Waltraud Kahle

A general repair, proportional-hazards, framework to model complex repairable systems

A system (machine) is observed to operate in 1 of 2 modes. The most common mode is loaded (or regular) operation. Occasionally the system is placed in an unloaded state, wherein while the system is mechanically still operating, it is assumed that the failure intensity is lower due to this reduction in operating intensity. A proportional hazards framework is used to capture this potential reduction in failure intensity due to switching of operating modes. In either operating condition, analyzed maintenance-records indicate that the system was occasionally shut down, and either a minor or a major repair was undertaken. Furthermore, despite such repairs, it is observed that both modes of operation (loaded or unloaded) resulted in random failures. On failure, 1 of 3 actions are taken: (1) failures were minimally repaired, (2) given a minor repair, or (3) given a major repair. Both minor and major repairs are assumed to impact the intensity following a virtual age process of the general form proposed by Kijima. This research develops a statistical model of such an operating/maintenance environment. Its purpose is to quantify the impacts of performing these repair actions on the failure intensities. Field data from an industrial-setting demonstrate that appropriate parameter estimates for such multiple phenomena can be obtained. Providing a richer, more detailed, modeling of the failure intensity of a system incorporating both operating conditions and repair effects has important ramifications for maintenance planning. This paper refers to related research, in which optimal timing of maintenance repairs depends fundamentally on the failure rate of the system.

Optimal maintenance policies in incomplete repair models

Abstract We consider an incomplete repair model, that is, the impact of repair is not minimal as in the homogeneous Poisson process and not “as good as new” as in renewal processes but lies between these boundary cases. The repairs are assumed to impact the failure intensity following a virtual age process of the general form proposed by Kijima. In previous works field data from an industrial setting were used to fit several models. In most cases the estimated rate of occurrence of failures was that of an underlying exponential distribution of the time between failures. In this paper, it is shown that there exist maintenance schedules under which the failure behavior of the failure-repair process becomes a homogeneous Poisson process.

Advances in Degradation Modeling: Applications to Reliability, Survival Analysis and Finance, Springer/ Birkhauser: Boston, 2010, 416p .

Review, Tutorials, and Perspective.- Trends in the Statistical Assessment of Reliability.- Degradation Processes: An Overview.- Defect Initiation, Growth, and Failure - A General Statistical Model and Data Analyses.- Properties of Lifetime Estimators Based on Warranty Data Consisting only of Failures.- Shock Models.- Shock Models.- Parametric Shock Models.- Poisson Approximation of Processes with Locally Independent Increments and Semi-Markov Switching - Toward Application in Reliability.- On Some Shock Models of Degradation.- Degradation Models.- The Wiener Process as a Degradation Model: Modeling and Parameter Estimation.- On the General Degradation Path Model: Review and Simulation.- A Closer Look at Degradation Models: Classical and Bayesian Approaches.- Optimal Prophylaxis Policy Under Non-monotone Degradation.- Deterioration Processes With Increasing Thresholds.- Failure Time Models Based on Degradation Processes.- Degradation and Fuzzy Information.- A New Perspective on Damage Accumulation, Marker Processes, and Weibulls Distribution.- Reliability Estimation and ALT.- Reliability Estimation of Mechanical Components Using Accelerated Life Testing Models.- Reliability Estimation from Failure-Degradation Data with Covariates.- Asymptotic Properties of Redundant Systems Reliability Estimators.- An Approach to System Reliability Demonstration Based on Accelerated Test Results on Components.- Survival Function Estimation.- Robust Versus Nonparametric Approaches and Survival Data Analysis.- Modelling Recurrent Events for Repairable Systems Under Worse Than Old Assumption.- Survival Models for Step-Stress Experiments With Lagged Effects.- Estimation of Density on Censored Data.- Competing Risk and Chaotic Systems.- Toward a Test for Departure of a Trajectory from a Neighborhood of a Chaotic System.- Probability Plotting with Independent Competing Risks.

Parameter Estimation in Damage Processes: Dependent Observation of Damage Increments and First Passage Time

In this paper we describe statistical methods for estimating the parameters of damage processes if in one realization both process increments and a failure time are observable. The likelihood function for such observations is developed and point estimates are compared with those for other models.

Statistical Analysis of Damage Processes

For analyzing reliability of products it is often important to investigate the damage process. The parameters of that process can be estimated from observations. If the limit level of damages is known, this leads to estimations of the parameters of the lifetime distributions. In the paper we describe several continuous and discrete models for damages. In a general shock model parameter estimators are given for various parametric models and various kinds of observation.1

Simultaneous confidence regions for the parameters of damage processes

In connection with the investigation of the reliability of products it is often necessary to consider the development of damage processes of such products to calculate parameters of reliability. The parameters of the damage process can be estimated by observations at discrete time points. If the limit level of damage is known the parameters of life distributions can be calculated by the estimated values of these parameters.

Statistical Analysis of Some Parametric Degradation Models

The applicability of purely lifetime based statistical analysis is limited due to several reasons. If the random event is the result of an underlying observable degradation process then it is possible to estimate the parameters of the resulting lifetime from observations of these process. In this paper we describe the degradation by a position-dependent marked doubly stochastic Poisson process. The intensity of such processes is a product of a deterministic function and a random variable Y which leads to an individual intensity for each realization. Our main interest consists in estimating the parameters of the distribution of Y under the assumption that the realization of Y is not observable.

The Wiener Process as a Degradation Model: Modeling and Parameter Estimation

In this chapter we describe a simple degradation model based on the Wiener process. A failure occurs when the degradation reaches a given level for the first time. In this case, the time to failure is inverse Gaussian distributed. The parameters of the lifetime distribution can be estimated from observation of degradation only, from observation of failures, or from observation of both degradation increments and failure times. In the chapter, statistical methods for estimating the parameters of degradation processes for different data structures are developed and compared.

On Parameter Estimation for a Position-Dependent Marking of a Doubly Stochastic Poisson Process

For analyzing reliability of technical systems it is often important to investigate damage processes. In this paper we describe a damage process (Zt) which is assumed to be generated by a positionependent marking of a doubly stochastic Poisson process. For some parametric intensity kernels of the corresponding marked point process we determine maximum-likelihood estimations. Censored observations are taken into account. Furthermore, the large sample case is considered.1

Bartlett corrections for the weibull distribution

We describe, for the calculation of Bartlett adjustments, a method which may be of use when a transformation to orthogonalized parameters can be found. This method is used to calculate the Bartlett adjustment for the Weibull distribution if two or only one of the parameters are of interest.