# On Wiener Process Degradation Model for Product Reliability Assessment: A Simulation Study

Authors: Herbert Hove; Farai Mlambo
Journal: Modelling and Simulation in Engineering
 (2022)

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## Abstract

This paper examines the application of the Wiener process as a degradation model. Its appropriateness as a degradation model is discussed and demonstrated with the aid of Monte Carlo simulations. In particular and for monotonically degrading systems, this paper demonstrates that the irreversible accumulation of damage can be modelled by the Wiener maximum process. First passage times of the Wiener and its maximum process are also revealed to coincide. Practical advantages of assessing system reliability from degradation data are highlighted by applying the Wiener process model to a real gallium arsenide (GaAs) laser data for telecommunication systems. The real data application results demonstrate that degradation analysis allows for conclusions about system reliability to be reached earlier without compromising estimation accuracy—a major practical advantage.


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## Body

## 1. Introduction

Assessing reliability of technical systems from failure time information is increasingly becoming a challenge. New technologies on the design for reliability continue to be developed. This has resulted in highly reliable systems that operate for long with few or no failures, even under accelerated conditions (Ye and Xie [1]). For such technical systems, collecting sufficient failure time information for reliability assessment is a costly exercise. Depending on the application, an alternative approach is to use gathered information on the state of the system and its performance while in operation, called degradation data. Through the use of suitable models and data analysis methods, registered degradation data can be converted to system reliability information which can be utilised for reliability assessment (Guo and Liao [2]).The rationale is based on the observation that ageing failures are linked to an underlying degradation process (Lehmann [3]; McLinn [4]). For most manufactured systems, the physical conditions degrade as the system ages, such as automobile tyre wear. For some systems however, degradation occurs in system performance such as the light intensity of a light-emitting diode (ELD) dropping with usage. The physical or system performance degradation has the interpretation of damage to the system. It accumulates with time or mission and ultimately causes failure when the accumulated damage reaches a failure threshold defined by industrial standards. The system failure time distribution FTt=PT≤t and its parameters are derived from the analysis of degradation data and the deterioration mechanism. Based on the derived FTt, reliability metrics of interest such as mean time to failure (MTTF) and 100pth percentiles are determined.Degradation models fall into two broad categories, namely, general path models and stochastic process models (Meeker et al. [5]). General path models have a well-developed theory. They are essentially mixed-effects regression models and can therefore incorporate covariates and random effects in a flexible way (Coble and Hines [6]). Their limitation is the inability to capture the time-varying behaviour of systems and the uncertainty ingrained in the evolution of system degradation over time. On the other hand, system degradation is naturally governed by a random mechanism that is best described by a stochastic process (Limon et al. [7]; Gorjian et al. [8]). In view of their random nature, stochastic process models allow for a natural explanation of the unexplained randomness of the degradation over time resulting from unobserved environmental factors. This study assumes a stochastic process model for system degradation paths. Its main objective is to investigate the application of the Wiener process in degradation modeling. With the help of Monte Carlo simulations, applications where the Wiener process is a suitable degradation model are demonstrated.


### 1.1. Overview

The remainder of the paper is organised as follows. In Section2, the basis of Wiener process as a degradation model and parameter estimation is reviewed. Known results are also demonstrated using Monte Carlo simulations. The application of Wiener maximum process for monotone degradation is the subject of Section 3. In Section 4, a real data application involving GaAs laser degradation data for telecommunication systems is presented. The study ends with concluding remarks in Section 5.


## 1.1. Overview

The remainder of the paper is organised as follows. In Section2, the basis of Wiener process as a degradation model and parameter estimation is reviewed. Known results are also demonstrated using Monte Carlo simulations. The application of Wiener maximum process for monotone degradation is the subject of Section 3. In Section 4, a real data application involving GaAs laser degradation data for telecommunication systems is presented. The study ends with concluding remarks in Section 5.


## 2. The Wiener Process as a Degradation Model

The Wiener processWt,t∈ℝ+ is (Kahle and Lehmann [9]; Wang [10]) the basic model for random accumulation of degradation over time. Its basis is that degradation increment in an immeasurably small time interval is the sum of a large number of small, independent random stress effects (additive superposition).Denote byBn the sum Bn=R1+R2+⋯+Rn where Ri is independent random variables with finite means ERi=μi and finite variances VarRi=σi2. Assume none of the Ri dominates the rest. Then from the central limit theorem, the standardisation of Bn denoted by
(1)Zn=Bn−EBnVarBn,converges under the Lindeberg condition (Beichelt [11]) to a normal distribution. That is,
(2)limn⟶∞PZn≤x=Φx=12π∫−∞xexp−u22du,where Φx is the standard normal distribution function. Thus, the degradation increments ΔWt=Wt+Δt−Wt over the interval t,t+Δt are normally distributed. Accordingly, the Wiener process Wt,t∈ℝ+ has the following properties:
(1)
For all0≤s<t, the degradation increment Ws,t=Wt−Ws is normally distributed with mean 0 and variance σ2t−s where σ2 is a variance parameter(2)
For any set of disjoint time intervals, increments are independent random variables distributed as described in property 1(3)
For any constanta and t1<t2, Wt2−Wt1=dWt2+a−Wt1+a. That is, Wt has stationary increments(4)
W0=0 almost surelySystem degradation generally has a nonzero mean. An obvious improvement of the Wiener process model is to include a drift measureν reflecting the rate of degradation. This yields a one dimensional Wiener process with drift
(3)Wt=νt+σBt,where σ is the diffusion parameter, and Bt is standard Brownian motion on 0,∞ capturing the stochastic evolution of the degradation process. Thus, EWt=νt and VarWt=σ2t. Consequently, Wt~Nνt,σ2t. Unless indicated otherwise, technical systems having the same design are assumed to have common drift and variance parameters.


### 2.1. Wiener Process Model for Nonmonotone Degradation

An assumption that is often valid in applications is that physical or performance degradation is a continuous process. Accordingly, sample paths of the stochastic process describing system degradation is ought to be restricted to continuous functions.Simulated sample paths of a Wiener process forν=3.5 and σ taking values 2.5, 2, 1.5, and 1 are presented in Figure 1. Process trajectories in Figure 1 are continuous functions. It is therefore not surprising that the Wiener process is the basic model for a degradation process. Describing system degradation by the Wiener process implies that physical or performance degradation can increase or decrease with time. While this might not be meaningful in many degradation applications, it is applicable to degradation processes whose levels vary bidirectionally over time when observed closely. Examples include
(1)
The gain of a transistor or the extent of propagation delay (Lu [12])(2)
Cracks healing and CD4 blood cell counts fluctuating (Singpurwalla [13])(3)
Resistance of the structure alternating with time in the framework of structural reliability (Dong and Cui [14])Figure 1
Simulated Wiener process sample paths.For monotone degradation processes such as wear-out, the application of the Wiener process is only as an approximation, which is especially good ifσ2≪ν. In this case, the trajectories are approximately monotone (see bottom-right panel of Figure 1) since the tooths in the evolving paths of the Wiener process are appreciably smoothed out. Alternatively, all factors contributing to nonmonotone behaviour in a Wiener process model may be attributed to pure noise and modelled accordingly.


### 2.2. Lifetime Estimation and Failure-Time Distribution

Assuming a Wiener process model, system lifetimeTs is the time Wt,t∈ℝ+ crosses the critical degradation level s for the first time. That is, Ts is the first passage time of the Wiener process to s. It is given by
(4)Ts=inft∈ℝ+:Wt≥s.It is well known (Chhikara and Folks [15]) that Ts is distributed as inverse Gaussian with probability density function (pdf)
(5)fTst=sσ2πt3exp−12σ2s−νt2t,t>0,ν>0.A useful reparameterisation of the density in Equation (5) in terms of the development of statistical properties analogous to those of the normal distribution (Tweedie [16]) is obtained by setting
(6)μ=sν;λ=s2σ2.This yields the reparameterised inverse Gaussian distributionTs~IGμ,λ with pdf and cumulative distribution function (cdf) given as
(7)fTst,μ,λ=λ2πt3exp−λ2μ2t−μ2t,t>0,(8)FTst=Φλttμ−1+exp2λμΦ−λttμ+1,respectively, where μ∈ℝ+ is the mean, and λ∈ℝ+ is the shape parameter. The inverse Gaussian distribution is right-skewed and bounded at zero. Figure 2 illustrates the probability density function for the inverse Gaussian with μ=3.5 and λ taking values 16, 45, and 100.Figure 2
Inverse Gaussian distribution as a function of shape parameter.The result thatTs is inverse Gaussian is demonstrated using Monte Carlo simulations. In particular, 103 sample paths were simulated and s is set to 250 to ensure all systems are tested to failure as illustrated in Figure 3.Figure 3
Simulated sample paths and threshold choice.System degradation is simulated at discrete times. Interpolation was done using splines method to ensure that the resultingTs values are continuous, and thus unique and more representative.Figure4 shows histograms of the resulting Ts values for different path parameter combinations. Additionally and based on the parameterisation in Equation (6), the theoretical inverse Gaussian pdf is also represented in green. It follows from Figure 4 that the histograms closely resemble the theoretical inverse Gaussian pdf. Hence, this illustrates that first passage times of a Wiener process with drift are indeed distributed as inverse Gaussian.Figure 4
Histograms of first passage times and theoretical distributions.


#### 2.2.1. Maximum Likelihood Estimation of Path Model Parameters

Path model parameters are estimated based on registered degradation data or derived first passage times. The former applies to highly reliable systems where failure does not interrupt observation of the degradation process, that is,s⟶∞. Denote by Witj the ith system degradation measure at inspection time tj, i=1,⋯,n; j=1,⋯,mi. Then, the degradation increment for the ith system is ΔWij=Witj−Witj−1. From Equation (3),
(9)ΔWij=νiΔtj+σiΔBtj,where Δtj=tj−tj−1 and ΔBtj=Btj−Btj−1 with ΔWij~NνiΔtj,σi2Δtj.Consequently,Δwij are the degradation increments for system i with pdf fΔwij;vi,σi2. Since the Wiener process has normally distributed increments, the likelihood function for system i is
(10)liνi,σi2Δwij=∏j=1mifiΔwijνi,σi2=∏j=1mi12πσi2Δtjexp−12σi2Δwij−νiΔtj2Δtj.The corresponding log-likelihood for theith system is given by
(11)Li=lnliνi,σi2Δwij=−mi2ln2π−milnσi−12∑j=1milnΔtj−12σi2∑j=1miΔwij−νiΔtj2Δtj.Taking partial derivatives of the log-likelihood function in Equation (11) with respect to νi and σi gives
(12)∂Li∂νi=∑j=1miΔtjΔwij−νiΔtjσi2Δtj=0,(13)∂Li∂σi=−miσi+1σi3∑j=1miΔwij−νiΔtj2Δtj=0.The maximum likelihood estimators (MLE)νi^ and σi^ are obtained by simultaneously solving Equations (12) and (13). They are
(14)νi^=∑j=1miΔwijΔtj,(15)σi^=1mi∑j=1miΔwij−νiΔtj2Δtj.Wiener process degradation incrementsΔwij are independent. Hence, the log-likelihood function of their full set is
(16)Lν,σ2Δwij=∑i=1nLνi,σi2Δwij,and MLE for model parameters ν and σ are
(17)ν^MLE=∑i=1n∑j=1miΔwijΔtj,(18)σ^MLE=1min∑i=1n∑j=1miΔwij−νiΔtj2Δtj.For most applications however,s is finite as determined by industrial standards. System lifetimes are the Ts values for the n sampled systems. Denote by τ1,⋯,τn these first passage times. Their density is given in Equation (5) for the underlying Wiener degradation process. The likelihood function is thus
(19)lν,σ2τ1,⋯,τn=∏i=1nsσ2πτi3exp−12σ2s−νt2τi,with log-likelihood function
(20)L=nlns−nlnσ−n2ln2π−32∑i=1nlnτi−12σ2∑i=1ns−ντi2τi.Maximising the log-likelihood function in Equation (20) in respect of process parameters ν and σ yields
(21)∂L∂ν=1σ2∑i=1ns−ντi=0,(22)∂L∂σ=−nσ+1σ3∑i=1ns−ντi2τi=0.MLE forν and σ is obtained by simultaneously solving Equations (21) and (22). They are
(23)ν^MLE=s1/n∑i=1nτi;σ^MLE=1n∑i=1ns−ν^τiτi.Values ofν^MLE and σ^MLE are obtained from Equation (23) based on the first passage times from the 103 simulated degradation paths. The results, together with true path parameter values, are presented in Table 1.Table 1
True and parameter estimates forIGν,σ.
IGν,σTrue valuesML estimatesνσν^MLEσ^MLE3.52.53.55192.43603.52.03.54902.05483.51.53.52981.59493.51.03.50531.0361They show that parameter estimates of the first passage time distribution in Equation (5) recover path parameters used in the simulation. This is equally true for parameter estimates of the transformed IGμ,λ distribution in Equation (7), and the results of which are presented in Table 2.Table 2
True and parameter estimates for transformedIGμ,λ.
IGμ,λTrue valuesML estimatesμλμ^MLEλ^MLEσ=2.571.42861000071.602910821σ=2.071.42861562571.442215223σ=1.571.42862777871.686125551σ=1.071.42866250071.463758291Thus, Monte Carlo simulations prove the correctness of the well-known result that first passage times of a Wiener process with drift obey inverse Gaussian law.


#### 2.2.2. Interval Estimation of Model Parameters

Maximum likelihood estimators are point estimates obtained from sample data. Accordingly, values ofν^ and σ^ are subject to sampling fluctuations and may or may not be close to the quantities being estimated. It is therefore important to quantify uncertainty associated with parameter estimates. Confidence intervals are very useful in quantifying uncertainty in point estimates due to sampling error arising from limited sample sizes. Exact confidence intervals can be constructed, if for example, the sampling distribution of θ^=ν^,σ^ is known. Otherwise, and for large samples, approximate confidence intervals are used. MLE is asymptotically normal. Hence, confidence intervals for θi, i=1,2 are constructed by the asymptotic normal approximation. Also called Wald confidence intervals, normal approximation confidence intervals are based on the Wald statistic:
(24)Zθ^i=θ^i−θise^θ^i~N0,1.The standard error ofθ^i is determined by the second derivative of the log-likelihood function with respect to θi which quantifies the curvature of the log-likelihood function. That is
(25)se^θ^i=−L′′θ∧i;τ−1,and is evaluated at θi=θ^i where τ is a vector of first passage times. The quantity −L′′θ^i;τ is the observed information. When constructing normal approximation confidence intervals however, −L′′θ^i;τ is often replaced by the expected or Fisher information
(26)Iθi=−EL′′θi;τ.The resulting1001−α% confidence interval for θi is given by
(27)θ^i±z1−α/2·se^θ^i.Alternatively, the statistic(28)Zlogθ^i=logθ^i−logθise^logθ^i~N0,1,is used instead. Observe that logθ^i is unrestricted in sign. Hence, Zlogθ^i is in general closer to a N0,1 distribution than is Zθ^i. Thus, and after transforming by exponentiation, the confidence interval for θi is
(29)θ^iexpz1−α/2×se^θ^i/θ^i,θ^i×expz1−α/2×se^θ^i/θ^i.Often in reliability, only a few system failures are observed. In this case, large sample normal theory is inexact. Rather, likelihood ratio confidence bounds method is often preferred. It is based on likelihood ratio equation:(30)−2·lnlθlθ^≥χα;k2,where lθ is the likelihood function for the unknown parameter θ, lθ^ is the likelihood function evaluated at θ^, and χα;k2 is the chi-squared statistic with probability α and k degrees of freedom, where k is the number of jointly estimated parameters. A rearrangement of Equations (30) yields
(31)lθ=lθ^·e−χα;k2/2,where terms on the right hand side are known exactly. The confidence limits for θi are the minimum and maximum calculated θi values for which Equation (31) holds.Contour plots are a useful way of simultaneously estimating likelihood ratio confidence bounds on the parameters. Equation (31) has no closed form solution, hence, a numerical solution is required instead. A crude approach is holding one parameter constant while iterating on the other until an acceptable solution is reached. Figure 5 gives contour plots for first passage times in Figure 4.Figure 5
Confidence bounds ofμ and λ for different path parameter combinations.
(a)
Path parameters:ν=3.5; σ=2.5(b)
Path parameters:ν=3.5; σ=2.0(c)
Path parameters:ν=3.5; σ=1.5(d)
Path parameters:ν=3.5; σ=1.0Confidence bounds forμ get narrower with decreasing volatility. This is expected because low variability results in smoother sample paths (Figure 3), hence, less variability in first passage times (Figure 4). The bounds for λ however get wider with decreasing volatility. This is a result of the scaling in Equation (6) and can be seen in the peakedness of the theoretical and empirical densities in Figures 2 and 4, respectively.


### 2.3. Sampling Distributions of First Passage Time Distribution Parameters

First passage times in Figure4 are based on 103 simulated sample paths. In practice however, a few systems are tested for economic reasons. Hence, trajectories for n=20 (small sample) and n=100 (large sample) are simulated. Values of μ^MLE and λ^MLE are obtained from the resulting first passage times. This procedure is repeated 104 times for each n, yielding respective sampling distributions in Figures 6 and 7.Figure 6
Sampling distributions ofμ^MLE and λ^MLE for n=20.Figure 7
Sampling distributions ofμ^MLE and λ^MLE for n=100.Sampling distributions forμ^MLE are fairly symmetrical regardless of sample size. Those of λ^MLE are right-skewed for small samples and close to normal for large samples. Therefore, approximate normal confidence intervals for λ may be appropriate for large samples whereas for μ, any sample size may apply.The performance ofμ^MLE and λ^MLE is also assessed for different sample sizes and volatility parameters. In particular, bias and empirical standard error (EmpSE) were reported together with their Monte Carlo standard errors (Monte Carlo SE). Bias is the amount by which ϕ^=μ^MLE,λ^MLE exceeds ϕ=μ,λ on average. Unbiasedness is a key property in frequentist theory. However, small biases maybe traded-off for other good properties. The EmpSE estimates the long-run standard deviation of ϕ^i for 104 replications. It is a measure of the precision (efficiency) of the estimator. Monte Carlo SE provides an estimate of the SE of the estimated performance measure as a result of using a finite number of replications (Morris et al. [17]). Another important measure is mean squared error (MSE), measuring the accuracy of ϕ^i used to estimate ϕi. It is a function of bias of ϕ^i and its variability.Expressions of these measures, together with their Monte Carlo SE, are given in Table3.Table 3
Expressions of performance measures and their Monte Carlo SE wherer is the number of replications.
Performance measureDefinitionEstimateMonte Carlo SEBiasEϕ^−ϕ∑i=1rϕ^ir−ϕ∑i=1rϕ∧i−ϕ¯2rr−1EmpSEVarϕ^∑i=1rϕ∧i−ϕ¯2r−1EmpSE^2r−1MSEEϕ∧−ϕ2∑i=1rϕ∧i−ϕ2r∑i=1rϕ∧i−ϕ2−MSE∧2rr−1Performance estimates for these measures are reported in Table4 for small and large number of sample paths and for different volatility parameters.Table 4
Performance estimates for measures of interest based on the104 replications and their Monte Carlo SE in parentheses.
σ=2.5MeasureQuantityn=20n=100Biasμ0.18660.01380.17580.0062λ1645.1843.924284.725917.2063EmpSEμ1.37670.00970.61960.0044λ4392.4231.06071720.6312.3167σ=1MeasureValuen=20n=100Biasμ0.02870.00540.02730.0024λ10808.6267.1051989.2293.6828EmpSEμ0.54370.00380.24240.0017λ26710.5188.8819368.2866.247As expected, bias, EmpSE, and their Monte Carlo SEs decrease as the sample size increases. Additionally, the estimation ofμ is better with smaller volatility. This is also expected since first passage times have more variability for large σ as can be seen from Figure 1. For λ however, the estimation is worse with smaller volatility. This is intuitive since a smaller σ leads to higher λ^MLE values for fixed s as explained by the inverse relation in Equation (6). Accordingly, the higher scale of λ^MLE values translates to higher values of performance measures. Estimates of MSE can be derived from those of bias and variability. Hence, they are not reported here.


## 2.1. Wiener Process Model for Nonmonotone Degradation

An assumption that is often valid in applications is that physical or performance degradation is a continuous process. Accordingly, sample paths of the stochastic process describing system degradation is ought to be restricted to continuous functions.Simulated sample paths of a Wiener process forν=3.5 and σ taking values 2.5, 2, 1.5, and 1 are presented in Figure 1. Process trajectories in Figure 1 are continuous functions. It is therefore not surprising that the Wiener process is the basic model for a degradation process. Describing system degradation by the Wiener process implies that physical or performance degradation can increase or decrease with time. While this might not be meaningful in many degradation applications, it is applicable to degradation processes whose levels vary bidirectionally over time when observed closely. Examples include
(1)
The gain of a transistor or the extent of propagation delay (Lu [12])(2)
Cracks healing and CD4 blood cell counts fluctuating (Singpurwalla [13])(3)
Resistance of the structure alternating with time in the framework of structural reliability (Dong and Cui [14])Figure 1
Simulated Wiener process sample paths.For monotone degradation processes such as wear-out, the application of the Wiener process is only as an approximation, which is especially good ifσ2≪ν. In this case, the trajectories are approximately monotone (see bottom-right panel of Figure 1) since the tooths in the evolving paths of the Wiener process are appreciably smoothed out. Alternatively, all factors contributing to nonmonotone behaviour in a Wiener process model may be attributed to pure noise and modelled accordingly.


## 2.2. Lifetime Estimation and Failure-Time Distribution

Assuming a Wiener process model, system lifetimeTs is the time Wt,t∈ℝ+ crosses the critical degradation level s for the first time. That is, Ts is the first passage time of the Wiener process to s. It is given by
(4)Ts=inft∈ℝ+:Wt≥s.It is well known (Chhikara and Folks [15]) that Ts is distributed as inverse Gaussian with probability density function (pdf)
(5)fTst=sσ2πt3exp−12σ2s−νt2t,t>0,ν>0.A useful reparameterisation of the density in Equation (5) in terms of the development of statistical properties analogous to those of the normal distribution (Tweedie [16]) is obtained by setting
(6)μ=sν;λ=s2σ2.This yields the reparameterised inverse Gaussian distributionTs~IGμ,λ with pdf and cumulative distribution function (cdf) given as
(7)fTst,μ,λ=λ2πt3exp−λ2μ2t−μ2t,t>0,(8)FTst=Φλttμ−1+exp2λμΦ−λttμ+1,respectively, where μ∈ℝ+ is the mean, and λ∈ℝ+ is the shape parameter. The inverse Gaussian distribution is right-skewed and bounded at zero. Figure 2 illustrates the probability density function for the inverse Gaussian with μ=3.5 and λ taking values 16, 45, and 100.Figure 2
Inverse Gaussian distribution as a function of shape parameter.The result thatTs is inverse Gaussian is demonstrated using Monte Carlo simulations. In particular, 103 sample paths were simulated and s is set to 250 to ensure all systems are tested to failure as illustrated in Figure 3.Figure 3
Simulated sample paths and threshold choice.System degradation is simulated at discrete times. Interpolation was done using splines method to ensure that the resultingTs values are continuous, and thus unique and more representative.Figure4 shows histograms of the resulting Ts values for different path parameter combinations. Additionally and based on the parameterisation in Equation (6), the theoretical inverse Gaussian pdf is also represented in green. It follows from Figure 4 that the histograms closely resemble the theoretical inverse Gaussian pdf. Hence, this illustrates that first passage times of a Wiener process with drift are indeed distributed as inverse Gaussian.Figure 4
Histograms of first passage times and theoretical distributions.


### 2.2.1. Maximum Likelihood Estimation of Path Model Parameters

Path model parameters are estimated based on registered degradation data or derived first passage times. The former applies to highly reliable systems where failure does not interrupt observation of the degradation process, that is,s⟶∞. Denote by Witj the ith system degradation measure at inspection time tj, i=1,⋯,n; j=1,⋯,mi. Then, the degradation increment for the ith system is ΔWij=Witj−Witj−1. From Equation (3),
(9)ΔWij=νiΔtj+σiΔBtj,where Δtj=tj−tj−1 and ΔBtj=Btj−Btj−1 with ΔWij~NνiΔtj,σi2Δtj.Consequently,Δwij are the degradation increments for system i with pdf fΔwij;vi,σi2. Since the Wiener process has normally distributed increments, the likelihood function for system i is
(10)liνi,σi2Δwij=∏j=1mifiΔwijνi,σi2=∏j=1mi12πσi2Δtjexp−12σi2Δwij−νiΔtj2Δtj.The corresponding log-likelihood for theith system is given by
(11)Li=lnliνi,σi2Δwij=−mi2ln2π−milnσi−12∑j=1milnΔtj−12σi2∑j=1miΔwij−νiΔtj2Δtj.Taking partial derivatives of the log-likelihood function in Equation (11) with respect to νi and σi gives
(12)∂Li∂νi=∑j=1miΔtjΔwij−νiΔtjσi2Δtj=0,(13)∂Li∂σi=−miσi+1σi3∑j=1miΔwij−νiΔtj2Δtj=0.The maximum likelihood estimators (MLE)νi^ and σi^ are obtained by simultaneously solving Equations (12) and (13). They are
(14)νi^=∑j=1miΔwijΔtj,(15)σi^=1mi∑j=1miΔwij−νiΔtj2Δtj.Wiener process degradation incrementsΔwij are independent. Hence, the log-likelihood function of their full set is
(16)Lν,σ2Δwij=∑i=1nLνi,σi2Δwij,and MLE for model parameters ν and σ are
(17)ν^MLE=∑i=1n∑j=1miΔwijΔtj,(18)σ^MLE=1min∑i=1n∑j=1miΔwij−νiΔtj2Δtj.For most applications however,s is finite as determined by industrial standards. System lifetimes are the Ts values for the n sampled systems. Denote by τ1,⋯,τn these first passage times. Their density is given in Equation (5) for the underlying Wiener degradation process. The likelihood function is thus
(19)lν,σ2τ1,⋯,τn=∏i=1nsσ2πτi3exp−12σ2s−νt2τi,with log-likelihood function
(20)L=nlns−nlnσ−n2ln2π−32∑i=1nlnτi−12σ2∑i=1ns−ντi2τi.Maximising the log-likelihood function in Equation (20) in respect of process parameters ν and σ yields
(21)∂L∂ν=1σ2∑i=1ns−ντi=0,(22)∂L∂σ=−nσ+1σ3∑i=1ns−ντi2τi=0.MLE forν and σ is obtained by simultaneously solving Equations (21) and (22). They are
(23)ν^MLE=s1/n∑i=1nτi;σ^MLE=1n∑i=1ns−ν^τiτi.Values ofν^MLE and σ^MLE are obtained from Equation (23) based on the first passage times from the 103 simulated degradation paths. The results, together with true path parameter values, are presented in Table 1.Table 1
True and parameter estimates forIGν,σ.
IGν,σTrue valuesML estimatesνσν^MLEσ^MLE3.52.53.55192.43603.52.03.54902.05483.51.53.52981.59493.51.03.50531.0361They show that parameter estimates of the first passage time distribution in Equation (5) recover path parameters used in the simulation. This is equally true for parameter estimates of the transformed IGμ,λ distribution in Equation (7), and the results of which are presented in Table 2.Table 2
True and parameter estimates for transformedIGμ,λ.
IGμ,λTrue valuesML estimatesμλμ^MLEλ^MLEσ=2.571.42861000071.602910821σ=2.071.42861562571.442215223σ=1.571.42862777871.686125551σ=1.071.42866250071.463758291Thus, Monte Carlo simulations prove the correctness of the well-known result that first passage times of a Wiener process with drift obey inverse Gaussian law.


### 2.2.2. Interval Estimation of Model Parameters

Maximum likelihood estimators are point estimates obtained from sample data. Accordingly, values ofν^ and σ^ are subject to sampling fluctuations and may or may not be close to the quantities being estimated. It is therefore important to quantify uncertainty associated with parameter estimates. Confidence intervals are very useful in quantifying uncertainty in point estimates due to sampling error arising from limited sample sizes. Exact confidence intervals can be constructed, if for example, the sampling distribution of θ^=ν^,σ^ is known. Otherwise, and for large samples, approximate confidence intervals are used. MLE is asymptotically normal. Hence, confidence intervals for θi, i=1,2 are constructed by the asymptotic normal approximation. Also called Wald confidence intervals, normal approximation confidence intervals are based on the Wald statistic:
(24)Zθ^i=θ^i−θise^θ^i~N0,1.The standard error ofθ^i is determined by the second derivative of the log-likelihood function with respect to θi which quantifies the curvature of the log-likelihood function. That is
(25)se^θ^i=−L′′θ∧i;τ−1,and is evaluated at θi=θ^i where τ is a vector of first passage times. The quantity −L′′θ^i;τ is the observed information. When constructing normal approximation confidence intervals however, −L′′θ^i;τ is often replaced by the expected or Fisher information
(26)Iθi=−EL′′θi;τ.The resulting1001−α% confidence interval for θi is given by
(27)θ^i±z1−α/2·se^θ^i.Alternatively, the statistic(28)Zlogθ^i=logθ^i−logθise^logθ^i~N0,1,is used instead. Observe that logθ^i is unrestricted in sign. Hence, Zlogθ^i is in general closer to a N0,1 distribution than is Zθ^i. Thus, and after transforming by exponentiation, the confidence interval for θi is
(29)θ^iexpz1−α/2×se^θ^i/θ^i,θ^i×expz1−α/2×se^θ^i/θ^i.Often in reliability, only a few system failures are observed. In this case, large sample normal theory is inexact. Rather, likelihood ratio confidence bounds method is often preferred. It is based on likelihood ratio equation:(30)−2·lnlθlθ^≥χα;k2,where lθ is the likelihood function for the unknown parameter θ, lθ^ is the likelihood function evaluated at θ^, and χα;k2 is the chi-squared statistic with probability α and k degrees of freedom, where k is the number of jointly estimated parameters. A rearrangement of Equations (30) yields
(31)lθ=lθ^·e−χα;k2/2,where terms on the right hand side are known exactly. The confidence limits for θi are the minimum and maximum calculated θi values for which Equation (31) holds.Contour plots are a useful way of simultaneously estimating likelihood ratio confidence bounds on the parameters. Equation (31) has no closed form solution, hence, a numerical solution is required instead. A crude approach is holding one parameter constant while iterating on the other until an acceptable solution is reached. Figure 5 gives contour plots for first passage times in Figure 4.Figure 5
Confidence bounds ofμ and λ for different path parameter combinations.
(a)
Path parameters:ν=3.5; σ=2.5(b)
Path parameters:ν=3.5; σ=2.0(c)
Path parameters:ν=3.5; σ=1.5(d)
Path parameters:ν=3.5; σ=1.0Confidence bounds forμ get narrower with decreasing volatility. This is expected because low variability results in smoother sample paths (Figure 3), hence, less variability in first passage times (Figure 4). The bounds for λ however get wider with decreasing volatility. This is a result of the scaling in Equation (6) and can be seen in the peakedness of the theoretical and empirical densities in Figures 2 and 4, respectively.


## 2.2.1. Maximum Likelihood Estimation of Path Model Parameters

Path model parameters are estimated based on registered degradation data or derived first passage times. The former applies to highly reliable systems where failure does not interrupt observation of the degradation process, that is,s⟶∞. Denote by Witj the ith system degradation measure at inspection time tj, i=1,⋯,n; j=1,⋯,mi. Then, the degradation increment for the ith system is ΔWij=Witj−Witj−1. From Equation (3),
(9)ΔWij=νiΔtj+σiΔBtj,where Δtj=tj−tj−1 and ΔBtj=Btj−Btj−1 with ΔWij~NνiΔtj,σi2Δtj.Consequently,Δwij are the degradation increments for system i with pdf fΔwij;vi,σi2. Since the Wiener process has normally distributed increments, the likelihood function for system i is
(10)liνi,σi2Δwij=∏j=1mifiΔwijνi,σi2=∏j=1mi12πσi2Δtjexp−12σi2Δwij−νiΔtj2Δtj.The corresponding log-likelihood for theith system is given by
(11)Li=lnliνi,σi2Δwij=−mi2ln2π−milnσi−12∑j=1milnΔtj−12σi2∑j=1miΔwij−νiΔtj2Δtj.Taking partial derivatives of the log-likelihood function in Equation (11) with respect to νi and σi gives
(12)∂Li∂νi=∑j=1miΔtjΔwij−νiΔtjσi2Δtj=0,(13)∂Li∂σi=−miσi+1σi3∑j=1miΔwij−νiΔtj2Δtj=0.The maximum likelihood estimators (MLE)νi^ and σi^ are obtained by simultaneously solving Equations (12) and (13). They are
(14)νi^=∑j=1miΔwijΔtj,(15)σi^=1mi∑j=1miΔwij−νiΔtj2Δtj.Wiener process degradation incrementsΔwij are independent. Hence, the log-likelihood function of their full set is
(16)Lν,σ2Δwij=∑i=1nLνi,σi2Δwij,and MLE for model parameters ν and σ are
(17)ν^MLE=∑i=1n∑j=1miΔwijΔtj,(18)σ^MLE=1min∑i=1n∑j=1miΔwij−νiΔtj2Δtj.For most applications however,s is finite as determined by industrial standards. System lifetimes are the Ts values for the n sampled systems. Denote by τ1,⋯,τn these first passage times. Their density is given in Equation (5) for the underlying Wiener degradation process. The likelihood function is thus
(19)lν,σ2τ1,⋯,τn=∏i=1nsσ2πτi3exp−12σ2s−νt2τi,with log-likelihood function
(20)L=nlns−nlnσ−n2ln2π−32∑i=1nlnτi−12σ2∑i=1ns−ντi2τi.Maximising the log-likelihood function in Equation (20) in respect of process parameters ν and σ yields
(21)∂L∂ν=1σ2∑i=1ns−ντi=0,(22)∂L∂σ=−nσ+1σ3∑i=1ns−ντi2τi=0.MLE forν and σ is obtained by simultaneously solving Equations (21) and (22). They are
(23)ν^MLE=s1/n∑i=1nτi;σ^MLE=1n∑i=1ns−ν^τiτi.Values ofν^MLE and σ^MLE are obtained from Equation (23) based on the first passage times from the 103 simulated degradation paths. The results, together with true path parameter values, are presented in Table 1.Table 1
True and parameter estimates forIGν,σ.
IGν,σTrue valuesML estimatesνσν^MLEσ^MLE3.52.53.55192.43603.52.03.54902.05483.51.53.52981.59493.51.03.50531.0361They show that parameter estimates of the first passage time distribution in Equation (5) recover path parameters used in the simulation. This is equally true for parameter estimates of the transformed IGμ,λ distribution in Equation (7), and the results of which are presented in Table 2.Table 2
True and parameter estimates for transformedIGμ,λ.
IGμ,λTrue valuesML estimatesμλμ^MLEλ^MLEσ=2.571.42861000071.602910821σ=2.071.42861562571.442215223σ=1.571.42862777871.686125551σ=1.071.42866250071.463758291Thus, Monte Carlo simulations prove the correctness of the well-known result that first passage times of a Wiener process with drift obey inverse Gaussian law.


## 2.2.2. Interval Estimation of Model Parameters

Maximum likelihood estimators are point estimates obtained from sample data. Accordingly, values ofν^ and σ^ are subject to sampling fluctuations and may or may not be close to the quantities being estimated. It is therefore important to quantify uncertainty associated with parameter estimates. Confidence intervals are very useful in quantifying uncertainty in point estimates due to sampling error arising from limited sample sizes. Exact confidence intervals can be constructed, if for example, the sampling distribution of θ^=ν^,σ^ is known. Otherwise, and for large samples, approximate confidence intervals are used. MLE is asymptotically normal. Hence, confidence intervals for θi, i=1,2 are constructed by the asymptotic normal approximation. Also called Wald confidence intervals, normal approximation confidence intervals are based on the Wald statistic:
(24)Zθ^i=θ^i−θise^θ^i~N0,1.The standard error ofθ^i is determined by the second derivative of the log-likelihood function with respect to θi which quantifies the curvature of the log-likelihood function. That is
(25)se^θ^i=−L′′θ∧i;τ−1,and is evaluated at θi=θ^i where τ is a vector of first passage times. The quantity −L′′θ^i;τ is the observed information. When constructing normal approximation confidence intervals however, −L′′θ^i;τ is often replaced by the expected or Fisher information
(26)Iθi=−EL′′θi;τ.The resulting1001−α% confidence interval for θi is given by
(27)θ^i±z1−α/2·se^θ^i.Alternatively, the statistic(28)Zlogθ^i=logθ^i−logθise^logθ^i~N0,1,is used instead. Observe that logθ^i is unrestricted in sign. Hence, Zlogθ^i is in general closer to a N0,1 distribution than is Zθ^i. Thus, and after transforming by exponentiation, the confidence interval for θi is
(29)θ^iexpz1−α/2×se^θ^i/θ^i,θ^i×expz1−α/2×se^θ^i/θ^i.Often in reliability, only a few system failures are observed. In this case, large sample normal theory is inexact. Rather, likelihood ratio confidence bounds method is often preferred. It is based on likelihood ratio equation:(30)−2·lnlθlθ^≥χα;k2,where lθ is the likelihood function for the unknown parameter θ, lθ^ is the likelihood function evaluated at θ^, and χα;k2 is the chi-squared statistic with probability α and k degrees of freedom, where k is the number of jointly estimated parameters. A rearrangement of Equations (30) yields
(31)lθ=lθ^·e−χα;k2/2,where terms on the right hand side are known exactly. The confidence limits for θi are the minimum and maximum calculated θi values for which Equation (31) holds.Contour plots are a useful way of simultaneously estimating likelihood ratio confidence bounds on the parameters. Equation (31) has no closed form solution, hence, a numerical solution is required instead. A crude approach is holding one parameter constant while iterating on the other until an acceptable solution is reached. Figure 5 gives contour plots for first passage times in Figure 4.Figure 5
Confidence bounds ofμ and λ for different path parameter combinations.
(a)
Path parameters:ν=3.5; σ=2.5(b)
Path parameters:ν=3.5; σ=2.0(c)
Path parameters:ν=3.5; σ=1.5(d)
Path parameters:ν=3.5; σ=1.0Confidence bounds forμ get narrower with decreasing volatility. This is expected because low variability results in smoother sample paths (Figure 3), hence, less variability in first passage times (Figure 4). The bounds for λ however get wider with decreasing volatility. This is a result of the scaling in Equation (6) and can be seen in the peakedness of the theoretical and empirical densities in Figures 2 and 4, respectively.


## 2.3. Sampling Distributions of First Passage Time Distribution Parameters

First passage times in Figure4 are based on 103 simulated sample paths. In practice however, a few systems are tested for economic reasons. Hence, trajectories for n=20 (small sample) and n=100 (large sample) are simulated. Values of μ^MLE and λ^MLE are obtained from the resulting first passage times. This procedure is repeated 104 times for each n, yielding respective sampling distributions in Figures 6 and 7.Figure 6
Sampling distributions ofμ^MLE and λ^MLE for n=20.Figure 7
Sampling distributions ofμ^MLE and λ^MLE for n=100.Sampling distributions forμ^MLE are fairly symmetrical regardless of sample size. Those of λ^MLE are right-skewed for small samples and close to normal for large samples. Therefore, approximate normal confidence intervals for λ may be appropriate for large samples whereas for μ, any sample size may apply.The performance ofμ^MLE and λ^MLE is also assessed for different sample sizes and volatility parameters. In particular, bias and empirical standard error (EmpSE) were reported together with their Monte Carlo standard errors (Monte Carlo SE). Bias is the amount by which ϕ^=μ^MLE,λ^MLE exceeds ϕ=μ,λ on average. Unbiasedness is a key property in frequentist theory. However, small biases maybe traded-off for other good properties. The EmpSE estimates the long-run standard deviation of ϕ^i for 104 replications. It is a measure of the precision (efficiency) of the estimator. Monte Carlo SE provides an estimate of the SE of the estimated performance measure as a result of using a finite number of replications (Morris et al. [17]). Another important measure is mean squared error (MSE), measuring the accuracy of ϕ^i used to estimate ϕi. It is a function of bias of ϕ^i and its variability.Expressions of these measures, together with their Monte Carlo SE, are given in Table3.Table 3
Expressions of performance measures and their Monte Carlo SE wherer is the number of replications.
Performance measureDefinitionEstimateMonte Carlo SEBiasEϕ^−ϕ∑i=1rϕ^ir−ϕ∑i=1rϕ∧i−ϕ¯2rr−1EmpSEVarϕ^∑i=1rϕ∧i−ϕ¯2r−1EmpSE^2r−1MSEEϕ∧−ϕ2∑i=1rϕ∧i−ϕ2r∑i=1rϕ∧i−ϕ2−MSE∧2rr−1Performance estimates for these measures are reported in Table4 for small and large number of sample paths and for different volatility parameters.Table 4
Performance estimates for measures of interest based on the104 replications and their Monte Carlo SE in parentheses.
σ=2.5MeasureQuantityn=20n=100Biasμ0.18660.01380.17580.0062λ1645.1843.924284.725917.2063EmpSEμ1.37670.00970.61960.0044λ4392.4231.06071720.6312.3167σ=1MeasureValuen=20n=100Biasμ0.02870.00540.02730.0024λ10808.6267.1051989.2293.6828EmpSEμ0.54370.00380.24240.0017λ26710.5188.8819368.2866.247As expected, bias, EmpSE, and their Monte Carlo SEs decrease as the sample size increases. Additionally, the estimation ofμ is better with smaller volatility. This is also expected since first passage times have more variability for large σ as can be seen from Figure 1. For λ however, the estimation is worse with smaller volatility. This is intuitive since a smaller σ leads to higher λ^MLE values for fixed s as explained by the inverse relation in Equation (6). Accordingly, the higher scale of λ^MLE values translates to higher values of performance measures. Estimates of MSE can be derived from those of bias and variability. Hence, they are not reported here.


## 3. Wiener Maximum Process Model for Monotone Degradation

System degradation often proceeds in one direction only, and hence monotone. This irreversible accumulation of damage can be explained by the Wiener maximum process(32)W+t=sup0≤u≤tWu,u≥0,which, by definition, is nondecreasing in its argument. It has initial condition W+0=0 since W0=0. Recall that Ts is the first time Wt,t∈ℝ+ passes the failure threshold s. Since Wt,t∈ℝ+ has continuous sample paths, the occurrence of the event Wt>s at time t suggests that the event Ts≤t has already been realised. That is,
(33)Wt>s=Ts≤tandWt>s.For the Wiener maximum process however,sup0≤u≤tWu≥s occurs if Wt,t∈ℝ+ crosses s at least once in the closed interval 0,t given that W0=0. That is,
(34)sup0≤u≤tWu≥sW0=0,if and only if Ts≤t. It follows therefore that
(35)sup0≤u≤tWu≥sW0=0=Ts≤t.Hence,sup0≤u≤tWu,u≥0 crosses s at exactly the same time that the process Wt,t∈ℝ+ crosses the same threshold. That is, the first passage time of the Wiener maximum process to s given by
(36)Ts=minu∈ℝ+:sup0≤u≤tWu=s,coincides with that of Wt,t∈ℝ+ to the same failure threshold as shown in Figure 8.Figure 8
Simulated sample paths for the Wiener processes with drift (red) and Wiener maximum processes (green)Figure9 shows the distribution of first passage times of both the Wiener and the Wiener maximum process based on 103 simulated sample paths. The Wiener maximum process is a more realistic model when it is important that the degradation process is monotone, as is often the case in practice. Figure 9 confirms the result that Wiener and Wiener maximum process first passage times have the same distribution. This is particularly the case when the volatility parameter σ is much smaller compared to the drift parameter ν. Hence, system failure times assuming a Wiener maximum process are also distributed as inverse Gaussian with density specified in Equation (5). Thus, the popularity of the Wiener process stems from the fact that it is applicable to both nonmonotone and monotone degradation.Figure 9
Histograms of Wiener and Wiener maximum process first passage times and theoretical distributions.


## 4. Real Data Application

Real data from a degradation test of a gallium arsenide (GaAs) laser for telecommunications systems is considered in this section. The laser uses a built-in feedback circuit to maintain a constant light output. As it ages, the laser requires more current to maintain the constant light output. The first time a 10% increase in current is needed to achieve the constant light output, the laser is considered to have failed. That is,s=10. The data is from an accelerated degradation test involving 15 randomly sampled lasers. The lasers were tested for 4000hours at an elevated temperature of 80∘C. The elevated temperature was estimated by engineers to accelerate failure by a factor of approximately 40. Table C.17 in Meeker and Escobar [18] contains more information about the test. The data are plotted in Figure 10.Figure 10
GaAs laser sample degradation paths.By the end of the test at4000hours, three lasers had failed; at 3374hours, 3521hours, and 3781hours. The laser has a desired lifetime of at least 200000hours at the use-level temperature of 20∘C. This amounts to a corresponding lifetime of 5000hours at the elevated temperature of 80∘C. Consequently, the estimation target is the laser’s unreliability at 5000hours, F5000. GaAs laser sample degradation paths in Figure 10 appear to be monotone. Hence, the Wiener maximum process sup0≤u≤tWu,u≥0 is a reasonable model. First passage times of sup0≤u≤tWu,u≥0 to s however coincides with those of the Wiener process with drift Wt,t∈ℝ+ to the same failure threshold as demonstrated in Figure 8. Hence, the simpler model is assumed, and path model parameter estimates θ^=ν^MLE,σ^MLE are obtained from Equation (23). The uncertainty associated with θ^i is quantified using bootstrap and jackknife methods (Tibshirani and Efron [19]). The former entails randomly drawing 103 samples of size 15 with replacement from the 15 lasers and estimating θ^. The bootstrap normal approximate 95% confidence interval for θ^i is
(37)Meanθ^i±1.96×Varθ^i.The jackknife sequentially leaves theith laser out for i=1,⋯,15 and estimates θ^ from degradation data on 14 lasers. The approximate 1−α% jackknife confidence interval is given by
(38)θ^i±t1−α;n−1×se^jack,where t1−α;n−1 is the 1−αth percentile to the t distribution having n−1 degrees of freedom, and se^jack is the jackknife standard error estimate given by
(39)se^jack=n−1n∑i=1nθ∧i−Meanθ∧i2.Table5 reports ν^MLE and σ^MLE from the degradation analysis of the 4000-hour laser data and their approximate 95% confidence intervals. An important advantage of assessing system reliability from degradation data is that conclusions are reached earlier without compromising estimation accuracy. Therefore, laser data available after only 3000hours, 2000hours, and 1000hours are also analysed. Results in Table 5 show that analysis of laser data available after the different test times yielded comparable path parameter estimates. These shorter tests allow for highly reliable systems to be released early and corrective action on the unreliable ones to be done sooner. Figure 11 shows first passage time densities, i.e., IGμ,λ derived from path parameter estimates in Table 5 using Equation (6). There does not appear to be major differences between IG densities, particularly for the 4000-hour, 3000-hour, and 1000-hour laser degradation data. This is not a surprising result as has already been reported by Hove [20] though a general path model was assumed instead.Table 5
Wiener degradation path parameter estimates and 95% confidence intervals.
4000-hour laser degradation dataνσ0.002053870.0128735Bootstrap0.00181624,0.002299970.01060790,0.0147418Jackknife0.00177658,0.002331230.01039790,0.015318603000-hour laser degradation dataνσ0.002072000.01292070Bootstrap0.00182906,0.002324000.01068170,0.01478000Jackknife0.00178687,0.002357130.01047070,0.015339102000-hour laser degradation dataνσ0.002123430.01317980Bootstrap0.00186774,0.002395520.01057440,0.01538360Jackknife0.00181998,0.002426880.01031880,0.016004201000-hour laser degradation dataνσ0.002088890.0142229Bootstrap0.00177494,0.002408770.0103320,0.01733370Jackknife0.00171651,0.002461270.00992490,0.01846310Figure 11
Inverse Gaussian probability densities of the time to failure for laser data available after the different test times.Estimates of the desired probability of failing by5000hours, F^5000 from the analysis of laser data available after the different test times are presented in Table 6.Table 6
Estimation of the probability of failing by5000hours and lower distribution percentiles.
F^5000t^0.1t^0.05t^0.014000-hour laser degradation data0.63344323418439373000-hour laser degradation data0.66984285414839032000-hour laser degradation data0.76054178404338031000-hour laser degradation data0.6885420140543792Lower percentiles, often useful when determining warranty period for example, are also reported. These results show that3000-hour and 1000-hour analyses yielded more comparable estimates to the 4000-hour analysis than the 2000-hour analysis. Path parameter estimates in Table 5 and IGμ,λ densities in Figure 11 reflect this finding. This is surprising since the 1000-hour analysis utilises less registered degradation data than the 2000-hour analysis.Further analyses (not reported here) of laser data available after1250hours, 1500hours, …, 3750hours revealed changes in F^5000 values for the different test times. It follows from the results presented in Figure 12 that shorter tests from 2750-hour data (in red) yield F^5000 values that are comparable to the 4000-hour data.Figure 12
Failure probability estimates for different test times.


## 5. Concluding Remarks

When assessing reliability for highly reliable systems, degradation tests are an attractive alternative to life tests that record only failure times. This is especially so when few or no failures are observed in life tests of practical length, and a close relationship exists between system failure and the level of degradation. In this paper, the use of Wiener process for reliability assessment is reviewed. Monte Carlo simulations are used to demonstrate known results and to quantify performance measures. In particular, the well-known result that first passage times of a Wiener process with drift to a fixed barrier are distributed as inverse Gaussian is demonstrated. Additionally, the performance of MLEs of inverse Gaussian parameters was also assessed. The findings are as follows:(1)
Performance ofμ^MLE suffers some small upwards bias. The small bias suggests that if the number of replications in the simulation study is increased unboundedly, then, the long run average of all μ^MLE will not be far from their true values. Bias values decrease with both increase in sample size and decrease in volatility. For λ^MLE however, bias values appear to be large but this is a result of the scale. They decrease with sample size but increase with decreasing volatility(2)
Variability ofμ^MLE and λ^MLE is significantly lower for large sample sizes, as expected. The seemingly large variability for λ^MLE is again a matter of scale, explaining why it increases with decreasing volatilityFirst passage times of the Wiener maximum process to a fixed threshold are shown to coincide with those of the Wiener process with drift. This is in line with the presented theoretical result and is important for explaining strictly monotone degradation processes. In the main, real data application demonstrated a considerable reduction in test duration without compromising estimation quality.



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