-type models that arise from cumulative damage arguments. The marginal posterior distributions for the acceleration model parameters as well as the marginal posterior distributions for the inverse Gaussian parameters are obtained based on two types of non-informative priors, namely Jeffreys' and reverse-reference priors. The intractable integrals required for the posterior distributions are computed using the Laplace approximation approach. The results are illustrated for an additive damage model fitted to a carbon fiber tensile strength data set.
KEY WORDS:
Strength distributions; Laplace approximation; Size effect; Prior distributions.
INVERSE GAUSSIAN ACCELERATED TEST MODELS BASED ON CUMULATIVE DAMAGE
A. Onar and W. J. Padgett
Department of Mathematics and Statistics, University of Surrey, UK, and Department of Statistics, University of South Carolina, Columbia, SC 29208
ABSTRACT
The failure of a system under environmental stress often can be described by an accelerated test model which incorporates the environmental variable L. Here, the failure of such a system at environmental level L is modeled as the first passage of accumulated damage to a critical threshold value. Assuming a discrete additive damage model leads to a Birnbaum-Saunders-type distribution for the failure time which can be closely approximated by an inverse Gaussian-type model. However, if a continuous damage model based on a Gaussian process is assumed, a more general family of inverse Gaussian accelerated test models is obtained. Three sets of failure data are discussed to illustrate the usefulness of this general family.
KEY WORDS:
Cumulative damage, Inverse Gaussian distribution, First passage time, Size effect, Carbon composite materials, Tensile strength.
A CUMULATIVE DAMAGE MODEL FOR SYSTEM FAILURE WITH APPLICATION TO CARBON FIBERS AND COMPOSITES
S. D. Durham and W. J. Padgett
Department of Statistics, University of South Carolina, Columbia, SC 29208
Abstract
To estimate the distribution of tensile strength of materials, such as carbon composites or carbon fibers, specimens tested are typically measured to a certain size, or "gauge length." The measured strength is generally dependent on the size of the specimen. The most commonly used model for tensile strength is the Weibull distribution, justified by the "weakest link of a chain" argument. However, in many cases the Weibull does not fit experimental data very well. Here, an intuitively appealing cumulative damage argument is used to develop a new model for the strength of a general system which accounts for the size effect. The resulting strength distribution is a Weibull mixture of normal distributions which can be represented as a three-parameter version of the Birnbaum-Saunders distribution, depending on the size variable. It also can be considered as an inverse Gaussian distribution arising from the first passage time of a Brownian motion with large positive drift, which is a more convenient form for parameter estimation. The model is fitted to experimental strength data for carbon fibers and small composite specimens at various gauge lengths and is shown to fit better than the usual Weibull model in most cases.
KEY WORDS:
Failure distribution, Tensile strength, Weibull, Inverse Gaussian.
FAMILIES OF SMOOTH CONFIDENCE BANDS FOR THE SURVIVAL FUNCTION UNDER THE GENERAL RANDOM CENSORSHIP MODEL
Sneh Gulati and W. J. Padgett
Department of Statistics, Florida International University, Miami, FL 33199, and Department of Statistics, University of South Carolina, Columbia, SC 29208
ABSTRACT
Randomly right censored data often arise in industrial life testing and clinical trials. Several authors have proposed asymptotic confidence bands for the survival function when data are randomly censored on the right. All of these bands are based on the empirical estimator of the survival function. In this paper, families of asymptotic (1-a)100% confidence bands are developed from the smoothed estimate of the survival function under the general random censorship model. The new bands are compared to empirical bands, and it is shown that for small sample sizes, the smooth bands have a higher coverage probability than the empirical counterparts.
KEY WORDS:
Right-censored data; Kaplan-Meier estimator; Kernel estimator; Bandwidths; Coverage probability.
NONPARAMETRIC FUNCTION ESTIMATION FROM INVERSELY SAMPLED RECORD-BREAKING DATA
Sneh Gulati and W. J. Padgett
Department of Statistics, Florida International University, Miami, FL 33199, and Department of Statistics, University of South Carolina, Columbia, SC 29208
ABSTRACT
Often, in industrial stress testing, meteorological data analysis, and other similar situations, measurements may be made sequentially and only values smaller than all previous ones are recorded. When the number of records is fixed in advance, the data are referred to as inversely sampled record-breaking data. This paper is concerned with nonparametric estimation of the distribution and density functions from such inversely sampled record-breaking data (successive minima). For a single record-breaking sample, consistent estimation is not possible except in the extreme left tail of the distribution. Hence, replication is required and for m such independent record-breaking samples, the estimators are shown to be strongly consistent and asymptotically normal. Computer simulations are used to investigate the effect of the bandwidth on the mean squared errors and biases of the smooth estimators, and are also used to provide a comparison of their performance with the analogous estimators obtained under random sampling for record values.
KEY WORDS:
Record values; Distribution function; Density function; Consistency; Asymptotic normality; Kernel estimators.