Analysis of ADT Data

One can find resourceful approaches with respect to degradation models, connections and differences between degradation models and failure-time models, statistical methods for data analysis and statistical inferences related to degradation data in books by Nelson (1990), Klinger (1992) and Meeker & Escobar (1998a). These concerns have also been reviewed in papers by Nelson (1981), Meeker & Escobar (1993a, b) and Chao (1999).

Lu & Meeker (1993), Boulanger & Escobar (1994), Tseng et al (1994), Hamada (1995), Chiao & Hamada (1996), and Meeker et al (1998) considered general degradation path models to obtain estimates of the percentile of a failure time distribution. Suzuki et al (1993) used a linear degradation model to study the increase of a resistance measurement over time. Carey & Koenig (1991) used concave degradation models to describe the degradation of electronic components. Meeker and Escobar (1998) used similar models to monitor the growth of failure-causing filaments of clorine-copper compound in printed-circuit boards. Dowling (1993) and Meeker & Escobar (1998) used convex degradation models to study the growth of fatigue cracks. Lu et al (1997) proposed a model with random regression coefficients and standard- deviation function to analyze linear degradation data from semiconductors. Yanagisawa (1997) used degradation models to estimate the degradation of amorphous silicon solar cells. Su et al (1999) considered a random coefficient degradation model with random sample size. A data set from a semiconductor application was used to illustrate their method. Wu & Shao (1999) established the asymptotic properties of the (weighted) least square estimators under the nonlinear mixed-effect model. They used

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these properties to obtain point estimates and approximate confidence intervals for percentiles of the failure time distribution of metal film resistors and metal fatigue cracks. Chinnam (1999) used finite-duration impulse response multi-layer perception neural network to model degradation measures and self-organizing maps to model degradation variation. This method reduced overall operation cost by facilitating optimal component-replacement and maintenance strategies. Multiple linear regression methodology was established for describing the relationship between random parameters and stresses in Crk (2000). Wu & Tsai (2000) used the optimal fuzzy clustering method. Their procedure could get more accurate estimation results if the patterns of a few degradation paths were different from those of most degradation paths in a test. Tseng & Wen (2000) proposed the CE model to analyze the LED degradation data and then described the life distribution. These papers all focus on estimating the parameters in a degradation model and the percentiles of a failure time distribution.

Parameters in the deterministic models are usually estimated by ML, LS, and other method such as MML (Su et al, 1999). However, for most models, it is not easy to obtain the estimates in a simple expression. One usually needs to get the estimates by numerical searching. Pinherio and Bates (2000) presented an estimation scheme that may provide approximate MLEs for the general deterministic models. They also gave a program to achieve the calculation. See also Bates & Watts (1988). Instead of ML and LS method, nonparametric and Bayesian approach could also be used (Shiau & Lin 1999, Robinson & Crowder 2000) to estimate the unknown parameters in a distribution function.

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The Wiener diffusion process is the normally used stochastic model to describe degradation paths (Goh et al 1989, Doksum 1991, Ebrahim & Ramallingam 1993, Lawless et al 1995, Whitmore 1995, Doksum & Normand 1995&1996, Whitmore & Schenkelberg 1997, Whitmore et al 1998, Cox 1999, Normand & Doksum 2000). A Wiener process with constant initial damage and constant beginning of damage time has been studied in Kahle &. Lehmann (1998). The failure time as the first passage time followes an IGD. By assuming in each realization of the damage process both the process increments and the failure time are observable, the MLEs of the drift and variance of the damage process, the constant initial damage, the beginning time of damage, and the boundary damage level have been estimated simultaneously. See also Kahle (1994) for the confidence regions of the MLE of these parameters. A limitation of this method is that this model is only applicable at use stress and the testing time is normally too long for the manufactures. An extended model for multiple CSADT analysis has been presented in Doksum & Hoyland (1992). The parameters in the lifetime distribution were estimated by assuming failure time was exactly observable and measurable. But the reliability information in degradation increments was ignored.

Wendet (1998) presented some other stochastic models for damage processes and gave the MLE of unknown parameters for the corresponding lifetime distributions. See also Kahle & Wendt (2000). Cinlar (1980), Singpurwalla & Youngren (1993), Singpurwalla (1997), Bagdonavicius & Nikulin (2000a, b) modeled degradation by a gamma process that includes possibly time-dependent covariates. Other models, for example, the poisson process (Mercer, 1961), the cumulative B-models (Bogdanoff &. Kozin, 1985), the markov and semimarkov process (Cinlar 1984, Kopnov & Kanajev 1994, Kopnov 1999) are also workable.

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In most of the above papers, the unknown parameters are estimated by ML method. And the MLEs could not be expressed in a closed form and need to be searched with numerical methods that are quite time-consuming. In chapter 3, we will describe a certain kind of SSADT and present to estimate the unknown parameters by LS method. The estimates are shown with a closed form that reduces labor of calculation.