A new class of survival distribution for degradation processes subject to shocks
Abstract
Many systems experience gradual degradation while simultaneously being exposed to a stream of random shocks of varying magnitudes that eventually cause failure when a shock exceeds the residual strength of the system. In this paper, we present a family of stochastic processes, called shockdegradation processes, that describe this failure mechanism. In our failure model, system strength follows a geometric degradation process. The degradation process itself is any Lévy process, that is, any stochastic process with stationary independent increments. The shock stream is a Fréchet stochastic process, a process derived from the Fréchet extremevalue distribution. Finally, the shockdegradation process is a convolution of the Fréchet shock process and any one of the candidate degradation processes. The system fails at the first occasion when a shock takes system strength across a threshold at zero. The paper presents results for Wiener diffusion processes and gamma processes as examples of Lévy degradation processes. The paper develops key statistical properties of the process model and its survival distribution, including several that are important for its practical application. As the failure mechanism is a first hitting time event, applications that require regression structures fall within the domain of threshold regression methodology.
Keywords
Degradation process Extreme value First hitting time Fréchet process Lévy process Shock stream Stationary independent increments System strength Threshold regressionIntroduction
Shock processes have been studied extensively to explain such diverse phenomena as device failures, insurance claims, earth quakes and business bankruptcies, to name but a few. Early work considered shocks arriving according to a stochastic counting process, combined with a probability of surviving the incident shock (Esary and Marshall 1973; Block and Savits 1978). The latter probability might increase or decrease as a function of the cumulative number of shocks. Later studies considered systems governed by bivariate stochastic processes consisting of one process that generated interarrival times for shocks and the other that generated the magnitudes of the shocks. The system is assumed to fail when the arriving shock exceeds some critical level (Shanthikumar and Sumita 1983; Gut and Hüsler 1999). The current literature continues to propose many extensions and refinements of shock models. In (Mercier and Hai Ha 2017), for example, a bivariate failure model is proposed for which some shocks may be fatal while others are damaging but not fatal.
Stochastic processes as models for system degradation have also been studied in depth. The fields of application and types of systems studied range widely over business, economics, engineering, health, material science, and many more. To cite a few of the numerous publications on the subject that have appeared in the last two decades, we mention that a Wiener diffusion process and a Gaussian process with drift are used in (Doksum and Normand 1995) to model a key biomarker for patients infected with human immunodeficiency virus (HIV). A Wiener diffusion process is a model for degradation of a selfregulating heating cable in (Whitmore and Schenkelberg 1997). Geometric Brownian motion and gamma processes are models of degradation investigated in (Park and Padgett 2005), with case demonstrations given for electrical resistors and metal fatigue. A good review of degradation models for systems reliability was given in (Ye and Xie 2015).
Many types of systems fail when they receive a shock that is sufficient to break them. For example, optical fiber may break under a momentary strain. On the other hand, in modeling human health, a hip may fracture in a fall, and an acute exacerbation may cause the death of a patient with cystic fibrosis (Castilone et al. 2000; Aaron et al. 2015; He et al. 2015). In this article, we are interested in the convolution of two processes that each contribute to failure. First, an underlying degradation process describes the slow weakening or deterioration of the system through time. Second, a shock process describes a random stream of irregular adverse impacts that are superimposed on the degrading system. Failure occurs when the strength of the system is reduced to the breaking point by the combination of degradation and a shock. We propose a general model which combines stochastic processes for both shocks and degradation to create a doubly stochastic process. We model the degradation process as a Lévy stochastic process (a process having stationary independent increments) and the shock stream by a Fréchet stochastic process (a classical extremevalue generating process). We refer to the convolution of these two processes as a shockdegradation process. The system fails in this model at the first moment that the system strength reaches zero. As this failure time is a first hitting time (FHT) of a threshold by the shockdegradation process, realworld applications of the model can use threshold regression methods to take account of regression covariates (Lee and Whitmore 2006; 2010; Lee et al. 2010).
In the following sections we define the component degradation and shock processes, describe their statistical properties, explain how they are integrated into a single composite process, and explore the nature and form of the survival distribution produced by the composite process.
Composite shockdegradation process
Degradation and system strength processes
The system strength process is denoted by {Y(t),t≥0}. Our model assumes that the initial system strength is Y(0)=y_{0}>0. Strength Y(t) tends to decline over time in physical systems but in other kinds of systems, such as social, economic and biological systems, strength can fluctuate or even increase over time. The measurement units of strength are those of the application context, such as lung capacity in liters, battery power in watthours, or financial reserves in millions of dollars. We refer to ‘degradation of strength’ in our general discussion but the possibility of strengthening through time must be contemplated in some contexts.
With stationary independent increments, the c.g.f. of W(t) for any t>0 equals tκ(u).
The connection between system strength and degradation defined in (1) implies that system strength changes in increments that are proportional to the residual strength at any moment. As a result, the system strength Y(t) never reaches a point of zero strength but may approach it asymptotically. Consequently, the system cannot fail through degradation alone. As we show shortly, the failure occurs when shocks exceed the residual strength.
 Wiener diffusion process. Consider a Wiener diffusion process {W(t),t≥0}, with W(0)=0, having mean parameter μ and variance parameter σ^{2}≥0. A Wiener process consists of independent normally distributed increments. Its sample paths are continuous but tend to meander up and down randomly through time. The c.g.f. of a Wiener process has the form:$$ \kappa(u)=u\mu+u^{2}\sigma^{2}/2 \quad \text{for }\infty < u < \infty. $$(3)
When W(t) is a Wiener process, the strength process Y(t)=y_{0} exp[W(t)] is a geometric Wiener process.
In most applications of the Wiener degradation process, the mean parameter μ is negative, indicating that system strength tends to decline. This situation is usual in physical and engineering systems but applications where μ is positive are encountered in social, economic and health systems. In our general mathematical development, we leave the sign of μ unrestricted.
 Gamma process. Consider a gamma process {X(t),t≥0}, with X(0)=0, having scale parameter η>0 and shape parameter ζ>0. A gamma process consists of independent gammadistributed increments. It has monotonically increasing sample paths that consist of random steps of random size. For our model here, we take W(t)=−X(t). The negative sign assures that the sample paths of the degradation process {W(t)} are monotonically decreasing. The negativegamma process finds applications in physical and engineering systems that degrade monotonically and undergo irregular wear and tear through time. A gamma model with monotone increasing sample paths can also be accommodated mathematically but is encountered less frequently in realworld applications. The c.g.f. of W(t) as a negativegamma process has the form:$$ \kappa(u)=\zeta \ln\left(\frac{\eta}{\eta+u}\right) \quad \text{for }\eta+u >0. $$(4)
The corresponding strength process has the following geometric negativegamma form: Y(t)=y_{0} exp[W(t)]=y_{0} exp[−X(t)].

Deterministic exponential process. In this family, system strength follows a deterministic exponential time path of the form Y(t)=y_{0} exp(λt) where λ denotes the degradation rate parameter. The c.g.f. for this deterministic process has the degenerate form κ(u)=λu. Note that this exponential function describes actual time decay if λ<0, no decay if λ=0 and system strengthening with time if λ>0.
This deterministic degradation process is obtained as a special instance of the Wiener model if we set σ^{2}=0, in which case λ=μ, the Wiener mean parameter. Likewise, the process is also a limiting case of the negativegamma degradation process if the ratio −ζ/η is fixed at λ and η is increased without limit.
Fréchet shock process
We look at shock streams through a different lens than previous investigators. In particular, we assume that shocks are generated according to a Fréchet process. This process is a family in the class of maxstable processes. For relevant mathematical background and properties, see (de Haan 1978; 1984; Stoev and Taqqu 2005). By adopting this process, we are not modeling the arrival pattern and magnitudes of individual shocks in an explicit fashion. Instead, the time scale is partitioned into intervals rather than viewed as a continuum of time points. The observed process is the sequence of maximum shocks occurring in the consecutive intervals of the partition. The underlying flow of shocks within each time interval remains latent and uncharacterized  only the maximum shock is potentially observable in each interval.
It follows from (5) that the maximum shock over interval (0,s], namely, V(0,s)= max{V_{1},…,V_{n}}, has the same form of c.d.f. as each of the component terms, namely, P[V(0,s)≤v]=G(v)^{s}.
where α is a scale parameter and β is a shape parameter. We define a Fréchet shock process as any sequence {V_{1},…,V_{n}} for a specified partition, as just described, where the V_{j} are generated from (5) using the Fréchet distribution in (6) as the generating distribution.
The Fréchet distribution is one of the classical extreme value distributions. It is the asymptotic distribution of sample maxima generated from any population distribution that is unbounded to the right and only has finite moments of order k>0 or less. Such populations are said to be of the Cauchy type (Gumbel 1953). A Fréchet distribution of order β is the asymptotic distribution for any Cauchytype distribution with k=β. The role of the Fréchet distribution as an extremevalue distribution gives a rationale for its use as a shock process. We might imagine a large random sample of shocks being drawn from a given Cauchytype population in each time interval of the process. The maximum values of such samples will be approximately distributed as a Fréchet distribution. As Cauchytype distribution families form a large and diverse collection (including the Fréchet distribution itself), this rationale suggests that this kind of shock process has potentially wide application.
Survival distributions of the shockdegradation process
We now combine the shock and degradation processes into a single model. We assume that the shock and degradation processes are independent. With this assumption and the mathematical forms we have chosen for the two component processes, we obtain a conjugate pairing that is mathematically tractable and allows us to generate a wide array of useful mathematical results.
The survival function
In this section we derive an expression for the survival function of our shockdegradation process. We denote the system survival time by S and the sample path of the degradation process W(t) over interval (0,s] by \({\boldsymbol {\mathcal {C}}}=\{W(t):0 \leq t \leq s, W(0)=0\}\).
The lower bound in (8) holds because inequality V_{j}≤Y_{Lj} is a sufficient condition for survival. This inequality assures that the maximum shock does not exceed system strength during the jth subinterval even if it occurs at the moment of least strength. The upper bound holds because the complementary inequality V_{j}>Y_{Uj} assures failure of the system at some moment during the jth subinterval. Thus, the inequality V_{j}≤Y_{Uj} is a necessary condition for survival.
where c=(α/y_{0})^{β}>0.
We note for future reference that the righthand expression in (14) is the Laplace transform of the stochastic integral Q(s) defined in (13), evaluated at c.
The notation V(t,t+dt) represents the maximum shock occurring in the differential time increment (t,t+dt]. Expression (15) states that survival time S is the first moment that the shock exceeds system strength. The conditional logsurvival probability in (12) and the survival function in (14) represent a twostep probabilistic evaluation of the stochastic differential expression in (15).
Lower bound for the survival function
Here κ is shorthand for κ(−β), the c.g.f. of the degradation process W(t) evaluated at −β. Parameter κ must be defined for this lower bound to exist. Its existence is not always assured. For example, for a gamma degradation process, we see from (4) that β<η is required for κ=κ(−β) to be defined. In other words, the shape parameter of the shock process cannot be too large relative to the scale parameter of the gamma process.
We find that this lower bound is quite tight for shockdegradation processes where the degradation process is lightly to moderately stochastic. Because the lower bound is often a good approximation and has a tractable closed form, we use it extensively later to explore features of the shockdegradation model.
Numerical evaluation of the survival function
Here κ_{r} is shorthand for the value of c.g.f. κ(u) at u=−rβ, for r=0,1,2,…. By definition, κ_{0}=0. Notice that the lower bound for the survival function given in (17) is obtained directly from the first expected moment \(m_{1}(s)=E_{{\boldsymbol {\mathcal {C}}}}\left [Q(s)\right ]\), which comes from (18) by setting ℓ=1 and equating κ_{1} with κ in (17).
By calculating successive expected moments m_{ℓ}(s) from (18), the survival function \(\overline {F}(s)\) in (14) can be approximated by finite expansions, as we sketch below and describe in more detail in Appendix A.3 Numerical approximations for the survival function. But caution is required because the larger expected moments of integral Q(s) may grow without limit or even be undefined so the evaluations must be done with care and judgment as we will show. Again, the gamma degradation process offers an example of the difficulty. For the gamma case, quantities κ_{r} are required to evaluate expected moments of order r or larger from (18) and yet these quantities are only defined if rβ<η. Higherorder expected moments of Q(s) for a Wiener diffusion process also grow without limit.
In the following numerical approximations for the survival function \(\overline {F}(s)\) in (14), we temporarily suppress the functional dependence of the random quantities on survival time s to simplify our notation. The approximations are presented in terms of expected central moments of Q which are more accurate. We note that exp(−cQ)= exp(−cm_{1}) exp[−c(Q−m_{1})], where \(m_{1}=E_{{\boldsymbol {\mathcal {C}}}}(Q)\) is the first expected moment of Q. Now we replace Q by Q_{∗}=Q−m_{1} and obtain approximations that are based on expected central moments \(m^{*}_{\ell }=E_{{\boldsymbol {\mathcal {C}}}}(Q_{*}^{\ell })\) that may be derived from (18) using the usual correspondence of uncentered and centered moments. Observe that \(m^{*}_{0}=1\) and \(m^{*}_{1}=E_{{\boldsymbol {\mathcal {C}}}}(Q_{*})=0\).
 Taylor series expansion. This approximation is based on expanding the expression for the survival function in a kth order Taylor series and then taking its expectation over \({\boldsymbol {\mathcal {C}}}\), as described in Appendix A.3 Numerical approximations for the survival function. This expansion results in the following approximation:$$ \overline{F}_{k}(s)=\exp({cm}_{1})\left[\sum_{\ell=0}^{k}(1)^{\ell}\frac{c^{\ell}}{\ell!}m^{*}_{\ell}\right]. $$(19)
 Euler productlimit expansion. This approximation is based on the Euler product limit expansion for exp[−cQ] for which details are given in Appendix A.3 Numerical approximations for the survival function. The kthorder approximation is as follows:$$ \overline{F}_{k}(s)=\exp({cm}_{1})\left[\sum_{\ell=0}^{k} (1)^{\ell} \frac{k!}{\ell!(k\ell)!}\left(\frac{c}{k}\right)^{\ell}m^{*}_{\ell} \right]. $$(20)
Because \(\exp ({cm}_{1})=E_{{\boldsymbol {\mathcal {C}}}}\left \{\exp \left [cQ(s)\right ]\right \}\) is a lower bound for survival function \(\overline {F}(s)\), it can be seen that the centralmoment approximations are basically adjusting the lower bound multiplicatively to improve the approximation. The lower bound itself is the 1storder approximation corresponding to k=1 in (19) and (20). As we have shown, the expected moments of Q can diverge or become undefined for larger k but we are free to choose k so this condition is avoided and the error of approximation is kept within acceptable limits.
Hazard and probability density functions
The hazard function and probability density function are companions of the survival function of a shockdegradation model. Approximations for these functions can be derived mathematically from the corresponding approximations for the survival function. Rather than presenting mathematical forms for these approximations, we find it more convenient to calculate them from the survival function itself using numerical approximations to the differential forms \(d\ln \overline {F}(s)/ds\) and \(d\overline {F}(s)/ds\), respectively.
The hazard is increasing, constant or decreasing exponentially according to whether κ is positive, zero, or negative, respectively. Thus, a roughly exponential shape for the empirical hazard function in a particular application would suggest that the shockdegradation model might be appropriate. Substituting s=0 into (21) gives h_{L}(0)=c=(α/y_{0})^{β}, which shows quite reasonably that the initial risk of failure depends on α/y_{0}, the ratio of the 37th percentile of shocks to the initial strength of the system, modulated by the shock shape parameter β.
The density function has a positive density of c at the origin and steadily declines as s increases if κ≤c. If κ>c, the density function has a single mode at ln(κ/c)/κ.
Special cases of the survival function
 Constant deterministic degradation rate. An immediate result of some interest is obtained in the case where system strength follows a deterministic exponential time path of the form Y(t)=y_{0} exp(λt). In this case, the logsurvival function can be obtained directly from (14) as:$$ \ln \overline{F}(s)=cQ(s)=c\int_{0}^{s} e^{\beta \lambda t}dt=c\left[\frac{e^{\beta\lambda s}1}{\beta\lambda}\right] $$(23)
This form of the survival function has already found application in two medical contexts (Aaron et al. 2015; He et al. 2015). By comparing (23) and (17), we see that the lowerbound survival function presented previously is mathematically identical to the survival function for this model with a constant deterministic degradation rate if κ=−βλ in the lower bound formula.
 Pure shock process. A special case of (23) is a pure shock process. For some systems, such as ceramic devices, degradation makes an insignificant contribution to the risk of failure in comparison to the impact of shocks. For this case, the initial strength y_{0} is a constant and the logsurvival function takes the following form:$$ \ln \overline{F}(s)=cs=(\alpha/y_{0})^{\beta} s $$(24)
The survival function here is a pure exponential distribution. Although a simple model, its failure rate has a variety of stochastic behaviors depending on the parameters of the shock process and the system strength y_{0}. To take some limiting cases, for very small β (approaching 0), the survival time is approximately exponentially distributed with a unit rate of decay. For very large β (approaching ∞), failure is almost immediate if y_{0}<α, is exponentially distributed with a unit rate of decay if y_{0}=α, and is long delayed if y_{0}>α.
Joint observation of survival and degradation
Estimation of the shockdegradation model in a practical setting will often involve data that represent longitudinal records on a sample of systems. Each data record consists of periodic readings on the underlying degradation process and ends with either censoring of the record or system failure. This kind of application requires new mathematical results, some of which we now present.
Longitudinal readings on the degradation process
For fixed values of α>0, β>0, v>0 and n, this formula gives a probability that approaches 1 as Δt approaches zero. Thus, in this limiting sense, the maximum shocks observed in n observation intervals can be made arbitrarily small by shrinking the widths of the n observation intervals. Expressed in more practical terms, the theoretical result implies that no material shocks are likely to be present in any finite number of observation intervals if each interval is sufficiently short. The lesson we learn is that material shocks in this kind of process occur sparingly and are rarely discovered at random moments of observation.
Markov decomposition of longitudinal data records
Given the preceding result, we now consider the data elements found in a longitudinal record consisting of periodic readings on system degradation. Our line of development follows the method of Markov decomposition proposed in (Lee and Whitmore 2006). We limit our study now to the lowerbound survival function in (17).
Set \({\boldsymbol {\mathcal {C}}}^{*}\) represents any sample path of the degradation path that starts at the origin W(0)=0 and ends at level w at time s or, equivalently, any strength sample path that starts at level y_{0} at time 0 and ends at level y=y_{0} exp(w) at time s. The condition describes a sample path that is pinned down at two end points but is otherwise free to vary.
P.d.f. g(w) is known from the specified form of the degradation process. The conditional survival function P{S>sw}, which we denote by \(\overline {F}(sw)\), is less straightforward and has yet to be derived in a general form for our shockdegradation model. Here we present a more limited mathematical result.
Survival, conditional on a closing degradation level
The inequality follows from Jensen’s inequality when the expectation and logarithmic operators are interchanged. Expectation \(E_{{\boldsymbol {\mathcal {C}}}^{*}}\left [Q(s)\right ]\) does not have a general closed form for the family of degradation processes with stationary independent increments; the quantity requires an evaluation of the conditional cumulant generating function \(E_{{\boldsymbol {\mathcal {C}}}^{*}}\left \{\exp \left [\beta W(t)\right ]\right \}\).
Thus, (30) and (31) are the two components we require for evaluating a tight lower bound for the joint probability in (28), namely, the probability of a system surviving beyond a censoring time s and having the degradation level w at that time.
Other application properties
Practical applications necessarily direct attention to other important properties of shockdegradation processes. We now discuss a couple of these properties.
Probability of no eventual failure (cure rate)
To illustrate two contrasting instances of (32), a look at the c.g.f. for a geometric Wiener strength process in (3) shows that κ<0 if μ−βσ^{2}/2>0. Thus, the degradation process in this situation must have μ both positive and large enough to override the variance offset βσ^{2}/2. In this situation, even if the Wiener process has no drift (so μ=0), the survival function will act as if there is a degradation trend toward failure. The tilt towards failure depends on parameter β of the shock distribution as well as the variance parameter σ^{2} of the Wiener process. In contrast, for a gamma degradation process with the c.g.f. defined in (4), we have parameter κ>0 and the degradation sample path decays monotonically and eventual failure is certain so P(S=∞)=0.
Powertransformed time scale
This family of survival distributions has already appeared in the distribution theory literature. A survival function with this mathematical form is referred to as an XTGG distribution in (Cordeiro et al. 2016), page 14, for the case where κ>0. Also, (Xie et al. 2002) refers to this type of distribution as a ‘modified Weibull extension.’ The latter article presents some properties, estimation methods, and engineering applications of this type of distribution. What is new here is its connection with the lower bound for survival functions of shockdegradation processes.
Concluding discussion
Our derivation of a family of shockdegradation processes is motivated by the ubiquity of realworld applications in which the strength of a system is degrading stochastically through time and is subjected simultaneously to random shocks of varying size. The system fails when a shock arrives that exceeds its remaining strength. The stochastic process involves a superposition of an intrinsic strength process and an independent shock process. We postulate that system strength decays geometrically according to a degradation process that possesses stationary independent increments and, further, that the shock stream follows a Fréchet process. The combination yields a realistic, tractable and flexible family of processes.
We have derived a number of properties for the shockdegradation process, including a series representation of the survival function and a lower bound for the function that is quite tight in many practical settings. The important situation where survival and the strength of a survivor are jointly observed is considered. For the important case of a Wiener degradation process, we have derived an elegant closed form for the probability of this joint event, which is a new and useful finding. Several additional mathematical results are derived that are important, including an expression for the probability of no eventual failure (which may occur if the system happens to be gradually strengthening rather than degrading). An expression is also obtained for the survival function if the time scale is subject to a power transformation. It is then discovered that the lowerbound survival function, with the transformed time scale, is a modified Weibull extension.
The special case of our shockdegradation model in which the strength process is a deterministic exponential function has been applied already in several studies. This version of the model was used in a study of osteoporotic hip fractures (He et al. 2015). The underlying strength process in the model represented skeletal health. The shock process represented external traumas, such as falls and stumbles, which taken together with chronic osteoporosis might trigger a fracture event. Threshold regression was used to associate time to fracture with baseline covariates of study participants. The same model was considered by (Aaron et al. 2015) to study mortality risks of cystic fibrosis patients. The model was estimated using patient data from a national cystic fibrosis registry. In this setting, acute exacerbations of the disease act as shocks. An exacerbation leads to death on the first occasion when its severity exceeds the respiratory strength of the patient. Practical issues around modelling and estimation for these two studies, as well as for a third study that investigated Norwegian divorces using the same model, were reviewed in the survey paper (Lee and Whitmore 2017).
All of these published applications that employed the simple deterministicexponential version of the model have used asymptotic maximum likelihood methods for parameter estimation and inferences. The data sets available in these applications were conventional censored survival data. There are important potential applications of the general model, however, that will have more complex types of data than plain censored survival data. For example, in some applications, measurements of both degradation and shock magnitudes prior to failure may be available for model estimation. As an illustration, for patients with chronic obstructive pulmonary disease (COPD), measurements of both lung function and severity levels of acute exacerbations may be gathered through time for each patient. Development of inference methods for the general shockdegradation model based on different data scenarios is a pressing need that we anticipate will be addressed in future research.
Although the shockdegradation model is conceptually clear and wellbehaved when the degradation process has only moderate variability, its stochastic behavior becomes chaotic and unpredictable when the degradation process is highly variable. In this situation, the right tail of the survival function becomes heavy and numerical calculation of tail probabilities becomes challenging. Our approximating series expansions are sensitive to this instability in the tails when the degradation process is volatile. An extremely volatile degradation process is not expected in designed or fabricated systems (such as technical devices, engineering structures, or economic and social organizations) but natural physical systems and phenomena frequently have chaotic failure processes. Examples of chaotic natural processes with event times that are extremely unpredictable include, for instance, the time for a cumulus cloud to disperse, an iceberg to collide with a stationary marine structure, or a soap bubble to burst in turbulent air.
A field of application that is the scene of intense current research on stochasticprocess models is financial engineering. It seems possible (indeed, likely) that research breakthroughs in stochastic financial models may very well further the development of our shockdegradation model. Business, economic and financial systems are characterized by both stochastic degradation and growth scenarios and these systems are also subject to a wide array of random shocks exactly as in our model. The mathematical advances described in (Hackmann 2015; Hackmann and Kuznetsov 2016), for example, are indicative of this rich source of development potential. Important extensions that will come from future research by these and other investigators might include more efficient numerical methods for calculating tail probabilities of the survival function and other probabilistic quantities for shockdegradation processes.
Appendix
A.1 Survival function for a Fréchet shockdegradation process
A.1.1 Definite integral representation of the conditional logsurvival function
Theorem 1
Proof
It is useful to review briefly the nature of sample paths of our degradation processes, which are Lévy processes. Every Lévy process can be represented as a superposition of processes that may include a Wiener process and independent Poisson processes having varying jump magnitudes. A pure Wiener process (including pure drift) is the only Lévy process with a continuous sample path. All others have discontinuous sample paths because of the Poisson jump component. Jumps of any specified size arrive according to a Poisson process with a sizedependent rate; a rate given by the Lévy measure of the process. For any fixed jump size, say ε>0, the process will experience (almost surely) only a finite number of jumps per unit time that exceed ε but may experience an infinite number of jumps per unit time that are all smaller than ε.
 1.
If our degradation process is a pure Wiener process then difference U_{π}(s)−L_{π}(s) will converge to 0 as norm Δ tends to zero because a Wiener process has continuous sample paths. Thus, degradation extremes W_{Lj} and W_{Uj} approach each other in each time interval (t_{j−1},t_{j}] as the interval shrinks to zero in the limit.
 2.If the degradation process has a jump component, we know the largest degradation jump experienced in interval (t_{j−1},t_{j}] of the time partition π cannot exceed \(W_{U_{j}}W_{L_{j}}\). Thus, for any given ε>0, we can divide the time intervals of the time partition π into the following two sets:$$J_{0}=\{j: W_{U_{j}}W_{L_{j}}<\epsilon\}, \quad J_{1}=\{j: W_{U_{j}}W_{L_{j}} \geq \epsilon\}$$We can then express the difference U_{π}(s)−L_{π}(s) as the sum A_{0π}(s)+A_{1π}(s), where:$$A_{0{\boldsymbol{{\pi}}}}(s)=c\sum_{J_{0}} (t_{j}t_{j1})[\exp(\beta W_{Lj})\exp(\beta W_{Uj})],$$$$A_{1{\boldsymbol{{\pi}}}}(s)=c\sum_{J_{1}} (t_{j}t_{j1})[\exp(\beta W_{Lj})\exp(\beta W_{Uj})].$$Sum A_{1π}(s) has a finite number of terms and thus will clearly converge to zero as norm Δ tends to zero. On the other hand, sum A_{0π}(s) may have an infinite number of terms. We note, however, that:$$A_{0{\boldsymbol{{\pi}}}}(s)\leq c [1\exp(\beta \epsilon)]\sum_{J_{0}} (t_{j}t_{j1})\exp(\beta W_{Lj})$$because \(W_{U_{j}}W_{L_{j}}<\epsilon \) and, therefore:$$\exp(\beta W_{Lj})\exp(\beta W_{Uj}) \leq [1\exp(\beta \epsilon)]\exp(\beta W_{Lj}) \quad \text{for all \(j\in J_{0}\)}.$$The sum \(\sum _{J_{0}} (t_{j}t_{j1})\exp (\beta W_{Lj})\) approaches the definite integral Q(s), defined in (35), as norm Δ approaches 0. This result is evident because time intervals in set J_{0} have aggregate measure (0,s] in the limit and W_{Lj} approaches W(t_{j−1}) as t_{j} approaches t_{j−1} in the limit, for each interval in J_{0}. Definite integral Q(s) exists because we have required that c.g.f. κ(u) exist for degradation process {W(t)}. Thus, we have shown that A_{0π}(s)≤c[1− exp(−βε)]Q(s) in the limit. Finally, as threshold ε>0 can be chosen arbitrarily, we can make it as small as we wish and so drive A_{0π}(s) to 0 in the limit. In conclusion, therefore, we have shown for a degradation process with a jump component that:$${\lim}_{\Delta \rightarrow 0} U_{{\boldsymbol{{\pi}}}}(s)L_{{\boldsymbol{{\pi}}}}(s)={\lim}_{\Delta \rightarrow 0} A_{0{\boldsymbol{{\pi}}}}(s)+A_{1{\boldsymbol{{\pi}}}}(s)=0.$$
With this last step, the proof is complete. □
A.1.2 Representation of the survival function as an infinite series
The unconditional survival function is obtained by taking the expectation of \(P(S>s{\boldsymbol {\mathcal {C}}})\) over all sample paths \({\boldsymbol {\mathcal {C}}}\), that is, \(\overline {F}(s)=E_{{\boldsymbol {\mathcal {C}}}}\left \{\exp \left [cQ(s)\right ]\right \}\).
A.2 Exact expressions for expected moments of Q(s)
where κ_{r} is shorthand for the value of c.g.f. κ(u) at u=−rβ, for r=0,1,2,…. By definition, κ_{0}=0. The derivation requires that the quantities κ_{r} exist. Their existence is assured if \(r\beta \in {\boldsymbol {\mathcal {Z}}}\) for all natural numbers r of interest. Here \({\boldsymbol {\mathcal {Z}}}\) is the open set on which the c.g.f. κ(u) of our degradation process {Y(t)} is defined.
A.3 Numerical approximations for the survival function
In developing numerical approximations for the survival function \(\overline {F}(s)=E_{{\boldsymbol {\mathcal {C}}}}\left \{\exp \left [cQ(s)\right ]\right \}\), we use notation m_{ℓ} for \(E_{{\boldsymbol {\mathcal {C}}}}\left (Q^{\ell }\right)\), the ℓth expected moment of Q. We also suppress the functional dependence of random quantity Q on survival time s to simplify our notation.
Taylor series expansion
Euler productlimit expansion
We note that the first term on the right side of equation (48) can be expanded as the following linear combination of expected moments of Q:
Comparing the counterpart terms in (46) and (50) shows that the two approximations differ only in their weights for the first k expected moments of Q.
Approximations based on expected central moments
Here m_{1} denotes the first expected moment of Q, that is, \(m_{1}=E_{{\boldsymbol {\mathcal {C}}}}(Q)\) and Q_{∗}=Q−m_{1} is the centered Q quantity.
By definition, we have \(m_{1}^{*}=E_{{\boldsymbol {\mathcal {C}}}}(Q_{*})=0\). Also, because \(\exp ({cm}_{1})=\exp \left \{{cE}_{{\boldsymbol {\mathcal {C}}}}\left [Q(s)\right ]\right \}\) is a lower bound for survival function \(\overline {F}(s)\), it can be seen that the centralmoment approximations in (53) and (54) are basically adjusting the lower bound multiplicatively to improve the approximation. In fact, the lower bound itself is the starting approximation with k=1 in the expansion for \(E_{{\boldsymbol {\mathcal {C}}}}\left [\exp \left (cQ^{*}\right)\right ]\).
Approximation bounds for the survival function
As k proceeds through odd and even numbers, the bounds in (56) and (57) become alternately upper and lower bounds and then lower and upper bounds for the remainder term. These bounds on the remainder can be tightened but we will not pursue this line of analysis further.
As anticipated earlier when we discussed the conditions for Fubini’s Theorem to apply, the expected moments of Q can diverge or become undefined for large k and therefore the expected central moments are similarly affected. Fortunately, we are free to choose the finite order of the approximation k as we please and it can be selected so the approximation error is minimized.
Approximations for the survival probability of an illustrative shockdegradation model for survival time s=30
Order k  Lower Bound  Taylor S. F. Estimate  Upper Bound  Euler S. F. Estimate 

Lower Bound on Survival Probability  
1  0.2867  0.2556  0.4449  0.2556 
Higherorder Approximations for the Survival Probability  
2  0.2364  0.3040  0.3005  0.2798 
3  0.2908  0.2867  0.3407  0.2840 
4  0.2627  0.3005  0.2989  0.2867 
5  0.2913  0.2908  0.3223  0.2883 
Derivation of the survival function conditional on a closing degradation level
and Φ(·) denotes the standard normal c.d.f..
Notes
Authors’ contributions
Both authors contributed equally. Both authors read and approved the final manuscript.
Funding
Lee’s work is supported in part by NIH Grant R01EY02445.
Competing interests
None of the authors has any competing interests in the manuscript.
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