Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes

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1 Lifetime Data Analysis, 11, , 2005 Ó 2005 Springer Science+Business Media, Inc. Manufactured in The Netherlands. Accelerated Degradation Models for Failure Based on Geometric Brownian Motion and Gamma Processes CHANSEOK PARK Department of Mathematical Sciences, Clemson University, Clemson, SC, 29634, USA W. J. PADGETT Department of Mathematical Sciences, Clemson University, Department of Statistics, University of South Carolina, Columbia, SC, 29208, USA Received May 8, 2005; Accepted June 15, 2005 Abstract. Based on a generalized cumulative damage approach with a stochastic process describing degradation, new accelerated life test models are presented in which both observed failures and degradation measures can be considered for parametric inference of system lifetime. Incorporating an accelerated test variable, we provide several new accelerated degradation models for failure based on the geometric Brownian motion or gamma process. It is shown that in most cases, our models for failure can be approximated closely by accelerated test versions of Birnbaum Saunders and inverse Gaussian distributions. Estimation of model parameters and a model selection procedure are discussed, and two illustrative examples using real data for carbon-film resistors and fatigue crack size are presented. Keywords: inverse Gaussian (Wald) distribution, degradation process, accelerated life test, geometric Brownian motion process, gamma process, censoring 1. Introduction Many materials and systems degrade over time before they break down or fail. To model such degradation over time and make inference about the failure of such systems or materials, we need to develop a methodology considering both observed failures and degradation data. Recently, Lu (1995), Pettit and Young (1999), and Padgett and Tomlinson (2004) introduced some methodology assuming that degradation is a Gaussian process. In this paper, a somewhat different approach is proposed in that the assumed degradation process can be different from a Gaussian process. The pitfall of the Gaussian assumption is that the process is not increasing and can possibly be negative. That is, at some time point before the degradation is measured, the degradation value can be larger than the measurement or can be even a negative value. This presents a very difficult physical interpretation and motivates the need for more physically realistic models for describing degradation. With the development of engineering and science technology, many modern products have longer lifetimes and greater reliability than those in the past. Thus, lifetime measurements and degradation measurements take much more time than

2 512 PARK AND PADGETT they used to. It is therefore difficult to observe failure times, or even degradation measurements, under normal use conditions. Since time-consuming tests under normal use conditions are costly, one needs to use accelerated tests. Accelerated life tests decrease the time to failure by exposing the products to higher levels of stress conditions (high-usage rates, increased levels of environmental variables) which cause earlier failures. Accelerated degradation tests expose the products to greater stress levels for degradation than the normal use stresses to obtain degradation measurements in a more timely fashion, and then time to failure is estimated by using the degradation measurements. Early work on degradation models is referenced by Nelson (1990), while more recent results are mentioned by Bagdonavicius and Nikulin (2002) and Meeker and Escobar (1998). In particular, degradation models based on Gaussian or other stochastic processes have been considered most recently by Doksum and Normand (1995), Lu (1995), Whitmore (1995), Whitmore and Schenkelberg (1997), Whitmore et al. (1998), Pettit and Young (1999), and Padgett and Tomlinson (2004). Bagdonavicius and Nikulin (2000) used a gamma degradation process that allowed covariates, but did not use exact failure times (first passage time to a damage threshold) for the likelihood, as we develop here. Lawless and Crowder (2004) also used the gamma process with covariates and random effects to model degradation under accelerated environments or degradation occurring at different rates in the same environment. Using regression-type methods, general degradation path models and special cases have been fitted and studied by several authors, including Lu and Meeker (1993), Boulanger and Escobar (1994), Hamada (1995), and Meeker et al. (1998). Also, degradation models applied to specific problems in engineering have been presented by several investigators including Carey and Koenig (1991), Yanagisawa (1997) and Meeker and Escobar (1998). In this paper, we develop several new models for degradation and failure data using a stochastic process, such as the geometric Brownian motion or gamma process, and incorporating an accelerated test variable. It is shown that in many cases, the models can be approximated closely by accelerated test versions of Birnbaum Saunders and inverse Gaussian distributions. Our framework is quite general, allowing for different degradation processes, as illustrated in Section 2 for geometric Brownian motion and gamma processes. Even though our approach is different, the specific model using the gamma degradation process with acceleration insection2isaspecialcaseoflawless and Crowder (2004) model, but in addition, we show that our resulting gamma process model can often be closely approximated by an inverse Gaussian model. To develop the general framework, suppose that as the tensile load or stress level on a material specimen or a system is increased, the cumulative damage X n+1 after n+1 increments of stress can be described by the cumulative damage model proposed by Durham and Padgett (1997), X nþ1 ¼ X n þ D n hðx n Þ; where D n denotes the damage incurred at the (n+1)st increment and h() is the damage model function. Recently, Park and Padgett (2005) proposed a new

3 ACCELERATED DEGRADATION MODELS 513 cumulative damage model using a damage accumulation function. This new model is given by cðx nþ1 Þ¼cðX n ÞþD n hðx n Þ; where c() is the damage accumulation function. In many cases, it is more appropriate to describe the damage process with a continuous process. A continuous version can be represented by dcðx u Þ¼hðX u ÞdD u : Then the cumulative damage or level of degradation of the system at the stress level (or time) t is given by Z t Z 1 t 0 hðx u Þ dcðx uþ¼ dd u ¼ D t D 0 : 0 Note that the above integral is a stochastic integral as defined by Jacod and Shiryaev (1987). By selecting various forms of the functions c() and h() with an appropriate stochastic process D u, several new models for degradation can be obtained. For example, assuming that D u is a Brownian motion process with h(u)=1 and c(u)=u, we obtain a degradation model based on a Gaussian process as studied by Lu (1995), Pettit and Young (1999), and Padgett and Tomlinson (2004). In this paper, we consider cðuþ ¼log u and h(u)=1 with a Brownian motion process D u, which results in a geometric Brownian motion process model for the degradation process X t and c(u)=u and h(u)=1 with a gamma process D u. In some applications, the load or stress on a unit causes failure and is recorded as the failure stress rather than the failure time. For example, a tensile load on a material specimen may be the cause of breakage of the specimen. Hence, when appropriate in the remainder of this paper, failure time may be interpreted as failure stress. 2. The Density Functions of the Failure Time and the Degradation Value In this section, we derive the probability density function (pdf) of the failure time (or first passage time) assuming a geometric Brownian motion or a gamma process for degradation. We also present the pdf of the level of degradation at the time observation of the process is terminated, i.e., the terminal value of the process, conditional on the event that the terminal value does not exceed the damage threshold C where failure occurs. Such a terminated process is commonly referred to as a truncated stochastic process.

4 514 PARK AND PADGETT 2.1. Geometric Brownian Motion Process In practice, it is often the case that a degradation process should be always positive. We first consider the geometric Brownian motion as a degradation process X t which is always positive, while the Brownian motion process is not. Using the damage model function h(u)=1 and the damage accumulation function c(u)=log u, we have the following cumulative damage or level of degradation of the system at the time t, log X t log X 0 ¼ D t : We assume that the stochastic process {D u :0 u t} is a Brownian motion process with positive drift coefficient a and diffusion b 2. This implies that the time at which log X t reaches a critical value can be considered as the threshold for failure (or first passage time) for a Brownian motion with positive drift a and diffusion b 2. Thus, the system degradation X t is a geometric Brownian motion process. The damage threshold level for failure is assumed to be a known positive constant C and the initial value of the process X t is given by X 0 =x 0. In what follows, we will derive the pdf of the first passage time to the threshold C and the pdf of the terminal value X t conditioning on the event that the terminal value does not exceed the threshold C during the process. For convenience, let Z t =log X t. Using the result of Cox and Miller ( ), we have the following probability that Z t z given that the initial value z 0 =log x 0, P½Z t zš ¼U z z 0 at p b ffiffi ; t where F() denotes the standard normal cumulative distribution function (cdf) and 1<z<1. It follows that P½X t xš ¼U log x log x 0 at p b ffiffi ; t where x>0. Thus, given that the process started at x 0 and terminated at level x at time t, the pdf for the geometric Brownian motion process X t is fðx; x 0 ; tþ ¼ 1 p b ffiffi / log x log x 0 at p t x b ffiffi ; ð1þ t where /() denotes the standard normal pdf. Any sample path within the interval (0, t) with the initial value X 0 =x 0 and the terminal value X t =x either does not exceed the threshold at C>0 or passes the threshold. Analogous to the approach of Lu (1995), the former event is denoted here by A and the latter by A c (complement of A). Therefore, the pdf f() consists of two parts representing the joint pdfs contributed by sample paths that do not exceed the threshold C, f A ðx; x 0 ; tþ, and sample paths passing the threshold, f A cðx; x 0 ; tþ. Hence, we have

5 ACCELERATED DEGRADATION MODELS 515 fðx; x 0 ; tþ ¼f A ðx; x 0 ; tþþf A cðx; x 0 ; tþ: Notice that the support of f A ðx; x 0 ; tþ is (0, C) and that of f A cðx; x 0 ; tþ is (C, ). The joint pdf f A () will be used for constructing the likelihood function for the degradation data. First, we consider the pdf of the first passage time. Let the random variable S be the first passage time of the geometric Brownian motion process X t to the threshold C. The pdf of S will be used for constructing the likelihood function for the observed failure data. In order for the random variable S to be the first passage time, we require: (i) X 0 =x 0, (ii) X t <C for all t 2ð0; SÞ, and (iii) X S C, that is, S ¼ inffujx u Cg. It is well known that for positive drift, the random variable S ¼ inffujx u Cg has an inverse Gaussian distribution since S ¼ inffuj log X u log Cg ¼inffuj log D u log C log x 0 g: For example, see Chhikara and Folks (1989) and Prabhu (1965, Theorem 4.2). The pdf of S is given by pffiffi! k kðs lþ2 g S ðs; x 0 ; CÞ ¼pffiffiffiffiffiffiffiffiffi exp 2ps 3 2l 2 ; ð2þ s where l ¼ðlog C log x 0 Þ=a and k ¼ðlog C log x 0 Þ 2 =b 2. The random variable S with the pdf (2) will be denoted as S IGðl; kþ. It is worth mentioning that when a b, the pdf of S in (2) can be approximated by the Birnbaum Saunders distribution, (Birnbaum and Saunders, 1969). Consider the discrete degradation process, that is, a material or a system degrades in a cyclic manner. Let Y i ¼ X i =X i 1 be independent and identically distributed (iid) and log Y i be normal with mean a and variance b 2, and let X n be the system degradation level after n time units have passed. Let N denote the discrete first passage time of the process X n to the threshold C. Since the process X n is approximately strictly increasing for a b, the probability of first passage after n time units is given by P½N > nš ffip½x n <CŠ: Since X n =Y n Y n)1 Y 1 x 0 and log Y i are iid, we have " # P½N > nš ffip Xn log Y i log C log x 0 : i¼1 It follows from the central limit theorem that P½N nš ffiu a p ffiffiffi n log C log x p 0 b b ffiffiffi : n

6 516 PARK AND PADGETT A continuous version of N can be represented by the Birnbaum Saunders distribution " G S ðs; x 0 ; CÞ ¼U 1 rffiffiffiffiffi r ffiffiffiffiffi!# s b a b ; s p where a ¼ ffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffi b=c 0 b=a, b ¼ C 0 =a, and C 0 ¼ log C log x 0. The difference between pthe ffiffi inverse Gaussian and Birnbaum Saunders distributions is negligible when k l p (Bhattacharyya ffiffiffi and Fries, 1982). Here l ¼ b and k ¼ b =a 2. Since the condition k l is equivalent to a b, the aforementioned distribution of S can be approximated closely by this Birnbaum Saunders distribution. Next, we will derive the joint pdf f A cðþ and then f A ðþ. By the integral identity in Lu (1995, see p. 19), the pdf f A cðþ is given by Z t f A cðx; x 0 ; tþ ¼ g S ðs; x 0 ; CÞfðx; C; t sþds 0 ¼ fðx; x 0 ; tþ exp 2ðlog C log x 0Þðlog C log xþ b 2 : t It follows that f A ðx; x 0 ; tþ ¼fðx; x 0 ; tþ 1 exp 2ðlog C log x 0Þðlog C log xþ b 2 ; t where 0<x<C and fðx; x 0 ; tþ is given in (1). This pdf is also obtained by changing the random variable. Lu (1995) and Padgett and Tomlinson (2004) provided the joint pdf f A ðx; x 0 ; tþ assuming a Brownian motion process as degradation. Thus, changing the variable to, say, x ¼ log x in their result gives the same pdf as above. Hence, the conditional pdf given the event A is given by fðx; x 0 ; tjaþ ¼ f Aðx; x 0 ; tþ ; PðAÞ Z C where P(A)= f A (x; x 0, t)dx Gamma Process 0 In certain physical situations, it is often the case that a degradation process should be always positive and strictly increasing. The geometric Brownian motion process is always positive, but it is not strictly increasing. Thus, we consider the gamma process as a degradation process X t which is always positive and strictly increasing. Assuming that the damage model function h(u)=1 and the damage accumulation function c(u)=u, we have the following cumulative damage or level of degradation of the system at the time t

7 ACCELERATED DEGRADATION MODELS 517 X t X 0 ¼ D t : We assume that the system degradation X t is a shifted gamma process with positive shape coefficient a, scale b and starting value X 0 =x 0. Let S be the first passage time of the gamma process X t to the threshold C. If the initial value of the process X t is x 0 and the threshold value is a known constant C, then X t ) x 0 is distributed as the gamma with shape coefficient a and scale b. Since X t is strictly increasing in t, wehave P½S > tš ¼P½X t <CŠ ¼P½X t x 0 <C x 0 Š Z C x0 1 ¼ 0 CðatÞb at xat 1 exp x dx b ¼ 1 Z Cb n at 1 e n dn; CðatÞ 0 where C b =(C ) x 0 )/b. Thus, the cdf and pdf of S are given by G S ðs; x 0 ; CÞ ¼ Cðas; C bþ CðasÞ g S ðs; x 0 ; CÞ ¼ d Cðas; C b Þ ; ds CðasÞ where G(a,z) is the incomplete gamma function defined by G(a,z)= R z n a)1 e )n dn. After tedious calculus and algebra, we obtain the following pdf of S, " g S ðs; x 0 ; CÞ¼ a CðasÞ Cðas; C bþ log C b þ Cas b ðasþ 2 2F 2 fas; asg; fas þ 1; as þ 1g; C b þ pðwð1 asþ p= tanðpasþ log C bþ a Cðas; C bþ WðasÞ; Cð1 asþ sinðpasþ CðasÞ ð3þ where WðzÞ ¼d=dz log CðzÞ is the digamma function and p F q ðþ is the generalized hypergeometric function or Barnes extended hypergeometric function, which is defined by X 1 pf q fa 1 ;...; a p g; fb 1 ;...; b q g; z ¼ k¼0 ða 1 Þ k ;...; ða p Þ k zk ðb 1 Þ k ;...; ðb q Þ k k! ; with ða i Þ k ¼ a i ða i þ 1Þða i þ k 1Þ. In the pdf of S, we used 2 F 2 ðþ which is given by X 1 as 2 ð C b Þ k 2F 2 fas; asg; fas þ 1; as þ 1g; C b ¼ 1 þ : as þ k k! k¼1

8 518 PARK AND PADGETT Using the following identities, Wð1 xþ ¼WðxÞþpcotðpxÞ p CðxÞCð1 xþ ¼ sinðpxþ ; we can simplify (3) as follows " # g S ðs; x 0 ; CÞ ¼ a CðasÞ Cðas; C bþ log C b þ Cas b ðasþ 2 2F 2 þ a WðasÞ log C b Cðas; C bþ WðasÞ CðasÞ Cðas; C b Þ ¼ a WðasÞ log C b 1 þ a C as b CðasÞ CðasÞ ðasþ 2 2F 2 ; ð4þ where 2 F 2 ¼ 1 þ P 2ð Cb 1 as Þ k k¼1 asþk k!. Although we obtained the exact distribution (4) of the first passage time to the threshold, the distribution is so complicated that it is very difficult to compute in practice. A simpler approximate distribution will be used for computation instead of (4). This approximate distribution can be obtained as follows. Consider the discrete version of S. Let N denote the discrete first passage time as before. Let Y i ¼ X i X i 1 be iid and Y i have gamma distributions with shape a and scale b. Since the process X n is strictly increasing, the probability of first passage after the n time units is given by P½N > nš ¼P½X n <CŠ: Using X n =Y n +Y n)1 ++Y 1 +x 0, we have " # P½N > nš ¼P Xn Y i <C x 0 : i¼1 It follows from the central limit theorem that P½N nš ffi1 U C x 0 abn pffiffi pffiffi : a b n A continuous version of N can be represented by the following Birnbaum Saunders distribution " G S ðs; x 0 ; CÞ ¼U 1 rffiffiffiffiffi r ffiffiffiffiffi! # s b a b ; s

9 ACCELERATED DEGRADATION MODELS 519 p ffiffiffiffiffiffi where a ¼ 1= C b, b p ¼ C b =a, and C b ¼ðC x 0 Þ=b. If C b = ffiffi pffiffi a Cb =a (i.e., a 1), then the above distribution is again approximated closely by the inverse Gaussian distribution with parameters l ¼ C b =a and k ¼ C 2 b =a. It is worth notingpthat ffiffi when l is reasonably large, then the approximation is fairly good even though k is not so much larger than l. As an illustration, pffiffiffi we consider the p ffiffiffi inverse Gaussian pffiffi and Birnbaum Saunders p ffiffiffi distributions with k ¼ 1, l ¼ 0:1 ( k =l ¼ 10) and k ¼ 10, l ¼ 10 ( k =l ¼ 1). We present the probability plots (Wilk and Gnanadesikan, 1968) of the quantiles of the Birnbaum Saunders versus inverse Gaussian distributions in Figure 1. They have almost the same approximations to the inverse Gaussian distributions, as seen in the figure. Hence, if the value of C 0 is reasonably large, then the true distribution is approximated closely by the above inverse Gaussian distribution, which allows easier parameter estimation. Next, we find the joint pdf of X t and A. The event A is again that any sample path within the interval (0, t) with the initial value X 0 =x 0 and the terminal value X t =x does not exceed the threshold C. The probability of the event A is then obtained by PðAÞ ¼P½S > tš ¼P max X u<c : 0<ut Since the gamma process X u is strictly increasing in u, we have max 0<ut X u ¼ X t so that P(A)=P[ X t < C]. By the law of total probability, we have Figure 1. The Q Q plot of the inverse Gaussian and Birnbaum Saunders distributions.

10 520 PARK AND PADGETT fðx; x 0 ; tþ ¼f A ðx; x 0 ; tþþf A cðx; x 0 ; tþ ¼ fðx; x 0 ; tjaþpðaþþfðx; x 0 ; tja c ÞPðA c Þ ¼ fðx; x 0 ; tþiðx CÞþfðx; x 0 ; tþiðx > CÞ; where IðÞ is the indicator function of a given condition. Notice that the support of f A ðx; x 0 ; tþ is (x 0, C) and that of f A cðx; x 0 ; tþ is (C, ). Thus, the joint pdf of X t and A is obtained by 1 f A ðx; x 0 ; tþ ¼ CðatÞb at ðx x 0Þ at 1 exp x x 0 ; b where x 0 < x < C. Hence, the conditional pdf given the event A is given by fðx; x 0 ; tjaþ ¼ f Aðx; x 0 ; tþ ; PðAÞ Z C where P(A)= f A (x; x 0, t)dx. 3. Parameter Estimation 0 For the degradation process X t described in Section 2, we assume that the drift coefficient a of the geometric Brownian motion process and the shape coefficient a of the gamma process are dependent on an acceleration variable L and thus we denote a L =a(l). Suppose that subjects are tested at the I different accelerated levels, L i, i=1,2,..., I. At each level L i, life or strength tests are performed for J i units so that the failure times, s ij, j=1,2,..., J i, are observed. In addition, for the jth unit at the level L i which does not fail by the termination of the test, Q i measurements of degradation are observed up to the termination time t ijkij, which result in degradation measurements x ijk at corresponding times t ijk for k=1,2,..., K ij and j=1,2,..., Q i. The measurements, x ij1, x ij2,..., x ijkij, are referred to as the degradation path data on the unit j at the accelerated test level L i. Note that if x ijk is being measured at t ijk, then t ijkij is the termination time of the unit being observed. For estimation over all of the I accelerated levels, the maximum likelihood estimator (MLE) of the unknown parameters in a given model can be obtained by maximizing the log-likelihood function. First, for the observed failures, the loglikelihood function is given by F ðhþ ¼ XI X J i i¼1 j¼1 log g S ðs ij ; x 0 ; CÞ; ð5þ

11 ACCELERATED DEGRADATION MODELS 521 where h denotes the vector of parameters to be estimated. The inverse Gaussian distribution is used for the above log-likelihood function because it is much simpler for computation of the MLEs for either the geometric Brownian motion or the gamma process degradation cases. Thus, the g(s; x 0,C) is taken to be the inverse Gaussian pdf with appropriate parameters l L and k L. Notice that except for the model based on the gamma process, the scale parameter k L is independent of the accelerated level L. The resulting models can all then be rewritten as a form of S _ IGðl L ; k L Þ: ð6þ The model based on the Brownian motion process proposed by Padgett and Tomlinson (2004) can also be of this form. The inverse Gaussian parameters of (6) under the various degradation models considered are summarized in Table 1. Next, we consider the degradation path measurements. Since the conditional pdf given the event A is proportional to the joint pdf f A (), the log-likelihood function for the degradation path measurements is given by D ðhþ / XI i¼1 X Q i X K ij j¼1 k¼1 log f A ðx ijk ; x ij;k 1 ; Dt ijk Þ; where Dt ijk ¼ t ijk t ij;k 1. Hence, the general log-likelihood function for both observed failures and degradation path measurements is given by ðhþ / F ðhþþ D ðhþ: ð7þ In what follows, we will incorporate the drift and shape coefficients with some acceleration functions linking the coefficients and the acceleration variable, and then we will obtain the MLEs of all the unknown parameters including those in the acceleration function. As described before, the drift coefficient depends on the acceleration variable L, and so does the shape coefficient for the geometric Brownian motion degradation process. There is a variety of acceleration (link) functions between the coefficients and the acceleration variable that can be used. These functions yield a large family of inverse-gaussian-type models (as in Onar and Padgett, 2000) with a covariate or acceleration variable that can be incorporated with degradation and failure data. For more details about the accelerated tests with the link functions, the reader is referred to Mann et al. (1974), Nelson (1990), Meeker and Escobar (1998), and other Table 1. Parameters of the inverse Gaussian distributions under the degradation process considered. Degradation process Parameters Brownian motion C 0 =C)x 0 l L ¼ C 0 =a L k ¼ C 2 0 =b2 Geometric Brownian motion C 0 ¼ log C log x 0 l L ¼ C 0 =a L k ¼ C 2 0 =b2 Gamma C b ¼ðC x 0 Þ=b l L ¼ C b =a L k L ¼ C 2 b =a L

12 522 PARK AND PADGETT references therein. Some popular acceleration functions are: (i) power rule model (a L ¼ nl g ), (ii) Arrhenius reaction rate model (a L ¼ ne g=l ), (iii) inverse-log model (a L ¼ nðlog LÞ g ), (iv) exponential model (a L ¼ ne gl ), and (v) inverse-linear model (a L ¼ n þ gl). It should be noted that except for the inverse linear model, the above models can be rewritten in the power rule model form by simply transforming the acceleration variable. The Arrhenius reaction rate model is obtained from the power rule model a V ¼ nv g with V ¼ e 1=L, the shifted inverse-log model is from V ¼ 1 þ log L, and the exponential model is from V ¼ e L. The original inverse-log model is given by V ¼ log L, which gives an infinite value of l L for L=1 (typical value for a normal use condition). To avoid this, we use a shifted version. Asymptotic confidence bounds for quantiles of the distribution of the failure time (or time to reach critical threshold) can be found by the same method used by Durham and Padgett (1997) and Padgett and Tomlinson (2004). That is, the information matrix can be computed and used to estimate the standard errors of the parameter estimates and then use Bonferroni inequalities. The Crame r s d method can also be used, if tractable, as illustrated by Owen and Padgett (1999) for accelerated Birnbaum Saunders models. 4. Examples In this section, we illustrate the models with two real-data examples. The data sets in the examples can be found in the mainstream statistical literature. For a measure of the model fit, we report the Akaike information criterion (Akaike, 1973,1974) defined by AIC ¼ 2 ðmaximum log-likelihoodþþ2m; where m is the number of independent model parameters. This Akaike information criterion (AIC) is frequently used in engineering and statistics literature to give a guideline for a model selection. When there are several potential models available, the one with the smallest AIC among them (Burnham and Anderson, 2002) can be selected as a good fitting model Carbon-Film Resistors Carbon-film resistors were subjected to higher than usual operating temperatures, and the percent increase in resistance for each was observed at t 0 =0, t 1 =452, t 2 =1030, t 3 =4341, and t 4 =8084 (in hours). Nine resistors were observed at 83 C ( K), and 10 observations were made at each of 133 C ( K) and 173 C ( K). For more details, see Shiomi and Yanagisawa (1979) and Suzuki et al. (1993). The raw data set is also explicitly given in Table C.3 of Meeker and Escobar (1998). It was assumed that the usual operating temperature was 50 C ( K); see Meeker and Escobar (1998). A threshold value for

13 ACCELERATED DEGRADATION MODELS 523 Table 2. Parameter estimates and AIC under consideration. Parameter estimates Acceleration model ^n ^g ^b AIC Brownian motion process Power 2: ) Arrhenius ) ) Inverse-log ) ) Exponential ) ) Geometric Brownian motion process Power ) )3 ) Arrhenius ) )3 ) Inverse-log ) )3 ) Exponential ) )3 ) percent increase in resistance was taken to be C=12 with x 0 =0, and the acceleration levels were given by the ratios to a normal operating temperature in kelvin so that L 0 =1, L 1 =1.102, L 2 =1.257 and L 3 = For the model based on the geometric Brownian motion process, it is reasonable and computationally easier to use the ratios in resistance to the initial value for each rather than the percent increase. That is, the value 0.28 of the percent increase was changed to , and so on. The MLEs with corresponding AIC are summarized in Table 2. For comparison purposes, we also include the model based on the Gaussian or Brownian motion process proposed by Padgett and Tomlinson (2004). Note that some degradation paths in this example are not strictly increasing and thus the gamma degradation process cannot be used here. The MLEs can be used to estimate the mean failure time at which the specified threshold C 0 =12 (Brownian motion process) and C 0 =log(1.12) (geometric Brownian motion process) is reached for resistors at a particular temperature level. For example, the mean failure time based on the Brownian motion process with the power rule model is estimated by ^l V ¼ C 0 =ð^nv^g Þ¼12=ð2: V 10:8926 Þ¼ 554:1957V 10:8926 (in thousands of hours), while that based on the geometric Brownian motion with the power rule model is estimated by ^l V ¼ C 0 =ð^nv^g Þ¼logð1:12Þ= ð2: V 10:6789 Þ¼504:624V 10:6789 (in thousands of hours). These parameter MLEs can also be used to estimate the quantiles of the distribution of failure times given the threshold. For the four temperatures, including the normal use temperature, the estimated 0.10, 0.50 and 0.90 quantiles are plotted in Figures 2 and 3. In the figures, the estimated quantiles under power rule, inverse-log and exponential models are compared to those under the Arrhenius reaction rate model. Note that in all cases the Arrhenius reaction rate model and inverse-log model give estimates that are quite close, while the power rule model and exponential model are quite different from these two at the two lower temperatures. All four acceleration models give very close estimated quantiles at the higher temperatures.

14 524 PARK AND PADGETT Figure 2. The 0.10, 0.50 and 0.90 quantiles (in thousands of hours) under consideration based on the Brownian motion Fatigue Crack Size We next use fatigue-crack-growth data from Hudak et al. (1978). This data set has been analyzed by Lu and Meeker (1993) and Meeker and Escobar (1998), who also explicitly provide the raw data set in Table C.14. Metal specimens with a 0.9 inch pre-crack (or notch) were stressed in a cyclic fashion for 0.12 million cycles or until crack size reached the critical length of 1.6 inches, resulting in failure. If the critical crack length was reached before the test ended, the number of cycles until failure was recorded as the specimen s lifetime. Otherwise, the crack length at the end of the test at 0.12 million cycles was recorded as a measurement of the specimen s degradation. The MLEs can be used to estimate the mean failure time at which the specified threshold C 0 =1.6 ) 0.9=0.7 (Brownian motion process), C 0 ¼ logð1:6þ logð0:9þ

15 ACCELERATED DEGRADATION MODELS 525 Figure 3. The 0.10, 0.50 and 0.90 quantiles (in thousands of hours) under consideration based on the geometric Brownian motion. Table 3. Parameter estimates and AIC under consideration with ML estimates of mean time to failure (in million cycles) to reach the threshold C under consideration. Parameter estimates Degradation process ^n ^b AIC Mean time to failure Degradation path only Brownian motion )1 ) Geometric Brownian )1 ) Gamma )2 ) Degradation path and failure observations Brownian motion )1 ) Geometric Brownian )1 ) Gamma )2 )

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