Case | a | b | c | d | L |
I | -0.25 | 1.20 | -0.80 | 1.60 | 5.15 |
II | -0.50 | 2.00 | -1.00 | 2.50 | 6.30 |
III | -0.75 | 3.00 | -1.20 | 3.20 | 7.80 |
IV | 0.30 | 1.00 | 0.20 | 0.60 | 8.20 |
Degradation data are an important source of products' reliability information. Though stochastic degradation models have been widely used for fitting degradation data, there is a lack of efficient and accurate methods to get their confidence intervals, especially in small sample cases. In this paper, based on the Wiener process, a doubly accelerated degradation test model is proposed, in which both the drift and diffusion parameters are affected by the stress level. The point estimates of model parameters are derived by constructing a regression model. Furthermore, based on the point estimates of model parameters, the interval estimation procedures are developed for the proposed model by constructing generalized pivotal quantities. First, the generalized confidence intervals of model parameters are developed. Second, based on the generalized pivotal quantities of model parameters, using the substitution method the generalized confidence intervals for some interesting quantities, such as the degradation rate μ0, the diffusion parameter σ20, the reliability function R(t0) and the mean lifetime E(T), are obtained. In addition, the generalized prediction intervals for degradation amount X0(t) and remaining useful life at the normal use stress level are also developed. Extensive simulations are conducted to investigate the performances of the proposed generalized confidence intervals in terms of coverage percentage and average interval length. Finally, a real data set is given to illustrate the proposed model.
Citation: Peihua Jiang, Xilong Yang. Reliability inference and remaining useful life prediction for the doubly accelerated degradation model based on Wiener process[J]. AIMS Mathematics, 2023, 8(3): 7560-7583. doi: 10.3934/math.2023379
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Degradation data are an important source of products' reliability information. Though stochastic degradation models have been widely used for fitting degradation data, there is a lack of efficient and accurate methods to get their confidence intervals, especially in small sample cases. In this paper, based on the Wiener process, a doubly accelerated degradation test model is proposed, in which both the drift and diffusion parameters are affected by the stress level. The point estimates of model parameters are derived by constructing a regression model. Furthermore, based on the point estimates of model parameters, the interval estimation procedures are developed for the proposed model by constructing generalized pivotal quantities. First, the generalized confidence intervals of model parameters are developed. Second, based on the generalized pivotal quantities of model parameters, using the substitution method the generalized confidence intervals for some interesting quantities, such as the degradation rate μ0, the diffusion parameter σ20, the reliability function R(t0) and the mean lifetime E(T), are obtained. In addition, the generalized prediction intervals for degradation amount X0(t) and remaining useful life at the normal use stress level are also developed. Extensive simulations are conducted to investigate the performances of the proposed generalized confidence intervals in terms of coverage percentage and average interval length. Finally, a real data set is given to illustrate the proposed model.
Accelerated life test (ALT) technology has been widely used in products' reliability analysis. However, even under ALT, little or no failure data can be acquired in a reasonable short life testing time for some high-reliability products. So, it is difficult to evaluate the reliability of those high-reliability products. In such a case, if there exists a quality characteristic (QC) related to reliability which degrades over time, an alternative approach is to collect the degradation data at the higher stress levels and then extrapolate the lifetime information and reliability metrics of products at the normal use stress level. Such an experiment is called an accelerated degradation test (ADT) [1,2]. For high-reliability products, the traditional ALT methods can not meet the requirement of product reliability evaluation, and the ADT technique provides another way to solve it [3].
As ADTs can greatly shorten the testing time, we often use them to quickly obtain more degradation data and make reliability analyses for products. In the past decades, the research of ADT models has become more and more popular [4,5,6]. In the literature, ADTs are broadly classified into constant-stress accelerated degradation test (CSADT), step-stress accelerated degradation test (SSADT) and progressive-step accelerated degradation test (PSADT), according to different stress loading methods. Among them, CSADT is the most popular ADT in real applications. In a CSADT, the testing units are divided into several groups, and each group of units is exposed to a certain severe stress condition to test and collect the degradation data.
In practice, the degradation path of products' QC over time is often described as stochastic {X(t);t≥0} to account for inherent randomness. Based on the assumption of additive accumulation of degradation, three kinds of degradation processes have been well exploited, i.e., the Gamma process [7,8,9,10], the Wiener process [6,11,12,13] and the inverse Gaussian process [3,14,15,16]. In most cases, the degradation paths of test units are monotonic, so the Gamma process and inverse Gaussian process are often used to model degradation data. However, in a few cases, the degradation path is not monotonic. A distinct feature of the Wiener process is that its sample path is not necessarily monotone, which might be meaningful in some degradation applications. In this situation, the Wiener degradation model may be a good choice.
In view of this, the Wiener process, as an important stochastic process, has been widely studied in degradation data analysis. Pan and Balakrishnan [17] discussed the multiple-step SSADT models based on the Wiener and Gamma processes, and they used the Bayesian Markov Chain Monte Carlo method to obtain the maximum likelihood estimates (MLEs) for such analytically intractable models and presented some computational results obtained from their implementation. Motivated by the observation that a unit with a higher degradation rate has a more volatile degradation path, Ye and Chen [18] proposed a new class of random effects models for the Wiener process and discussed the statistical inference of the model. Wang et al. [13] mainly studied the accurate reliability inference for the Wiener degradation model with random drift parameter and developed an exact test method to test whether there exists population heterogeneity. Other research about the Wiener degradation model with random effects can be found in [11,19,20]. Guan et al. [21] used the Bayesian method to study the Wiener ADT model. Wang et al. [22] discussed the reliability analysis for accelerated degradation data based on the Wiener process with random effects. By constructing pivotal quantities (PQs), they developed generalized confidence intervals (GCIs) for model parameters and some quantities, and the generalized prediction interval (GPI) for future degradation amount at designed stress level was also developed. Hong et al. [23] developed interval estimation procedures for the Wiener degradation process with fixed-effects and mixed-effects using the generalized pivotal quantity (GPQ) method. Pan et al. [24] studied a reliability estimation approach via Wiener degradation model with measurement errors. Jiang et al. [12] proposed a Wiener CSADT model and obtained the exact confidence intervals (ECIs) of model parameters. In addition, they proposed a new optimization criterion from the perspective of degradation prediction, and provided an optimal experimental design scheme.
To construct an ADT model, first is to reasonably determine the relations between model parameters and accelerated stress. For the Wiener ADT model, there are three possible parameter-stress relations. The first is that the drift parameter of the Wiener process depends on the stress level while the diffusion parameter is independent of the stress level [12,20,21,22,25]. The second is that the diffusion parameter depends on the stress level while the drift parameter is independent of it. In the second case, however, the stress level does not have any effect on the degradation speed while the degradation volatility increases with the stress level. This case is not common in reality. The third is that both the drift and diffusion parameters are affected by the stress level [26]. Clearly, the first two relations are special cases of the third one. In addition, due to the complexity of the Wiener ADT model with the third parameter-stress relation, few scholars studied such models, especially for the aspect of interval estimation for reliability metrics. In this paper, we consider a Wiener doubly ADT model in which the parameter-stress relation follows the third one. The relations between parameters and accelerated stress are given by μ=a+bξ,σ2=exp(c+dξ), and ξ is the accelerated stress. We mainly focus on the interval estimations of model parameters and some reliability metrics based on the Wiener doubly ADT model.
The remainder of the paper is organized as follows. In Section 2, the general framework of the Wiener doubly ADT model is outlined. In Section 3, the point estimates of model parameters are derived. In Section 4, we mainly study the interval estimation of model parameters and some reliability metrics. In Section 5, a simulation study is conducted to evaluate the performance of the proposed GCIs/GPIs. In Section 6, an example is provided to illustrate the proposed model and GPQ method. Finally, we provide some final conclusions in Section 7.
Suppose that one accelerated stress ξ is used in a CSADT, and under the stress ξ the degradation path {X(t),t≥0} of testing unit follows a Wiener process given by
X(t)=μt+σB(t) | (2.1) |
where B(t) is a standard Brownian motion, μ is the drift parameter, and σ2 is the diffusion parameter. The unit's lifetime T under the stress ξ is defined as the first-passage-time of X(t) to a pre-specified threshold L. It is well known that T follows IG(L/μ,L2/σ2) distribution with cumulative distribution function (CDF)
FT(t,μ,σ2)=Φ(μt−Lσ√t)+exp(2μLσ2)Φ(−μt+Lσ√t),t>0, | (2.2) |
where Φ(⋅) is the CDF of a standard normal distribution.
For this study, statistical inference of the Wiener doubly accelerated degradation test model is usually based on the following assumptions:
(A1) The CSADT is conducted using a single stress, which has K levels: ξ1<ξ2<⋯<ξK. ξ0 and ξK are the normal use stress level and the highest stress level used in the ADT.
(A2) For each stress level ξi, the degradation path of a test unit can be described as a Wiener process Xi(t),
Xi(t)=μit+σiB(t). |
(A3) Both the drift and the diffusion parameters are affected by the stress levels through the parameter-stress relationships
μi=a+bξi,σ2i=exp(c+dξi), |
where, a,b,c and d are unknown parameters. The degradation rate and the diffusion parameter under normal use condition can be obtained by μ0=a+bξ0,σ20=exp(c+dξ0), respectively.
Suppose that ni units are tested under the stress level ξi, and ri,j is the number of measurements for the jth test unit under the stress level ξi. The degradation characteristics of the jth test unit are measured at the times ti,j={ti,j,k;k=0,1,…,ri,j} under the stress level ξi. Moreover, let Xi,j={Xi,j(ti,j,0),Xi,j(ti,j,1),…,Xi,j(ti,j,ri,j)} denote the observed degradation characteristics of the jth test unit under the stress level ξi. Define ΔXi,j,k=Xi,j(ti,j,k)−Xi,j(ti,j,k−1), and Δti,j,k=ti,j,k−ti,j,k−1, where ti,j,0=0. The data collected from stress level ξi is Di={(ti,j,Xi,j);j=1,2,…,ni}, and the data from the whole double ADT is D=⋃Ki=1Di. Let N=∑Ki=1ni be the total number of test units and Mi=Σnij=1ri,j be the total number of measurements under stress level ξi. Further, define M=∑Ki=1Mi as the total number of measurements in the whole ADT and T=∑Ki=1∑nij=1ti,j,ri,j as the total test time.
In this study, we mainly focus on the interval estimation of the proposed model. In order to develop the interval estimation procedures, we need to obtain the point estimations of model parameters first. That is because the point estimates are the basis for constructing PQs of interval estimations. Suppose that ni units are tested under the stress level ξi, and the degradation data Di is provided. Based on Di, the log-likelihood function is given by
l(μi,σ2i|Di)=−12ni∑j=1ri,j∑k=1[ln(2πσ2iΔi,j,k)+(ΔXi,j,k−μiΔti,j,k)2σ2iΔti,j,k]. |
Let Xi=Σnij=1Σri,jk=1ΔXi,j,k, Ti=Σnij=1Σri,jk=1Δti,j,k, and then the MLEs of μi and σ2i are given as
ˆμi=XiTi;S2i=1Mi−1ni∑j=1ri,j∑k=1(ΔXi,j,k−ˆμiΔti,j,k)2Δti,j,k. | (3.1) |
Notice that
ΔXi,j,k−μiΔti,j,k=(ΔXi,j,k−ˆμiΔti,j,k)+(ˆμiΔti,j,k−μiΔti,j,k), |
and we have the following factorization:
ni∑j=1ri,j∑k=1(ΔXi,j,k−μiΔti,j,k)2σ2iΔti,j,k=(Mi−1)S2iσ2i+(ˆμi−μi)2Tiσ2i. |
According to Cochran's theorem in [27], we can easily prove the following facts:
(ⅰ) ˆμi∼N(a+bξi,σ2i/Ti);
(ⅱ) (Mi−1)S2i/σ2i∼χ2(Mi−1);
(ⅲ) ˆμi and S2i are mutually independent.
Let Yi=Xi/Ti,i=1,2,…,K, and then Yi∼N(μi,σ2i/Ti). So, the mean and variance of Yi are given by E(Yi)=a+bξi and Var(Yi)=σ2i/Ti, respectively. In order to get the estimates of parameters a and b, the following linear regression model is considered.
Yi=a+bξi+εi,εi∼N(0,σ2i/Ti). | (3.2) |
Based on the linear regression model (3.2), the estimates of parameters a and b are provided in the following Theorem 3.1.
Theorem 3.1. Under the linear regression model (3.2), given the degradation data D,
(1) The estimates of parameters a and b are given as
ˆa=GH−IMFG−I2,ˆb=FM−IHFG−I2, | (3.3) |
where
F=K∑i=1Tiσ2i,I=K∑i=1Tiξiσ2i; |
G=K∑i=1Tiξ2iσ2i,H=K∑i=1Xiσ2i,M=K∑i=1ξiXiσ2i. |
(2) The estimates ˆa and ˆb follow the normal distributions, that is,
ˆa∼N(a,GFG−I2),ˆb∼N(b,FFG−I2). |
(3) The covariance of the estimates ˆa and ˆb is given by
Cov(ˆa,ˆb)=−IFG−I2. |
In addition, the degradation rate μ0 at normal use stress level ξ0 can be estimated by ˆμ0=ˆa+ˆbξ0, and the estimate ˆμ0 is also unbiased and has the variance Var(ˆμ0)=(G−2Iξ0+Fξ20)/(FG−I2).
Notice that (Mi−1)S2i/σ2i∼χ2(Mi−1), and let Ωiˆ=log[(Mi−1)S2i/σ2i]. By calculating, the moment generating function (MGF) of Ωi is derived.
MΩi(t)=2tΓ(Mi−12+t)Γ(Mi−12). |
Using the MGF MΩi(t), the mean and variance of Ωi are given by
E(Ωi)=ψ(Mi−12)+log2,Var(Ωi)=ψ′(Mi−12), |
respectively, where
ψ(x)=dlog(Γ(x))/dx,ψ′(x)=d2log(Γ(x))/dx2. |
Let Uiˆ=log[(Mi−1)S2i/2]−ψ(Mi−12), and by calculation we find that
E(Ui)=log(σ2i)=c+dξi,Var(Ui)=ψ′(Mi−12). | (3.4) |
To get the estimates of parameters c and d, the following linear regression model is constructed.
Ui=c+dξi+δi,E(δi)=0,Var(δi)=ψ′(Mi−12). | (3.5) |
Some properties about the estimates of c and d are given in the following Theorem 3.2.
Theorem 3.2. Under the linear regression model (3.5), given the degradation data D.
(1) The estimates of parameters c and d are given as
˜c=G1H1−I1M1F1G1−I21,˜d=F1M1−I1H1F1G1−I21, | (3.6) |
where
F1=K∑i=1[ψ′(Mi−12)]−1,I1=K∑i=1ξi[ψ′(Mi−12)]−1; |
G1=K∑i=1ξ2i[ψ′(Mi−12)]−1,H1=K∑i=1Ui[ψ′(Mi−12)]−1,M1=K∑i=1ξiUi[ψ′(Mi−12)]−1. |
(2) The estimates ˜c and ˜d are unbiased, that is, E(˜c)=c,E(˜d)=d.
(3) The variance and covariance of the estimates ˜c and ˜d are given by
Var(˜c)=G1F1G1−I21,Var(˜d)=F1F1G1−I21,Cov(˜c,˜d)=−I1F1G1−I21. |
Based on estimates ˜c and ˜d, the diffusion parameter σ20 at the normal use stress level ξ0 can be estimated by ˜σ20=exp(˜c+˜dξ0). However, the estimate ˜σ20 is biased. The following Theorem 3.3 gives an unbiased estimate of σ20.
Theorem 3.3. Let ˜c and ˜d be the estimates of c and d defined in (3.6), and
Diˆ=[G1−(ξ0+ξi)I1+ξ0ξiF1]/[ψ′(Mi−12)(F1G1−I21)]. |
Then,
(1) If (Mi−1)/2+Di>0(i=1,2,…,K), an unbiased estimate of σ20 can be given by
˜σ20u=˜σ20exp(K∑i=1Diψ(Mi−12))K∏i=1Γ(Mi−12)Γ(Di+Mi−12). | (3.7) |
(2) If (Mi−1)/2+2Di>0(i=1,2,…,K), the variance of ˜σ20u is given by
Var(˜σ20u)=σ20K∏i=1[Γ(Mi−12)Γ(2Di+Mi−12)−Γ2(Di+Mi−12)Γ2(Di+Mi−12)]. | (3.8) |
(3) If (Mi−1)/2+2Di>0(i=1,2,…,K), the estimate ˜σ20u has a smaller mean square error than that of ˜σ20.
In this subsection, we try to derive the GCIs of parameters a and b. In order to get the GPQs of a,b, we first develop the GPQs of parameters σ2i(i=1,2,…,K). As (Mi−1)S2i/σ2i∼χ2(Mi−1), generating a copy Q0,i from the χ2(Mi−1) distribution, the GPQ of σ2i can be obtained by
Wi=(Mi−1)S2iQ0,i,i=1,2,…,K. | (4.1) |
It is worth emphasizing that Q0,i is treated as a known quantity in generalized inference [28].
Based on the model (3.2), substituting S2i for the unknown parameter σ2i, the following weighted sum of squares is considered.
U(a,b)=K∑i=1TiS2i(Yi−a−bξi)2 | (4.2) |
By minimizing (4.2), the estimates of a,b are given as
˜a=(∑Ki=1Tiξ2iS2i)(∑Ki=1XiS2i)−(∑Ki=1TiξiS2i)(∑Ki=1ξiXiS2i)(∑Ki=1TiS2i)(∑Ki=1Tiξ2iS2i)−(∑Ki=1TiξiS2i)2,˜b=(∑Ki=1TiS2i)(∑Ki=1ξiXiS2i)−(∑Ki=1TiξiS2i)(∑Ki=1XiS2i)(∑Ki=1TiS2i)(∑Ki=1Tiξ2iS2i)−(∑Ki=1TiξiS2i)2. |
Let V1=˜a−a,V2=˜b−b, and then V1,V2 can be presented as
V1=(∑Ki=1Tiξ2iS2i)(∑Ki=1Ziσi√TiS2i)−(∑Ki=1TiξiS2i)(∑Ki=1ξiZiσi√TiS2i)(∑Ki=1TiS2i)(∑Ki=1Tiξ2iS2i)−(∑Ki=1TiξiS2i)2, | (4.3) |
V2=(∑Ki=1TiS2i)(∑Ki=1ξiZiσi√TiS2i)−(∑Ki=1TiξiS2i)(∑Ki=1Ziσi√TiS2i)(∑Ki=1TiS2i)(∑Ki=1Tiξ2iS2i)−(∑Ki=1TiξiS2i)2, | (4.4) |
where
Ziˆ=[Xi−(a+bξi)Ti]/(σi√Ti)∼N(0,1). |
It is obvious that in (4.3) and (4.4) the distributions of V1 and V2 only depend on the unknown parameter σ2i. So, generating a series of copies Z∗i from the standard normal distribution N(0,1), replace the unknown parameter σi by its GPQ √Wi in (4.3) and (4.4) to get the quantities V∗1 and V∗2.
V∗1=(∑Ki=1Tiξ2iS2i)(∑Ki=1Z∗i√Wi√TiS2i)−(∑Ki=1TiξiS2i)(∑Ki=1ξiZ∗i√Wi√TiS2i)(∑Ki=1TiS2i)(∑Ki=1Tiξ2iS2i)−(∑Ki=1TiξiS2i)2, | (4.5) |
V∗2=(∑Ki=1TiS2i)(∑Ki=1ξiZ∗i√Wi√TiS2i)−(∑Ki=1TiξiS2i)(∑Ki=1Z∗i√Wi√TiS2i)(∑Ki=1TiS2i)(∑Ki=1Tiξ2iS2i)−(∑Ki=1TiξiS2i)2. | (4.6) |
According to the substitute method given in [28,29], the GPQs of a and b are obtained by
G1=˜a−V∗1,G2=˜b−V∗2. | (4.7) |
Based on G1 and G2, the confidence intervals of a and b can be constructed. Because the distributions of G1 and G2 are very complicated, a simulation procedure can be used. Let Gi,α be the α percentile of Gi, and then [G1,α/2,G1,1−α/2] and [G2,α/2,G2,1−α/2] are the 1−α level GCIs of a and b, respectively.
In this subsection, we will derive the GCIs of model parameters c and d. Based on (3.6) in Theorem 3.2, let V3=˜c−c, V4=˜d−d, and then V3 and V4 can be represented as
V3=(∑Ki=1ξ2i[ψ′(Mi−12)]−1)[∑Ki=1[ψ′(Mi−12)]−1(logQi2−ψ(Mi−12))]F1G1−I21−(∑Ki=1ξi[ψ′(Mi−12)]−1)[∑Ki=1[ψ′(Mi−12)]−1ξi(logQi2−ψ(Mi−12))]F1G1−I21. | (4.8) |
V4=(∑Ki=1[ψ′(Mi−12)]−1)[∑Ki=1[ψ′(Mi−12)]−1ξi(logQi2−ψ(Mi−12))]F1G1−I21−(∑Ki=1ξi[ψ′(Mi−12)]−1)[∑Ki=1[ψ′(Mi−12)]−1(logQi2−ψ(Mi−12))]F1G1−I21, | (4.9) |
where Qiˆ=(Mi−1)S2i/σ2i∼χ2(Mi−1). From (4.8) and (4.9) we know that the distribution of V3 and V4 only depend on the unknown parameters σ2i through Qi. So, we can generating a series of Q∗i from the distribution χ2(Mi−1) to replace Qi and get V∗3 and V∗4.
V∗3=(∑Ki=1ξ2i[ψ′(Mi−12)]−1)[∑Ki=1[ψ′(Mi−12)]−1(logQ∗i2−ψ(Mi−12))]F1G1−I21−(∑Ki=1ξi[ψ′(Mi−12)]−1)[∑Ki=1[ψ′(Mi−12)]−1ξi(logQ∗i2−ψ(Mi−12))]F1G1−I21. | (4.10) |
V∗4=(∑Ki=1[ψ′(Mi−12)]−1)[∑Ki=1[ψ′(Mi−12)]−1ξi(logQ∗i2−ψ(Mi−12))]F1G1−I21−(∑Ki=1ξi[ψ′(Mi−12)]−1)[∑Ki=1[ψ′(Mi−12)]−1(logQ∗i2−ψ(Mi−12))]F1G1−I21. | (4.11) |
Hence, the GPQs of parameters c and d can be given by
G3=˜c−V∗3,G4=˜d−V∗4. | (4.12) |
Let Gi,α denote the α percentile of Gi, and then [G3,α/2,G3,1−α/2] and [G4,α/2,G4,1−α/2] are the 1−α level GCIs of c and d, respectively.
In practical applications, some important quantities for the Wiener double ADT model at the normal use stress level ξ0, such as μ0, σ20 and the reliability function of lifetime T, may be of more interest than the model parameters (a,b,c,d). However, because these quantities involve two or more parameters, their interval estimations tend to be difficult. Similar to the cases of parameters (a,b,c,d), we can develop the GCIs for these quantities.
Notice that μ0,σ20 and the reliability function of lifetime T are given by μ0=a+bξ0,σ20=exp(c+dξ0) and R(t0)=1−FT(t0|μ0,σ20), respectively. According to the substitution method given in [28], the GPQs for μ0,σ20 and R(t0) are given by
G5=G1+G2ξ0, | (4.13) |
G6=exp(G3+G4ξ0), | (4.14) |
G7=1−FT(t0|G5,G6), | (4.15) |
respectively.
Let Gi,α denote the α percentile of Gi. Then, [Gi,α/2,Gi,1−α/2],i=5,6,7, are the 1−α level GCIs of μ0,σ20, and R(t0), respectively. The percentiles of Gi,i=1,2,⋯,7 can be obtained by the following simulation, Algorithm 1.
Algorithm 1: GCIs for a,b,c,d and quantities μ0,σ20 and R(t0).
(1) Given data set {(ΔXi,j,k,Δti,j,k,ξi),i=1,⋯,K;j=1,⋯,ni;k=1,⋯,ri,j}, compute Xi,S2i and Ui.
(2) Generate a series of {Q0,i}Ki=1 from χ2(Mi−1), and then compute a series of {Wi}Ki=1 through Eq (4.1).
(3) Generate a series of {Z∗i}Ki=1 from N(0,1), based on {Wi}Ki=1, and through Eq (4.7) compute G1 and G2.
(4) Generate a series of {Q∗i}Ki=1 from χ2(Mi−1), and through Eq (4.12) compute G3 and G4.
(5) Based on G1,G2,G3 and G4, through Eqs (4.13)–(4.15) compute G5,G6 and G7.
(6) Repeat (2)–(5) B times, and then B values of Gi,i=1,2,⋯,7 are obtained, respectively.
(7) Arrange all Gi values in ascending order: Gi,(1)<Gi,(2)<⋯<Gi,(B),i=1,2,⋯,7. Then, the α percentile of Gi is estimated by Gi,(αB).
Remark 1. As is known to all, at the normal use stress level ξ0, the unit's mean lifetime E(T)=L/μ0 is the monotonic function of μ0. Therefore, the GCI of E(T) can be derived from the GCI of μ0.
In practical applications, the prediction intervals of the degradation characteristic, the lifetime and the remaining useful lifetime (RUL) of unit under normal use condition may be more practical and interesting for the product designer and user. So, it is important and meaningful to discuss the prediction interval of degradation characteristic X0(t) and RUL(τ). Unfortunately, for the proposed Wiener double ADT model, it is hard to obtain the exact prediction interval of these quantities, so we develop the GPI for them.
At the normal use stress level ξ0, the degradation characteristic X0(t) can be presented as
X0(t)=μ0t+σ0B(t),μ0=a+bξ0,σ20=exp(c+dξ0). |
Notice that given the failure threshold L, the product's lifetime T under the normal operating condition follows an inverse Gaussian distribution IG(L/μ0,L2/σ20). Given a fixed time τ, if the degradation characteristic X0(τ)=xτ is known, the remaining useful life RUL(τ) is defined as
RUL(τ)=inf{t|X0(t+τ)−X0(τ)≥L−xτ,t≥0}. |
Note that X0(t) has stationary independent increments, so RUL(τ)∼IG((L−xτ)/μ0,(L−xτ)2/σ20). Based on GPQs G5 and G6, using the substitution method given in [33], the GPQs of X0(t) and RUL(τ) are obtained by
G8=G5t+√G6tZ,Z∼N(0,1), | (4.16) |
G9∼IG((L−xτ)/G5,(L−xτ)2/G6). | (4.17) |
Let Gi,α be the α percentile of Gi, and then [G8,α/2,G8,1−α/2] and [G9,α/2,G9,1−α/2] are the 1−α level GPIs of degradation characteristic X0(t) and RUL. The percentiles of G9 and G10 can be obtained by the following simulation, Algorithm 2.
Algorithm 2: GPIs for X0(t) and RUL(τ).
(1) Given data set {(ΔXi,j,k,Δti,j,k,ξi),i=1,⋯,K;j=1,⋯,ni;k=1,⋯,ri,j}, compute Xi,S2i and Ui.
(2) Generate a series of {Q0,i}Ki=1 from χ2(Mi−1), and then compute a series of {Wi}Ki=1 through Eq (4.1).
(3) Generate a series of {Z∗i}Ki=1 from N(0,1), based on {Wi}Ki=1, and through Eq (4.7) compute the G1 and G2.
(4) Generate a series of {Q∗i}Ki=1 from χ2(Mi−1), and through Eq (4.12) compute G3 and G4.
(5) Based on G1,G2,G3 and G4, through Eqs (4.13) and (4.14) compute G5 and G6.
(6) Based on G5,G6, through Eqs (4.16) and (4.17) compute G8 and G9.
(7) Repeat (2)–(6) B times, and B values of G8,G9 are obtained, respectively.
(8) Arrange all Gi values in ascending order: Gi,(1)<Gi,(2)<⋯<Gi,(B),i=8,9. Then, the α percentile of Gi is estimated by Gi,(αB).
Remark 2. Note that at the normal use stress level ξ0, the unit's lifetime T follows the IG(L/μ0,L2/σ20) distribution. So, the GCI of lifetime T can be obtained by the GPQ G10, where G10∼IG(L/G5,L2/G6).
In this section, a Monte Carlo simulation study is implemented to evaluate the proposed GCIs/GPIs of model parameters and some quantities in terms of coverage percentage (CP) and average interval length (AL). Without loss of generality, we consider an ADT with three stress levels ξ1=1,ξ2=2 and ξ3=3, and the normal use stress level is ξ0=0.5. Four parameter settings are selected and given in Table 1. For convenience, the values of ni,ri,j,Δti,j,k are chosen to be n1=⋯=nKˆ=n=5,8,10;ri,jˆ=r=4,6,8; and Δti,j,k=1. Three combinations of (n,r,Δti,j,k) are considered in the simulation: (5,4,1),(8,6,1),(10,8,1). A total of 12 combinations of (a,b,c,d,L) and (n,r,Δti,j,k) are examined in the simulation study. We take B=5000 in the simulation study, and all the results are based on 5000 replications. The simulation results are provided in Tables 2–4.
Case | a | b | c | d | L |
I | -0.25 | 1.20 | -0.80 | 1.60 | 5.15 |
II | -0.50 | 2.00 | -1.00 | 2.50 | 6.30 |
III | -0.75 | 3.00 | -1.20 | 3.20 | 7.80 |
IV | 0.30 | 1.00 | 0.20 | 0.60 | 8.20 |
(n,r) | 0.9 | 0.95 | 0.9 | 0.95 | |
a | b | ||||
(5,4) | 0.8980(5.7245) | 0.9535(6.8696) | 0.8910(5.0503) | 0.9490(6.0723) | |
(8,6) | 0.8930(3.6243) | 0.9530(4.3272) | 0.8935(3.1994) | 0.9505(3.8228) | |
(10,8) | 0.8945(2.7886) | 0.9420(3.3271) | 0.8920(2.4650) | 0.9460(2.9447) | |
c | d | ||||
(5,4) | 0.9065(1.6709) | 0.9520(1.9989) | 0.9050(0.7740) | 0.9455(0.9269) | |
(8,6) | 0.9080(1.0463) | 0.9565(1.2489) | 0.9085(0.4847) | 0.9570(0.5787) | |
(10,8) | 0.8955(0.8043) | 0.9430(0.9593) | 0.9005(0.3723) | 0.9440(0.4443) | |
μ0 | σ20 | ||||
(5,4) | 0.9030(2.0925) | 0.9530(2.4557) | 0.9015(1.9745) | 0.9515(2.4745) | |
(8,6) | 0.8960(1.4858) | 0.9500(1.7236) | 0.9090(1.1410) | 0.9585(1.3864) | |
(10,8) | 0.8920(1.2317) | 0.9410(1.4231) | 0.8935(0.8448) | 0.9455(1.0188) | |
R(5) | E(T) | ||||
(5,4) | 0.9090(0.7848) | 0.9585(0.8661) | 0.9025(5.4195×104) | 0.9525(5.8182×104) | |
(8,6) | 0.8960(0.6604) | 0.9505(0.7522) | 0.8955(5.0215×104) | 0.9505(5.4538×104) | |
(10,8) | 0.9105(0.5661) | 0.9585(0.6571) | 0.8915(4.5678×104) | 0.9405(5.2177×104) | |
X0(10) | RUL(4) | ||||
(5,4) | 0.9090(26.0166) | 0.9595(31.1272) | 0.9075(0.8362×103) | 0.9580(3.2902×103) | |
(8,6) | 0.9085(20.1517) | 0.9535(24.0043) | 0.9095(0.6120×103) | 0.9570(2.3715×103) | |
(10,8) | 0.9040(17.7753) | 0.9580(21.1568) | 0.9080(0.4568×103) | 0.9535(1.7337×103) |
(n,r) | 0.9 | 0.95 | 0.9 | 0.95 | |
a | b | ||||
(5,4) | 0.8965(10.3095) | 0.9475(12.3940) | 0.9005(9.6583) | 0.9485(11.6339) | |
(8,6) | 0.9020(6.5242) | 0.9500(7.8010) | 0.9015(6.1281) | 0.9515(7.3373) | |
(10,8) | 0.8930(5.0067) | 0.9425(5.9777) | 0.8920(4.7051) | 0.9440(5.6210) | |
c | d | ||||
(5,4) | 0.9020(1.6707) | 0.9530(1.9987) | 0.9010(0.7740) | 0.9470(0.9267) | |
(8,6) | 0.8995(1.0471) | 0.9495(1.2498) | 0.8995(0.4847) | 0.9530(0.5789) | |
(10,8) | 0.9045(0.8040) | 0.9515(0.9589) | 0.8960(0.3723) | 0.9495(0.4441) | |
μ0 | σ20 | ||||
(5,4) | 0.8960(3.4859) | 0.9430(4.0890) | 0.9020(2.3736) | 0.9505(2.9752) | |
(8,6) | 0.8975(2.4609) | 0.9505(2.8554) | 0.9020(1.3195) | 0.9505(1.6046) | |
(10,8) | 0.8905(1.9922) | 0.9420(2.3054) | 0.9015(0.9827) | 0.9530(1.1847) | |
R(5) | E(T) | ||||
(5,4) | 0.9075(0.8565) | 0.9525(0.9185) | 0.8960(6.8449×104) | 0.9435(7.2617×104) | |
(8,6) | 0.8920(0.7815) | 0.9475(0.8611) | 0.8975(6.3119×104) | 0.9515(6.9116×104) | |
(10,8) | 0.8910(0.7136) | 0.9415(0.8020) | 0.8915(6.0068×104) | 0.9420(6.5718×104) | |
X0(10) | RUL(4) | ||||
(5,4) | 0.9075(39.8173) | 0.9535(47.4507) | 0.9085(0.9394×103) | 0.9580(3.7096×103) | |
(8,6) | 0.9095(29.3593) | 0.9540(34.8016) | 0.9090(0.7060×103) | 0.9585(2.7778×103) | |
(10,8) | 0.9070(24.8711) | 0.9565(29.4950) | 0.9075(0.5405×103) | 0.9570(2.1125×103) |
(n,r) | 0.9 | 0.95 | 0.9 | 0.95 | |
a | b | ||||
(5,4) | 0.8925(2.0082) | 0.9470(2.4155) | 0.9065(1.1180) | 0.9500(1.3433) | |
(8,6) | 0.8940(1.2607) | 0.9460(1.5079) | 0.8945(0.7043) | 0.9510(0.8417) | |
(10,8) | 0.9020(0.9708) | 0.9500(1.1593) | 0.8990(0.5423) | 0.9515(0.6473) | |
c | d | ||||
(5,4) | 0.8935(1.6708) | 0.9475(1.9989) | 0.8975(0.7740) | 0.9525(0.9268) | |
(8,6) | 0.9070(1.0463) | 0.9560(1.2489) | 0.9080(0.4847) | 0.9570(0.5787) | |
(10,8) | 0.8975(0.8042) | 0.9505(0.9592) | 0.8980(0.3725) | 0.9455(0.4443) | |
μ0 | σ20 | ||||
(5,4) | 0.8945(1.3499) | 0.9440(1.5745) | 0.8945(2.6428) | 0.9475(3.3125) | |
(8,6) | 0.8910(0.9320) | 0.9455(1.1026) | 0.9090(1.4650) | 0.9590(1.7801) | |
(10,8) | 0.9000(0.7308) | 0.9460(0.8707) | 0.9025(1.0978) | 0.9500(1.3239) | |
R(5) | E(T) | ||||
(5,4) | 0.9005(0.4893) | 0.9500(0.5779) | 0.8940(3.8581×104) | 0.9440(4.9431×104) | |
(8,6) | 0.8980(0.3236) | 0.9535(0.3857) | 0.8910(1.0701×104) | 0.9455(1.7598×104) | |
(10,8) | 0.9080(0.2481) | 0.9495(0.2961) | 0.9000(2.1828×103) | 0.9465(4.1033×103) | |
X0(10) | RUL(4) | ||||
(5,4) | 0.9040(19.7837) | 0.9535(23.7148) | 0.9095(172.1151) | 0.9575(611.4828) | |
(8,6) | 0.9085(16.5558) | 0.9560(19.8108) | 0.9000(42.7509) | 0.9510(108.4594) | |
(10,8) | 0.9000(15.4118) | 0.9535(18.4244) | 0.8935(22.2517) | 0.9425(38.8602) |
Tables 2–4 summarize the CPs and ALs of the two-sided equal-tailed 90% and 95% GCIs/GPIs for model parameters and some quantities under the cases of Ⅱ, Ⅲ and Ⅳ. It is observed from the simulation results that the CPs of the proposed GCIs/GPIs are quite close to the nominal levels, even for small sample sizes. Based on the normal approximation to the binomial distribution, CPs between 94% and 96% are considered appropriate for the 95% confidence intervals. For fixed parameter settings, when the sample size n and the number of measurements r increase, the ALs of GCIs/GPIs decrease as expected. These findings show that the proposed confidence interval procedures work well, and the performances of the proposed GCIs/GPIs are satisfactory with respect to the CPs.
As is known to all, the parametric bootstrap method is a classic approach to obtain confidence intervals for model parameters. In order to fully evaluate the performances of the GCIs/GPIs, we also consider the bootstrap CIs for the Wiener double ADT model. For comparison, the confidence limits (CLs), such as lower confidence limit (LCL) and upper confidence limit (UCL), for model parameters and some quantities are also examined. We performed a comparative analysis of the CIs, LCLs and UCLs obtained by the GPQ method and the bootstrap-p method. For saving space, we only give the simulation results under the case Ⅰ, and they are provided in Tables 5–7. The bootstrap-p procedure is also based on 5,000 bootstrap samples.
(n,r) | parameter | GCI/GPI | bootstrap-p CI | |||
0.9 | 0.95 | 0.9 | 0.95 | |||
(5,4) | a | 0.8970(2.7521) | 0.9495(3.3031) | 0.8955(2.6992) | 0.9460(3.2173) | |
b | 0.9020(2.0580) | 0.9500(2.4701) | 0.9065(2.0494) | 0.9425(2.4435) | ||
c | 0.8965(1.6708) | 0.9475(1.9989) | 0.9050(1.6712) | 0.9430(1.9996) | ||
d | 0.8975(0.7740) | 0.9525(0.9268) | 0.9130(0.7739) | 0.9545(0.9269) | ||
μ0 | 0.8990(1.2206) | 0.9480(1.4245) | 0.8900(1.1859) | 0.9435(1.3769) | ||
σ20 | 0.8965(1.6029) | 0.9475(2.0091) | 0.9070(1.4805) | 0.9425(1.7927) | ||
R(5) | 0.9035(0.6619) | 0.9530(0.7557) | 0.9080(0.6433) | 0.9575(0.7340) | ||
E(T) | 0.8995(4.3254×104) | 0.9480(4.6507×104) | 0.8900(4.2568×104) | 0.9425(4.6248×104) | ||
X0(10) | 0.9045(17.4588) | 0.9535(20.9393) | 0.9080(17.0667) | 0.9585(20.4164) | ||
RUL(4) | 0.9095(0.7176×103) | 0.9560(2.7925×103) | 0.9150(1.0513×103) | 0.9595(4.0867×103) | ||
(8,6) | a | 0.9025(1.7323) | 0.9470(2.0689) | 0.9075(1.7223) | 0.9560(2.0526) | |
b | 0.8965(1.3058) | 0.9460(1.5596) | 0.9060(1.3045) | 0.9555(1.5551) | ||
c | 0.8960(1.0462) | 0.9490(1.2487) | 0.9085(1.0471) | 0.9545(1.2495) | ||
d | 0.8965(0.4847) | 0.9455(0.5786) | 0.9090(0.4849) | 0.9550(0.5787) | ||
μ0 | 0.9040(0.8624) | 0.9475(0.9978) | 0.9060(0.8752) | 0.9555(1.0068) | ||
σ20 | 0.8970(0.8852) | 0.9490(1.0761) | 0.9085(0.8621) | 0.9465(1.0335) | ||
R(5) | 0.9030(0.4646) | 0.9495(0.5502) | 0.9040(0.4719) | 0.9465(0.5564) | ||
E(T) | 0.9035(3.8440×104) | 0.9475(4.2632×104) | 0.9070(3.8062×104) | 0.9565(4.3076×104) | ||
X0(10) | 0.9065(14.1289) | 0.9560(16.8762) | 0.9095(14.1436) | 0.9575(16.8773) | ||
RUL(4) | 0.9030(0.4656×103) | 0.9475(1.7509×103) | 0.9185(0.4966×103) | 0.9565(1.8667×103) | ||
(10,8) | a | 0.9020(1.3350) | 0.9520(1.5926) | 0.8915(1.3304) | 0.9450(1.5851) | |
b | 0.9005(1.0058) | 0.9505(1.1999) | 0.9055(1.0064) | 0.9525(1.1990) | ||
c | 0.8975(0.8042) | 0.9505(0.9592) | 0.9040(0.8042) | 0.9470(0.9593) | ||
d | 0.8980(0.3725) | 0.9465(0.4443) | 0.9065(0.3724) | 0.9560(0.4442) | ||
μ0 | 0.9030(0.7263) | 0.9535(0.8376) | 0.8960(0.7174) | 0.9470(0.8268) | ||
σ20 | 0.9025(0.6659) | 0.9500(0.8030) | 0.8955(0.6544) | 0.9485(0.7827) | ||
R(5) | 0.9035(0.3685) | 0.9525(0.4397) | 0.8950(0.3635) | 0.9455(0.4331) | ||
E(T) | 0.9025(3.3101×104) | 0.9520(3.9055×104) | 0.8945(3.2908×104) | 0.9450(3.8446×104) | ||
X0(10) | 0.9030(13.0539) | 0.9505(15.5771) | 0.9050(12.9732) | 0.9555(15.4825) | ||
RUL(4) | 0.9080(0.3250×103) | 0.9510(1.1652×103) | 0.9245(0.3730×103) | 0.9655(1.3665×103) |
(n,r) | parameter | LCL in GPQ method | LCL in bootstrap-p method | |||
0.9 | 0.95 | 0.9 | 0.95 | |||
(5,4) | a | 0.9035(-1.3223) | 0.9470(-1.6337) | 0.8925(-1.3213) | 0.9445(-1.6200) | |
b | 0.9010(0.4076) | 0.9545(0.1749) | 0.8935(0.4172) | 0.9555(0.1905) | ||
c | 0.8940(-1.4176) | 0.9435(-1.5874) | 0.9130(-1.4733) | 0.9640(-1.6779) | ||
d | 0.9010(1.2911) | 0.9535(1.2043) | 0.9040(1.3052) | 0.9570(1.2183) | ||
μ0 | 0.9025(0.0831) | 0.9485(0.0441) | 0.9035(0.0862) | 0.9470(0.0468) | ||
σ20 | 0.9035(0.6658) | 0.9425(0.5840) | 0.9175(0.6349) | 0.9640(0.5390) | ||
R(5) | 0.9430(0.4179) | 0.9570(0.3116) | 0.9245(0.4328) | 0.9595(0.3310) | ||
E(T) | 0.9035(0.1479×104) | 0.9510(0.0675×104) | 0.8920(0.1548×104) | 0.9430(0.0677×104) | ||
X0(10) | 0.8915(-1.3915) | 0.9425(-3.0086) | 0.8805(-1.3434) | 0.9375(-2.9459) | ||
RUL(4) | 0.9410(2.2967) | 0.9575(1.6463) | 0.9260(2.4000) | 0.9650(1.7000) | ||
(8,6) | a | 0.9005(-0.9378) | 0.9495(-1.1313) | 0.9035(-0.9058) | 0.9485(-1.0969) | |
b | 0.8915(0.7058) | 0.9455(0.5602) | 0.9135(0.6798) | 0.9565(0.5356) | ||
c | 0.9005(-1.2072) | 0.9495(-1.3165) | 0.9180(-1.2202) | 0.9660(-1.3435) | ||
d | 0.8940(1.4130) | 0.9465(1.3590) | 0.9050(1.4146) | 0.9435(1.3605) | ||
μ0 | 0.8965(0.0942) | 0.9475(0.0551) | 0.9025(0.0961) | 0.9490(0.0549) | ||
σ20 | 0.9015(0.7492) | 0.9515(0.6879) | 0.9165(0.7385) | 0.9655(0.6693) | ||
R(5) | 0.9095(0.5941) | 0.9560(0.5075) | 0.9185(0.5863) | 0.9580(0.4999) | ||
E(T) | 0.9030(0.0551×104) | 0.9535(0.0189×104) | 0.9135(0.0472×104) | 0.9575(0.0136×104) | ||
X0(10) | 0.8950(-1.2837) | 0.9430(-2.7351) | 0.8870(-1.1942) | 0.9405(-2.6476) | ||
RUL(4) | 0.9060(2.7260) | 0.9530(2.0512) | 0.9145(2.7046) | 0.9625(2.0479) | ||
(10,8) | a | 0.9015(-0.7763) | 0.9525(-0.9249) | 0.9055(-0.7809) | 0.9560(-0.9278) | |
b | 0.8995(0.8159) | 0.9490(0.7041) | 0.8905(0.8152) | 0.9500(0.7039) | ||
c | 0.8955(-1.1031) | 0.9445(-1.1882) | 0.9065(-1.1142) | 0.9560(-1.2075) | ||
d | 0.9025(1.4501) | 0.9525(1.4087) | 0.9065(1.4532) | 0.9550(1.4119) | ||
μ0 | 0.9030(0.1066) | 0.9535(0.0640) | 0.9040(0.1079) | 0.9565(0.0659) | ||
σ20 | 0.8965(0.8010) | 0.9485(0.7494) | 0.9070(0.7934) | 0.9555(0.7366) | ||
R(5) | 0.9050(0.6699) | 0.9520(0.6012) | 0.8940(0.6735) | 0.9485(0.6061) | ||
E(T) | 0.9065(0.0268×104) | 0.9515(0.0111×104) | 0.8885(0.0353×104) | 0.9430(0.0162×104) | ||
X0(10) | 0.8975(-1.2025) | 0.9465(-2.5920) | 0.8960(-1.2022) | 0.9450(-2.5857) | ||
RUL(4) | 0.8930(2.8285) | 0.9525(2.1574) | 0.9120(2.8721) | 0.9590 (2.1891) |
(n,r) | parameter | UCL in GPQ method | UCL in bootstrap-p method | |||
0.9 | 0.95 | 0.9 | 0.95 | |||
(5,4) | a | 0.9045(0.8079) | 0.9490(1.1184) | 0.8925(0.7807) | 0.9470(1.0792) | |
b | 0.9030(2.0010) | 0.9525(2.2329) | 0.8950(2.0136) | 0.9480(2.2399) | ||
c | 0.9025(-0.1204) | 0.9540(0.0834) | 0.8925(-0.1762) | 0.9410(-0.0067) | ||
d | 0.8955(1.8914) | 0.9455(1.9784) | 0.9070(1.9054) | 0.9575(1.9922) | ||
μ0 | 0.9055(1.0601) | 0.9505(1.2647) | 0.8920(1.0396) | 0.9430(1.2327) | ||
σ20 | 0.9025(1.8569) | 0.9540(2.1869) | 0.8870(1.7710) | 0.9390(2.0194) | ||
R(5) | 0.8925(0.9535) | 0.9450(0.9735) | 0.8855(0.9539) | 0.9440(0.9743) | ||
E(T) | 0.9020(0.0000×104) | 0.9485(4.3930×104) | 0.9025(0.0000×104) | 0.9470(4.3245×104) | ||
X0(10) | 0.9425(0.0000) | 0.9580(14.4502) | 0.9330(0.0000) | 0.9705(14.1208) | ||
RUL(4) | 0.9030(256.4904) | 0.9545(984.8106) | 0.9035(274.7000) | 0.9550(1053.0000) | ||
(8,6) | a | 0.9020(0.4084) | 0.9540(0.6010) | 0.9150(0.4357) | 0.9570(0.6254) | |
b | 0.9025(1.7209) | 0.9450(1.8660) | 0.9045(1.6961) | 0.9445(1.8400) | ||
c | 0.8950(-0.3928) | 0.9445(-0.2703) | 0.8930(-0.4057) | 0.9425(-0.2964) | ||
d | 0.8925(1.7898) | 0.9515(1.8437) | 0.9135(1.7916) | 0.9615(1.8454) | ||
μ0 | 0.9030(0.7902) | 0.9565(0.9176) | 0.9135(0.8048) | 0.9570(0.9301) | ||
σ20 | 0.8940(1.4264) | 0.9445(1.5732) | 0.8930(1.4061) | 0.9410(1.5314) | ||
R(5) | 0.8980(0.9570) | 0.9455(0.9721) | 0.8970(0.9563) | 0.9450(0.9719) | ||
E(T) | 0.8975(0.0000×104) | 0.9480(3.8629×104) | 0.9035 (0.0000×104) | 0.9465(3.8199×104) | ||
X0(10) | 0.9095(0.0000) | 0.9580(11.3939) | 0.9315(0.0000) | 0.9665(11.4959) | ||
RUL(4) | 0.9040(148.1373) | 0.9540(532.5225) | 0.9050 (139.0714) | 0.9560(498.6037) | ||
(10,8) | a | 0.8990(0.2619) | 0.9520(0.4101) | 0.8915(0.2559) | 0.9400(0.4026) | |
b | 0.9020(1.5987) | 0.9515(1.7099) | 0.9045(1.5994) | 0.9525(1.7103) | ||
c | 0.9030(-0.4769) | 0.9540(-0.3841) | 0.8960(-0.4880) | 0.9450(-0.4033) | ||
d | 0.8975(1.7399) | 0.9485(1.7812) | 0.9035(1.7430) | 0.9515(1.7843) | ||
μ0 | 0.9060(0.6921) | 0.9515(0.7903) | 0.8885(0.6862) | 0.9425(0.7833) | ||
σ20 | 0.9045(1.3143) | 0.9540(1.4152) | 0.8950(1.3016) | 0.9430(1.3910) | ||
R(5) | 0.9035(0.9563) | 0.9590(0.9697) | 0.8965(0.9560) | 0.9485(0.9696) | ||
E(T) | 0.9020(0.0000×104) | 0.9540(3.3212×104) | 0.9025(0.0000×104) | 0.9545(3.3070×104) | ||
X0(10) | 0.9090(0.0000) | 0.9555(10.4619) | 0.9170(0.0000) | 0.9590(10.3875) | ||
RUL(4) | 0.9085(103.6819) | 0.9560(345.9644) | 0.9160(111.2404) | 0.9655(375.1649) |
Tables 5–7 show that the CPs of the CIs, LCLs and UCLs obtained by the GPQ method are all very close to the nominal levels, even for small sample sizes. However, the CPs of the CIs, LCLs and UCLs obtained by the bootstrap-p method are not close to the nominal levels for some parameters and quantities. In particular, from Table 5 we find that the bootstrap-p CIs of RUL(4) are not close to the nominal levels. In addition, from Tables 6 and 7 we also find that the LCLs and UCLs obtained in the bootstrap-p method perform badly. For example, the CPs of model parameters c,σ20, reliability function R(5), mean lifetime E(T), degradation amount X0(10) and RUL(4) deviate from the nominal levels.
As the sample size n increases, the CPs of the bootstrap-p CIs/PIs approach the nominal levels. Tables 5–7 also indicate that, for fixed parameter settings, as the sample size n increases, the ALs become shorter, the LCLs become larger, and the UCLs become smaller for both GPQ method and bootstrap-p method as expected. These findings indicate that the CIs, LCLs and UCLs obtained in the GPQ method perform better than the corresponding bootstrap-p ones in terms of CP. Therefore, we recommend the proposed CIs, LCLs and UCLs in the GPQ method for the proposed Wiener double ADT model, especially in small sample cases.
In this section, we provide the ADT data of commercial white LEDs to illustrate our proposed methods. Degradation in lumen maintenance is the main failure mechanism for LEDs [30]. An LED is defined as a failure when the lumen maintenance decreases by 30% of its initial level.
In the ADT, 16 retrofit LED tubes based on a low-power LED are assigned to be tested at stress levels s1=25∘C and s2=55∘C, respectively. In general, the normal operating temperature of the LED is s0=25∘C. The tubes are placed in the climate chamber to ensure the stability of the ambient operating temperature, and the lumen outputs of the tubes are measured regularly. Note that there exists a very short period of increase in the lumen output due to incomplete burn-in [31], so this period is identified from the original data and is discarded in the analysis. For each test unit, the decrease in lumen output is normalized by the initial output. In addition, there is an unexpected catastrophic failure, and only 15 tubes under 55∘C are recorded. Figure 1 shows the degradation paths of the LEDs. Similar to Hong et al. [23], to protect proprietary information, a power-law time-scale transformation with exponent 0.70 is used to linearize the degradation paths.
The degradation data are not monotone, so the Gamma process and inverse Gaussian process are not suitable to deal with them. Thus, we choose the Wiener process to model the degradation data. In order to assess the goodness-of-fit, a standard normal Q-Q plot for the average standardized degradation increments is given in Figure 2. The points scatter around the line nicely except one point at the left tail, probably due to measurement errors for some observations.
Since temperature acts as the accelerated stress in this example, the Arrhenius model is used to describe the relationship between parameters and the accelerated stress. Hence, we take
ξi=exp(−11605/(si+273.15)). |
Similar to Lim and Yum [32], the standardized stress levels are given by
ζi=ξi−ξ0ξ2−ξ0,i=0,1,2. |
Since s0=s1, the temperature levels are standardized as ζ0=ζ1=0,ζ2=1.
Let σ2i be replaced by its estimate S2i in (3.3), and the point estimates of the parameters a and b are given by ˜a=0.2026,˜b=0.0619. Based on Eq (3.6), the point estimates of the parameters c and d are obtained by ˜c=3.9533,˜d=0.2960. Notice that μi=a+bζi,σ2i=exp(c+d)ζi,i=1,2. The estimates of μi and σ2i are given by ˜μ1=0.2044,˜μ2=0.2645,˜σ21=52.5497,˜σ22=70.0571, respectively.
Notice that the mean degradation path at the stress level si is E(Xi(t))=μit, and the estimate of E(Xi(t)) is then given by ~E(Xi(t))=˜μit (considering the transformed time scale, here the time t should be replaced by t0.7). The degradation paths and the estimates of these mean degradation paths under transformed time scale are given in Figure 1. It is obvious that these degradation data are fitted well by the proposed Wiener doubly accelerated degradation model.
To illustrate the GPQ method described in section 4, we use the degradation data in Figure 1 to derive the GCIs/GPIs of model parameters and some reliability metrics. Based on the Gi,i=1,2,⋯,6, the GCIs of model parameters a,b,c,d and quantities μ0,σ20 are obtained, for nominal levels 90% and 95%. According to G7, the GCIs of reliability function R(100) are also obtained. In addition, using G8 and G9, the GPIs of degradation amount X0(150) and remaining useful life RUL(126) are developed. All the results are based on 10,000 replications and presented in Table 8.
Parameter | 90% | Length | 95% | Length |
a | (-0.0739, 0.4786) | 0.5525 | (-0.1286, 0.5323) | 0.6609 |
b | (-0.3882, 0.5149) | 0.9031 | (-0.4743, 0.6016) | 1.0759 |
c | (3.7885, 4.1246) | 0.3361 | (3.7577, 4.1592) | 0.4015 |
d | (0.0953, 0.4927) | 0.3974 | (0.0571, 0.5302) | 0.4731 |
μ0 | (0.0000, 0.4786) | 0.4786 | (0.0000, 0.5323) | 0.5323 |
σ20 | (44.1879, 61.8413) | 17.6534 | (42.8486, 64.0231) | 21.1745 |
R(100) | (0.4704, 0.6095) | 0.1391 | (0.4570, 0.6196) | 0.1626 |
T | (0.0008, 1.3891)×104 | 1.3883×104 | (0.0005, 4.3066)×104 | 4.3061×104 |
RUL(126) | (0.0004, 0.9336)×104 | 0.9332×104 | (0.0003, 2.9032)×104 | 2.9029×104 |
X0(150) | (-62.2527, 76.5406) | 138.7933 | (-76.1094, 90.2592) | 166.3686 |
In this paper we proposed a doubly accelerated degradation test model of Wiener process, in which both the degradation rate and the diffusion parameter are affected by the stress level. The point estimates of model parameters are obtained by constructing a regression model. Furthermore, based on the point estimates of model parameters, the GCIs of model parameters are developed by constructing GPQs. Utilizing the substitute method, the GCIs of some quantities and reliability metrics are derived, and the GPIs of degradation amount X0(t) and RUL for units under normal use stress level are also developed.
Extensive simulations are carried out to evaluate the performances of the proposed procedures. The simulation results reveal that the proposed confidence intervals performed well in terms of CPs, and compared with the traditional bootstrap method. Finally, an illustrative example is provided to demonstrate the proposed procedures. As is known to all, the Gamma process and inverse Gaussian process are also widely used in analysis of accelerated degradation data. In the future, we shall mainly focus on studying the small sample inferential methods for doubly accelerated degradation test models based on the Gamma and inverse Gaussian process.
The authors thank the Editor and the reviewers for their detailed comments and suggestions, which considerably helped improve the manuscript. The work is supported by the Pre-research Project of the National Science Foundation of Anhui Polytechnic University (Xjky08201903), the Talent Cultivation and Research Start-up Foundation of Anhui Polytechnic University (S022022014), the Support program for outstanding young talents in colleges and universities of Anhui Province (gxyqZD2022046) and the National Social Science Foundation of China (18BTJ034, 20BTJ048).
The authors declare that they have no conflicts of interest.
Let V=diag(σ21/T1,⋯,σ2K/TK),Y=(Y1,⋯,YK)T, and
Z=(11⋯1⋯1s1s2⋯si⋯sK)T. |
Then, the estimates (ˆa,ˆb) are given by
(ˆaˆb)=(ZTV−1Z)−1ZTV−1Y=1FG−I2(GH−IMFM−IH). |
So, we have the expectation of ˆa and ˆb
E(ˆaˆb)=1FG−I2(GE(H)−IE(M)FE(M)−IE(H))=(ab). |
Furthermore, the covariance matrix of the estimates (ˆa,ˆb) is given by
Var(ˆaˆb)=(ZTV−1Z)−1=1FG−I2(G−I−IF). |
So, the variance and covariance of the estimates ˆa and ˆb are obtained by
Var(ˆa)=GFG−I2,Var(ˆb)=FFG−I2,Cov(ˆa,ˆb)=−IFG−I2. |
Notice that the Xi's are normal distributions and independent of each other and ˆa and ˆb are linear combinations of them, so the estimates ˆa and ˆb are also normal distributions. That is,
ˆa∼N(a,GFG−I2),ˆb∼N(b,FFG−I2). |
The proof of Theorem 3.1 is completed.
Let V=diag(ψ′(M1−12),⋯,ψ′(MK−12)),Y=(U1,⋯,UK)T. Similar to the proof of Theorem 3.1, Theorem 3.2 can be easily proved, and here we neglect the detailed proof.
Let ζi=log((Mi−1)S2i2σ2i)−ψ(Mi−12), and then E(ζi)=0,Var(ζi)=ψ′(Mi−12). Notice that
H1=K∑i=1[ψ′(Mi−12)]−1ζi+cK∑i=1[ψ′(Mi−12)]−1+dK∑i=1ξi[ψ′(Mi−12)]−1=:H0+cF1+dI1.M1=K∑i=1[ψ′(Mi−12)]−1ξiζi+cK∑i=1ξi[ψ′(Mi−12)]−1+dK∑i=1ξ2i[ψ′(Mi−12)]−1=:M0+cI1+dG1. |
So, we have
˜c=G1H0−I1M0F1G1−I21+c,˜d=F1M0−I1H0F1G1−I21+d. |
So,
log˜σ20−logσ20=(˜c−c)+(˜d−d)ξ0=G1H0−I1M0F1G1−I21+F1M0−I1H0F1G1−I21ξ0=K∑i=1G1−(ξ0+ξi)I1+ξ0ξiF1ψ′(Mi−12)(F1G1−I21)ζi=K∑i=1Dilog((Mi−1)S2i2σ2i)−K∑i=1Diψ(Mi−12). |
Because (Mi−1)S2iσ2i∼χ2(Mi−1), and
˜σ20/σ20=K∏i=1((Mi−1)S2i2σ2i)Diexp[−K∑i=1Diψ(Mi−12)], |
we have
E(˜σ20σ20)=exp[−K∑i=1Diψ(Mi−12)]E[K∏i=1((Mi−1)S2i2σ2i)Di]=exp[−K∑i=1Diψ(Mi−12)]K∏i=1Γ(Di+Mi−12)Γ(Mi−12). |
E((˜σ20)2σ40)=exp[−2K∑i=1Diψ(Mi−12)]K∏i=1Γ(2Di+Mi−12)Γ(Mi−12). |
From the above formulas, the unbiased estimate of σ20 and its variance Var(˜σ20u) can be given as
˜σ20u=˜σ20exp[ΣKi=1Diψ(Mi−12)]K∏i=1Γ(Mi−12)Γ(Di+Mi−12). |
Var(˜σ20u)=σ20K∏i=1[Γ(Mi−12)Γ(2Di+Mi−12)−Γ2(Di+Mi−12)Γ2(Di+Mi−12)]. |
Similar to the proof of Theorem 6 in Wang and Yu [33], we can prove that ˜σ20u has a smaller mean square error than that of ˜σ20, where we neglect the detailed proof. The proof of Theorem 3.3 is completed.
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Case | a | b | c | d | L |
I | -0.25 | 1.20 | -0.80 | 1.60 | 5.15 |
II | -0.50 | 2.00 | -1.00 | 2.50 | 6.30 |
III | -0.75 | 3.00 | -1.20 | 3.20 | 7.80 |
IV | 0.30 | 1.00 | 0.20 | 0.60 | 8.20 |
(n,r) | 0.9 | 0.95 | 0.9 | 0.95 | |
a | b | ||||
(5,4) | 0.8980(5.7245) | 0.9535(6.8696) | 0.8910(5.0503) | 0.9490(6.0723) | |
(8,6) | 0.8930(3.6243) | 0.9530(4.3272) | 0.8935(3.1994) | 0.9505(3.8228) | |
(10,8) | 0.8945(2.7886) | 0.9420(3.3271) | 0.8920(2.4650) | 0.9460(2.9447) | |
c | d | ||||
(5,4) | 0.9065(1.6709) | 0.9520(1.9989) | 0.9050(0.7740) | 0.9455(0.9269) | |
(8,6) | 0.9080(1.0463) | 0.9565(1.2489) | 0.9085(0.4847) | 0.9570(0.5787) | |
(10,8) | 0.8955(0.8043) | 0.9430(0.9593) | 0.9005(0.3723) | 0.9440(0.4443) | |
μ0 | σ20 | ||||
(5,4) | 0.9030(2.0925) | 0.9530(2.4557) | 0.9015(1.9745) | 0.9515(2.4745) | |
(8,6) | 0.8960(1.4858) | 0.9500(1.7236) | 0.9090(1.1410) | 0.9585(1.3864) | |
(10,8) | 0.8920(1.2317) | 0.9410(1.4231) | 0.8935(0.8448) | 0.9455(1.0188) | |
R(5) | E(T) | ||||
(5,4) | 0.9090(0.7848) | 0.9585(0.8661) | 0.9025(5.4195×104) | 0.9525(5.8182×104) | |
(8,6) | 0.8960(0.6604) | 0.9505(0.7522) | 0.8955(5.0215×104) | 0.9505(5.4538×104) | |
(10,8) | 0.9105(0.5661) | 0.9585(0.6571) | 0.8915(4.5678×104) | 0.9405(5.2177×104) | |
X0(10) | RUL(4) | ||||
(5,4) | 0.9090(26.0166) | 0.9595(31.1272) | 0.9075(0.8362×103) | 0.9580(3.2902×103) | |
(8,6) | 0.9085(20.1517) | 0.9535(24.0043) | 0.9095(0.6120×103) | 0.9570(2.3715×103) | |
(10,8) | 0.9040(17.7753) | 0.9580(21.1568) | 0.9080(0.4568×103) | 0.9535(1.7337×103) |
(n,r) | 0.9 | 0.95 | 0.9 | 0.95 | |
a | b | ||||
(5,4) | 0.8965(10.3095) | 0.9475(12.3940) | 0.9005(9.6583) | 0.9485(11.6339) | |
(8,6) | 0.9020(6.5242) | 0.9500(7.8010) | 0.9015(6.1281) | 0.9515(7.3373) | |
(10,8) | 0.8930(5.0067) | 0.9425(5.9777) | 0.8920(4.7051) | 0.9440(5.6210) | |
c | d | ||||
(5,4) | 0.9020(1.6707) | 0.9530(1.9987) | 0.9010(0.7740) | 0.9470(0.9267) | |
(8,6) | 0.8995(1.0471) | 0.9495(1.2498) | 0.8995(0.4847) | 0.9530(0.5789) | |
(10,8) | 0.9045(0.8040) | 0.9515(0.9589) | 0.8960(0.3723) | 0.9495(0.4441) | |
μ0 | σ20 | ||||
(5,4) | 0.8960(3.4859) | 0.9430(4.0890) | 0.9020(2.3736) | 0.9505(2.9752) | |
(8,6) | 0.8975(2.4609) | 0.9505(2.8554) | 0.9020(1.3195) | 0.9505(1.6046) | |
(10,8) | 0.8905(1.9922) | 0.9420(2.3054) | 0.9015(0.9827) | 0.9530(1.1847) | |
R(5) | E(T) | ||||
(5,4) | 0.9075(0.8565) | 0.9525(0.9185) | 0.8960(6.8449×104) | 0.9435(7.2617×104) | |
(8,6) | 0.8920(0.7815) | 0.9475(0.8611) | 0.8975(6.3119×104) | 0.9515(6.9116×104) | |
(10,8) | 0.8910(0.7136) | 0.9415(0.8020) | 0.8915(6.0068×104) | 0.9420(6.5718×104) | |
X0(10) | RUL(4) | ||||
(5,4) | 0.9075(39.8173) | 0.9535(47.4507) | 0.9085(0.9394×103) | 0.9580(3.7096×103) | |
(8,6) | 0.9095(29.3593) | 0.9540(34.8016) | 0.9090(0.7060×103) | 0.9585(2.7778×103) | |
(10,8) | 0.9070(24.8711) | 0.9565(29.4950) | 0.9075(0.5405×103) | 0.9570(2.1125×103) |
(n,r) | 0.9 | 0.95 | 0.9 | 0.95 | |
a | b | ||||
(5,4) | 0.8925(2.0082) | 0.9470(2.4155) | 0.9065(1.1180) | 0.9500(1.3433) | |
(8,6) | 0.8940(1.2607) | 0.9460(1.5079) | 0.8945(0.7043) | 0.9510(0.8417) | |
(10,8) | 0.9020(0.9708) | 0.9500(1.1593) | 0.8990(0.5423) | 0.9515(0.6473) | |
c | d | ||||
(5,4) | 0.8935(1.6708) | 0.9475(1.9989) | 0.8975(0.7740) | 0.9525(0.9268) | |
(8,6) | 0.9070(1.0463) | 0.9560(1.2489) | 0.9080(0.4847) | 0.9570(0.5787) | |
(10,8) | 0.8975(0.8042) | 0.9505(0.9592) | 0.8980(0.3725) | 0.9455(0.4443) | |
μ0 | σ20 | ||||
(5,4) | 0.8945(1.3499) | 0.9440(1.5745) | 0.8945(2.6428) | 0.9475(3.3125) | |
(8,6) | 0.8910(0.9320) | 0.9455(1.1026) | 0.9090(1.4650) | 0.9590(1.7801) | |
(10,8) | 0.9000(0.7308) | 0.9460(0.8707) | 0.9025(1.0978) | 0.9500(1.3239) | |
R(5) | E(T) | ||||
(5,4) | 0.9005(0.4893) | 0.9500(0.5779) | 0.8940(3.8581×104) | 0.9440(4.9431×104) | |
(8,6) | 0.8980(0.3236) | 0.9535(0.3857) | 0.8910(1.0701×104) | 0.9455(1.7598×104) | |
(10,8) | 0.9080(0.2481) | 0.9495(0.2961) | 0.9000(2.1828×103) | 0.9465(4.1033×103) | |
X0(10) | RUL(4) | ||||
(5,4) | 0.9040(19.7837) | 0.9535(23.7148) | 0.9095(172.1151) | 0.9575(611.4828) | |
(8,6) | 0.9085(16.5558) | 0.9560(19.8108) | 0.9000(42.7509) | 0.9510(108.4594) | |
(10,8) | 0.9000(15.4118) | 0.9535(18.4244) | 0.8935(22.2517) | 0.9425(38.8602) |
(n,r) | parameter | GCI/GPI | bootstrap-p CI | |||
0.9 | 0.95 | 0.9 | 0.95 | |||
(5,4) | a | 0.8970(2.7521) | 0.9495(3.3031) | 0.8955(2.6992) | 0.9460(3.2173) | |
b | 0.9020(2.0580) | 0.9500(2.4701) | 0.9065(2.0494) | 0.9425(2.4435) | ||
c | 0.8965(1.6708) | 0.9475(1.9989) | 0.9050(1.6712) | 0.9430(1.9996) | ||
d | 0.8975(0.7740) | 0.9525(0.9268) | 0.9130(0.7739) | 0.9545(0.9269) | ||
μ0 | 0.8990(1.2206) | 0.9480(1.4245) | 0.8900(1.1859) | 0.9435(1.3769) | ||
σ20 | 0.8965(1.6029) | 0.9475(2.0091) | 0.9070(1.4805) | 0.9425(1.7927) | ||
R(5) | 0.9035(0.6619) | 0.9530(0.7557) | 0.9080(0.6433) | 0.9575(0.7340) | ||
E(T) | 0.8995(4.3254×104) | 0.9480(4.6507×104) | 0.8900(4.2568×104) | 0.9425(4.6248×104) | ||
X0(10) | 0.9045(17.4588) | 0.9535(20.9393) | 0.9080(17.0667) | 0.9585(20.4164) | ||
RUL(4) | 0.9095(0.7176×103) | 0.9560(2.7925×103) | 0.9150(1.0513×103) | 0.9595(4.0867×103) | ||
(8,6) | a | 0.9025(1.7323) | 0.9470(2.0689) | 0.9075(1.7223) | 0.9560(2.0526) | |
b | 0.8965(1.3058) | 0.9460(1.5596) | 0.9060(1.3045) | 0.9555(1.5551) | ||
c | 0.8960(1.0462) | 0.9490(1.2487) | 0.9085(1.0471) | 0.9545(1.2495) | ||
d | 0.8965(0.4847) | 0.9455(0.5786) | 0.9090(0.4849) | 0.9550(0.5787) | ||
μ0 | 0.9040(0.8624) | 0.9475(0.9978) | 0.9060(0.8752) | 0.9555(1.0068) | ||
σ20 | 0.8970(0.8852) | 0.9490(1.0761) | 0.9085(0.8621) | 0.9465(1.0335) | ||
R(5) | 0.9030(0.4646) | 0.9495(0.5502) | 0.9040(0.4719) | 0.9465(0.5564) | ||
E(T) | 0.9035(3.8440×104) | 0.9475(4.2632×104) | 0.9070(3.8062×104) | 0.9565(4.3076×104) | ||
X0(10) | 0.9065(14.1289) | 0.9560(16.8762) | 0.9095(14.1436) | 0.9575(16.8773) | ||
RUL(4) | 0.9030(0.4656×103) | 0.9475(1.7509×103) | 0.9185(0.4966×103) | 0.9565(1.8667×103) | ||
(10,8) | a | 0.9020(1.3350) | 0.9520(1.5926) | 0.8915(1.3304) | 0.9450(1.5851) | |
b | 0.9005(1.0058) | 0.9505(1.1999) | 0.9055(1.0064) | 0.9525(1.1990) | ||
c | 0.8975(0.8042) | 0.9505(0.9592) | 0.9040(0.8042) | 0.9470(0.9593) | ||
d | 0.8980(0.3725) | 0.9465(0.4443) | 0.9065(0.3724) | 0.9560(0.4442) | ||
μ0 | 0.9030(0.7263) | 0.9535(0.8376) | 0.8960(0.7174) | 0.9470(0.8268) | ||
σ20 | 0.9025(0.6659) | 0.9500(0.8030) | 0.8955(0.6544) | 0.9485(0.7827) | ||
R(5) | 0.9035(0.3685) | 0.9525(0.4397) | 0.8950(0.3635) | 0.9455(0.4331) | ||
E(T) | 0.9025(3.3101×104) | 0.9520(3.9055×104) | 0.8945(3.2908×104) | 0.9450(3.8446×104) | ||
X0(10) | 0.9030(13.0539) | 0.9505(15.5771) | 0.9050(12.9732) | 0.9555(15.4825) | ||
RUL(4) | 0.9080(0.3250×103) | 0.9510(1.1652×103) | 0.9245(0.3730×103) | 0.9655(1.3665×103) |
(n,r) | parameter | LCL in GPQ method | LCL in bootstrap-p method | |||
0.9 | 0.95 | 0.9 | 0.95 | |||
(5,4) | a | 0.9035(-1.3223) | 0.9470(-1.6337) | 0.8925(-1.3213) | 0.9445(-1.6200) | |
b | 0.9010(0.4076) | 0.9545(0.1749) | 0.8935(0.4172) | 0.9555(0.1905) | ||
c | 0.8940(-1.4176) | 0.9435(-1.5874) | 0.9130(-1.4733) | 0.9640(-1.6779) | ||
d | 0.9010(1.2911) | 0.9535(1.2043) | 0.9040(1.3052) | 0.9570(1.2183) | ||
μ0 | 0.9025(0.0831) | 0.9485(0.0441) | 0.9035(0.0862) | 0.9470(0.0468) | ||
σ20 | 0.9035(0.6658) | 0.9425(0.5840) | 0.9175(0.6349) | 0.9640(0.5390) | ||
R(5) | 0.9430(0.4179) | 0.9570(0.3116) | 0.9245(0.4328) | 0.9595(0.3310) | ||
E(T) | 0.9035(0.1479×104) | 0.9510(0.0675×104) | 0.8920(0.1548×104) | 0.9430(0.0677×104) | ||
X0(10) | 0.8915(-1.3915) | 0.9425(-3.0086) | 0.8805(-1.3434) | 0.9375(-2.9459) | ||
RUL(4) | 0.9410(2.2967) | 0.9575(1.6463) | 0.9260(2.4000) | 0.9650(1.7000) | ||
(8,6) | a | 0.9005(-0.9378) | 0.9495(-1.1313) | 0.9035(-0.9058) | 0.9485(-1.0969) | |
b | 0.8915(0.7058) | 0.9455(0.5602) | 0.9135(0.6798) | 0.9565(0.5356) | ||
c | 0.9005(-1.2072) | 0.9495(-1.3165) | 0.9180(-1.2202) | 0.9660(-1.3435) | ||
d | 0.8940(1.4130) | 0.9465(1.3590) | 0.9050(1.4146) | 0.9435(1.3605) | ||
μ0 | 0.8965(0.0942) | 0.9475(0.0551) | 0.9025(0.0961) | 0.9490(0.0549) | ||
σ20 | 0.9015(0.7492) | 0.9515(0.6879) | 0.9165(0.7385) | 0.9655(0.6693) | ||
R(5) | 0.9095(0.5941) | 0.9560(0.5075) | 0.9185(0.5863) | 0.9580(0.4999) | ||
E(T) | 0.9030(0.0551×104) | 0.9535(0.0189×104) | 0.9135(0.0472×104) | 0.9575(0.0136×104) | ||
X0(10) | 0.8950(-1.2837) | 0.9430(-2.7351) | 0.8870(-1.1942) | 0.9405(-2.6476) | ||
RUL(4) | 0.9060(2.7260) | 0.9530(2.0512) | 0.9145(2.7046) | 0.9625(2.0479) | ||
(10,8) | a | 0.9015(-0.7763) | 0.9525(-0.9249) | 0.9055(-0.7809) | 0.9560(-0.9278) | |
b | 0.8995(0.8159) | 0.9490(0.7041) | 0.8905(0.8152) | 0.9500(0.7039) | ||
c | 0.8955(-1.1031) | 0.9445(-1.1882) | 0.9065(-1.1142) | 0.9560(-1.2075) | ||
d | 0.9025(1.4501) | 0.9525(1.4087) | 0.9065(1.4532) | 0.9550(1.4119) | ||
μ0 | 0.9030(0.1066) | 0.9535(0.0640) | 0.9040(0.1079) | 0.9565(0.0659) | ||
σ20 | 0.8965(0.8010) | 0.9485(0.7494) | 0.9070(0.7934) | 0.9555(0.7366) | ||
R(5) | 0.9050(0.6699) | 0.9520(0.6012) | 0.8940(0.6735) | 0.9485(0.6061) | ||
E(T) | 0.9065(0.0268×104) | 0.9515(0.0111×104) | 0.8885(0.0353×104) | 0.9430(0.0162×104) | ||
X0(10) | 0.8975(-1.2025) | 0.9465(-2.5920) | 0.8960(-1.2022) | 0.9450(-2.5857) | ||
RUL(4) | 0.8930(2.8285) | 0.9525(2.1574) | 0.9120(2.8721) | 0.9590 (2.1891) |
(n,r) | parameter | UCL in GPQ method | UCL in bootstrap-p method | |||
0.9 | 0.95 | 0.9 | 0.95 | |||
(5,4) | a | 0.9045(0.8079) | 0.9490(1.1184) | 0.8925(0.7807) | 0.9470(1.0792) | |
b | 0.9030(2.0010) | 0.9525(2.2329) | 0.8950(2.0136) | 0.9480(2.2399) | ||
c | 0.9025(-0.1204) | 0.9540(0.0834) | 0.8925(-0.1762) | 0.9410(-0.0067) | ||
d | 0.8955(1.8914) | 0.9455(1.9784) | 0.9070(1.9054) | 0.9575(1.9922) | ||
μ0 | 0.9055(1.0601) | 0.9505(1.2647) | 0.8920(1.0396) | 0.9430(1.2327) | ||
σ20 | 0.9025(1.8569) | 0.9540(2.1869) | 0.8870(1.7710) | 0.9390(2.0194) | ||
R(5) | 0.8925(0.9535) | 0.9450(0.9735) | 0.8855(0.9539) | 0.9440(0.9743) | ||
E(T) | 0.9020(0.0000×104) | 0.9485(4.3930×104) | 0.9025(0.0000×104) | 0.9470(4.3245×104) | ||
X0(10) | 0.9425(0.0000) | 0.9580(14.4502) | 0.9330(0.0000) | 0.9705(14.1208) | ||
RUL(4) | 0.9030(256.4904) | 0.9545(984.8106) | 0.9035(274.7000) | 0.9550(1053.0000) | ||
(8,6) | a | 0.9020(0.4084) | 0.9540(0.6010) | 0.9150(0.4357) | 0.9570(0.6254) | |
b | 0.9025(1.7209) | 0.9450(1.8660) | 0.9045(1.6961) | 0.9445(1.8400) | ||
c | 0.8950(-0.3928) | 0.9445(-0.2703) | 0.8930(-0.4057) | 0.9425(-0.2964) | ||
d | 0.8925(1.7898) | 0.9515(1.8437) | 0.9135(1.7916) | 0.9615(1.8454) | ||
μ0 | 0.9030(0.7902) | 0.9565(0.9176) | 0.9135(0.8048) | 0.9570(0.9301) | ||
σ20 | 0.8940(1.4264) | 0.9445(1.5732) | 0.8930(1.4061) | 0.9410(1.5314) | ||
R(5) | 0.8980(0.9570) | 0.9455(0.9721) | 0.8970(0.9563) | 0.9450(0.9719) | ||
E(T) | 0.8975(0.0000×104) | 0.9480(3.8629×104) | 0.9035 (0.0000×104) | 0.9465(3.8199×104) | ||
X0(10) | 0.9095(0.0000) | 0.9580(11.3939) | 0.9315(0.0000) | 0.9665(11.4959) | ||
RUL(4) | 0.9040(148.1373) | 0.9540(532.5225) | 0.9050 (139.0714) | 0.9560(498.6037) | ||
(10,8) | a | 0.8990(0.2619) | 0.9520(0.4101) | 0.8915(0.2559) | 0.9400(0.4026) | |
b | 0.9020(1.5987) | 0.9515(1.7099) | 0.9045(1.5994) | 0.9525(1.7103) | ||
c | 0.9030(-0.4769) | 0.9540(-0.3841) | 0.8960(-0.4880) | 0.9450(-0.4033) | ||
d | 0.8975(1.7399) | 0.9485(1.7812) | 0.9035(1.7430) | 0.9515(1.7843) | ||
μ0 | 0.9060(0.6921) | 0.9515(0.7903) | 0.8885(0.6862) | 0.9425(0.7833) | ||
σ20 | 0.9045(1.3143) | 0.9540(1.4152) | 0.8950(1.3016) | 0.9430(1.3910) | ||
R(5) | 0.9035(0.9563) | 0.9590(0.9697) | 0.8965(0.9560) | 0.9485(0.9696) | ||
E(T) | 0.9020(0.0000×104) | 0.9540(3.3212×104) | 0.9025(0.0000×104) | 0.9545(3.3070×104) | ||
X0(10) | 0.9090(0.0000) | 0.9555(10.4619) | 0.9170(0.0000) | 0.9590(10.3875) | ||
RUL(4) | 0.9085(103.6819) | 0.9560(345.9644) | 0.9160(111.2404) | 0.9655(375.1649) |
Parameter | 90% | Length | 95% | Length |
a | (-0.0739, 0.4786) | 0.5525 | (-0.1286, 0.5323) | 0.6609 |
b | (-0.3882, 0.5149) | 0.9031 | (-0.4743, 0.6016) | 1.0759 |
c | (3.7885, 4.1246) | 0.3361 | (3.7577, 4.1592) | 0.4015 |
d | (0.0953, 0.4927) | 0.3974 | (0.0571, 0.5302) | 0.4731 |
μ0 | (0.0000, 0.4786) | 0.4786 | (0.0000, 0.5323) | 0.5323 |
σ20 | (44.1879, 61.8413) | 17.6534 | (42.8486, 64.0231) | 21.1745 |
R(100) | (0.4704, 0.6095) | 0.1391 | (0.4570, 0.6196) | 0.1626 |
T | (0.0008, 1.3891)×104 | 1.3883×104 | (0.0005, 4.3066)×104 | 4.3061×104 |
RUL(126) | (0.0004, 0.9336)×104 | 0.9332×104 | (0.0003, 2.9032)×104 | 2.9029×104 |
X0(150) | (-62.2527, 76.5406) | 138.7933 | (-76.1094, 90.2592) | 166.3686 |
Case | a | b | c | d | L |
I | -0.25 | 1.20 | -0.80 | 1.60 | 5.15 |
II | -0.50 | 2.00 | -1.00 | 2.50 | 6.30 |
III | -0.75 | 3.00 | -1.20 | 3.20 | 7.80 |
IV | 0.30 | 1.00 | 0.20 | 0.60 | 8.20 |
(n,r) | 0.9 | 0.95 | 0.9 | 0.95 | |
a | b | ||||
(5,4) | 0.8980(5.7245) | 0.9535(6.8696) | 0.8910(5.0503) | 0.9490(6.0723) | |
(8,6) | 0.8930(3.6243) | 0.9530(4.3272) | 0.8935(3.1994) | 0.9505(3.8228) | |
(10,8) | 0.8945(2.7886) | 0.9420(3.3271) | 0.8920(2.4650) | 0.9460(2.9447) | |
c | d | ||||
(5,4) | 0.9065(1.6709) | 0.9520(1.9989) | 0.9050(0.7740) | 0.9455(0.9269) | |
(8,6) | 0.9080(1.0463) | 0.9565(1.2489) | 0.9085(0.4847) | 0.9570(0.5787) | |
(10,8) | 0.8955(0.8043) | 0.9430(0.9593) | 0.9005(0.3723) | 0.9440(0.4443) | |
μ0 | σ20 | ||||
(5,4) | 0.9030(2.0925) | 0.9530(2.4557) | 0.9015(1.9745) | 0.9515(2.4745) | |
(8,6) | 0.8960(1.4858) | 0.9500(1.7236) | 0.9090(1.1410) | 0.9585(1.3864) | |
(10,8) | 0.8920(1.2317) | 0.9410(1.4231) | 0.8935(0.8448) | 0.9455(1.0188) | |
R(5) | E(T) | ||||
(5,4) | 0.9090(0.7848) | 0.9585(0.8661) | 0.9025(5.4195×104) | 0.9525(5.8182×104) | |
(8,6) | 0.8960(0.6604) | 0.9505(0.7522) | 0.8955(5.0215×104) | 0.9505(5.4538×104) | |
(10,8) | 0.9105(0.5661) | 0.9585(0.6571) | 0.8915(4.5678×104) | 0.9405(5.2177×104) | |
X0(10) | RUL(4) | ||||
(5,4) | 0.9090(26.0166) | 0.9595(31.1272) | 0.9075(0.8362×103) | 0.9580(3.2902×103) | |
(8,6) | 0.9085(20.1517) | 0.9535(24.0043) | 0.9095(0.6120×103) | 0.9570(2.3715×103) | |
(10,8) | 0.9040(17.7753) | 0.9580(21.1568) | 0.9080(0.4568×103) | 0.9535(1.7337×103) |
(n,r) | 0.9 | 0.95 | 0.9 | 0.95 | |
a | b | ||||
(5,4) | 0.8965(10.3095) | 0.9475(12.3940) | 0.9005(9.6583) | 0.9485(11.6339) | |
(8,6) | 0.9020(6.5242) | 0.9500(7.8010) | 0.9015(6.1281) | 0.9515(7.3373) | |
(10,8) | 0.8930(5.0067) | 0.9425(5.9777) | 0.8920(4.7051) | 0.9440(5.6210) | |
c | d | ||||
(5,4) | 0.9020(1.6707) | 0.9530(1.9987) | 0.9010(0.7740) | 0.9470(0.9267) | |
(8,6) | 0.8995(1.0471) | 0.9495(1.2498) | 0.8995(0.4847) | 0.9530(0.5789) | |
(10,8) | 0.9045(0.8040) | 0.9515(0.9589) | 0.8960(0.3723) | 0.9495(0.4441) | |
μ0 | σ20 | ||||
(5,4) | 0.8960(3.4859) | 0.9430(4.0890) | 0.9020(2.3736) | 0.9505(2.9752) | |
(8,6) | 0.8975(2.4609) | 0.9505(2.8554) | 0.9020(1.3195) | 0.9505(1.6046) | |
(10,8) | 0.8905(1.9922) | 0.9420(2.3054) | 0.9015(0.9827) | 0.9530(1.1847) | |
R(5) | E(T) | ||||
(5,4) | 0.9075(0.8565) | 0.9525(0.9185) | 0.8960(6.8449×104) | 0.9435(7.2617×104) | |
(8,6) | 0.8920(0.7815) | 0.9475(0.8611) | 0.8975(6.3119×104) | 0.9515(6.9116×104) | |
(10,8) | 0.8910(0.7136) | 0.9415(0.8020) | 0.8915(6.0068×104) | 0.9420(6.5718×104) | |
X0(10) | RUL(4) | ||||
(5,4) | 0.9075(39.8173) | 0.9535(47.4507) | 0.9085(0.9394×103) | 0.9580(3.7096×103) | |
(8,6) | 0.9095(29.3593) | 0.9540(34.8016) | 0.9090(0.7060×103) | 0.9585(2.7778×103) | |
(10,8) | 0.9070(24.8711) | 0.9565(29.4950) | 0.9075(0.5405×103) | 0.9570(2.1125×103) |
(n,r) | 0.9 | 0.95 | 0.9 | 0.95 | |
a | b | ||||
(5,4) | 0.8925(2.0082) | 0.9470(2.4155) | 0.9065(1.1180) | 0.9500(1.3433) | |
(8,6) | 0.8940(1.2607) | 0.9460(1.5079) | 0.8945(0.7043) | 0.9510(0.8417) | |
(10,8) | 0.9020(0.9708) | 0.9500(1.1593) | 0.8990(0.5423) | 0.9515(0.6473) | |
c | d | ||||
(5,4) | 0.8935(1.6708) | 0.9475(1.9989) | 0.8975(0.7740) | 0.9525(0.9268) | |
(8,6) | 0.9070(1.0463) | 0.9560(1.2489) | 0.9080(0.4847) | 0.9570(0.5787) | |
(10,8) | 0.8975(0.8042) | 0.9505(0.9592) | 0.8980(0.3725) | 0.9455(0.4443) | |
μ0 | σ20 | ||||
(5,4) | 0.8945(1.3499) | 0.9440(1.5745) | 0.8945(2.6428) | 0.9475(3.3125) | |
(8,6) | 0.8910(0.9320) | 0.9455(1.1026) | 0.9090(1.4650) | 0.9590(1.7801) | |
(10,8) | 0.9000(0.7308) | 0.9460(0.8707) | 0.9025(1.0978) | 0.9500(1.3239) | |
R(5) | E(T) | ||||
(5,4) | 0.9005(0.4893) | 0.9500(0.5779) | 0.8940(3.8581×104) | 0.9440(4.9431×104) | |
(8,6) | 0.8980(0.3236) | 0.9535(0.3857) | 0.8910(1.0701×104) | 0.9455(1.7598×104) | |
(10,8) | 0.9080(0.2481) | 0.9495(0.2961) | 0.9000(2.1828×103) | 0.9465(4.1033×103) | |
X0(10) | RUL(4) | ||||
(5,4) | 0.9040(19.7837) | 0.9535(23.7148) | 0.9095(172.1151) | 0.9575(611.4828) | |
(8,6) | 0.9085(16.5558) | 0.9560(19.8108) | 0.9000(42.7509) | 0.9510(108.4594) | |
(10,8) | 0.9000(15.4118) | 0.9535(18.4244) | 0.8935(22.2517) | 0.9425(38.8602) |
(n,r) | parameter | GCI/GPI | bootstrap-p CI | |||
0.9 | 0.95 | 0.9 | 0.95 | |||
(5,4) | a | 0.8970(2.7521) | 0.9495(3.3031) | 0.8955(2.6992) | 0.9460(3.2173) | |
b | 0.9020(2.0580) | 0.9500(2.4701) | 0.9065(2.0494) | 0.9425(2.4435) | ||
c | 0.8965(1.6708) | 0.9475(1.9989) | 0.9050(1.6712) | 0.9430(1.9996) | ||
d | 0.8975(0.7740) | 0.9525(0.9268) | 0.9130(0.7739) | 0.9545(0.9269) | ||
μ0 | 0.8990(1.2206) | 0.9480(1.4245) | 0.8900(1.1859) | 0.9435(1.3769) | ||
σ20 | 0.8965(1.6029) | 0.9475(2.0091) | 0.9070(1.4805) | 0.9425(1.7927) | ||
R(5) | 0.9035(0.6619) | 0.9530(0.7557) | 0.9080(0.6433) | 0.9575(0.7340) | ||
E(T) | 0.8995(4.3254×104) | 0.9480(4.6507×104) | 0.8900(4.2568×104) | 0.9425(4.6248×104) | ||
X0(10) | 0.9045(17.4588) | 0.9535(20.9393) | 0.9080(17.0667) | 0.9585(20.4164) | ||
RUL(4) | 0.9095(0.7176×103) | 0.9560(2.7925×103) | 0.9150(1.0513×103) | 0.9595(4.0867×103) | ||
(8,6) | a | 0.9025(1.7323) | 0.9470(2.0689) | 0.9075(1.7223) | 0.9560(2.0526) | |
b | 0.8965(1.3058) | 0.9460(1.5596) | 0.9060(1.3045) | 0.9555(1.5551) | ||
c | 0.8960(1.0462) | 0.9490(1.2487) | 0.9085(1.0471) | 0.9545(1.2495) | ||
d | 0.8965(0.4847) | 0.9455(0.5786) | 0.9090(0.4849) | 0.9550(0.5787) | ||
μ0 | 0.9040(0.8624) | 0.9475(0.9978) | 0.9060(0.8752) | 0.9555(1.0068) | ||
σ20 | 0.8970(0.8852) | 0.9490(1.0761) | 0.9085(0.8621) | 0.9465(1.0335) | ||
R(5) | 0.9030(0.4646) | 0.9495(0.5502) | 0.9040(0.4719) | 0.9465(0.5564) | ||
E(T) | 0.9035(3.8440×104) | 0.9475(4.2632×104) | 0.9070(3.8062×104) | 0.9565(4.3076×104) | ||
X0(10) | 0.9065(14.1289) | 0.9560(16.8762) | 0.9095(14.1436) | 0.9575(16.8773) | ||
RUL(4) | 0.9030(0.4656×103) | 0.9475(1.7509×103) | 0.9185(0.4966×103) | 0.9565(1.8667×103) | ||
(10,8) | a | 0.9020(1.3350) | 0.9520(1.5926) | 0.8915(1.3304) | 0.9450(1.5851) | |
b | 0.9005(1.0058) | 0.9505(1.1999) | 0.9055(1.0064) | 0.9525(1.1990) | ||
c | 0.8975(0.8042) | 0.9505(0.9592) | 0.9040(0.8042) | 0.9470(0.9593) | ||
d | 0.8980(0.3725) | 0.9465(0.4443) | 0.9065(0.3724) | 0.9560(0.4442) | ||
μ0 | 0.9030(0.7263) | 0.9535(0.8376) | 0.8960(0.7174) | 0.9470(0.8268) | ||
σ20 | 0.9025(0.6659) | 0.9500(0.8030) | 0.8955(0.6544) | 0.9485(0.7827) | ||
R(5) | 0.9035(0.3685) | 0.9525(0.4397) | 0.8950(0.3635) | 0.9455(0.4331) | ||
E(T) | 0.9025(3.3101×104) | 0.9520(3.9055×104) | 0.8945(3.2908×104) | 0.9450(3.8446×104) | ||
X0(10) | 0.9030(13.0539) | 0.9505(15.5771) | 0.9050(12.9732) | 0.9555(15.4825) | ||
RUL(4) | 0.9080(0.3250×103) | 0.9510(1.1652×103) | 0.9245(0.3730×103) | 0.9655(1.3665×103) |
(n,r) | parameter | LCL in GPQ method | LCL in bootstrap-p method | |||
0.9 | 0.95 | 0.9 | 0.95 | |||
(5,4) | a | 0.9035(-1.3223) | 0.9470(-1.6337) | 0.8925(-1.3213) | 0.9445(-1.6200) | |
b | 0.9010(0.4076) | 0.9545(0.1749) | 0.8935(0.4172) | 0.9555(0.1905) | ||
c | 0.8940(-1.4176) | 0.9435(-1.5874) | 0.9130(-1.4733) | 0.9640(-1.6779) | ||
d | 0.9010(1.2911) | 0.9535(1.2043) | 0.9040(1.3052) | 0.9570(1.2183) | ||
μ0 | 0.9025(0.0831) | 0.9485(0.0441) | 0.9035(0.0862) | 0.9470(0.0468) | ||
σ20 | 0.9035(0.6658) | 0.9425(0.5840) | 0.9175(0.6349) | 0.9640(0.5390) | ||
R(5) | 0.9430(0.4179) | 0.9570(0.3116) | 0.9245(0.4328) | 0.9595(0.3310) | ||
E(T) | 0.9035(0.1479×104) | 0.9510(0.0675×104) | 0.8920(0.1548×104) | 0.9430(0.0677×104) | ||
X0(10) | 0.8915(-1.3915) | 0.9425(-3.0086) | 0.8805(-1.3434) | 0.9375(-2.9459) | ||
RUL(4) | 0.9410(2.2967) | 0.9575(1.6463) | 0.9260(2.4000) | 0.9650(1.7000) | ||
(8,6) | a | 0.9005(-0.9378) | 0.9495(-1.1313) | 0.9035(-0.9058) | 0.9485(-1.0969) | |
b | 0.8915(0.7058) | 0.9455(0.5602) | 0.9135(0.6798) | 0.9565(0.5356) | ||
c | 0.9005(-1.2072) | 0.9495(-1.3165) | 0.9180(-1.2202) | 0.9660(-1.3435) | ||
d | 0.8940(1.4130) | 0.9465(1.3590) | 0.9050(1.4146) | 0.9435(1.3605) | ||
μ0 | 0.8965(0.0942) | 0.9475(0.0551) | 0.9025(0.0961) | 0.9490(0.0549) | ||
σ20 | 0.9015(0.7492) | 0.9515(0.6879) | 0.9165(0.7385) | 0.9655(0.6693) | ||
R(5) | 0.9095(0.5941) | 0.9560(0.5075) | 0.9185(0.5863) | 0.9580(0.4999) | ||
E(T) | 0.9030(0.0551×104) | 0.9535(0.0189×104) | 0.9135(0.0472×104) | 0.9575(0.0136×104) | ||
X0(10) | 0.8950(-1.2837) | 0.9430(-2.7351) | 0.8870(-1.1942) | 0.9405(-2.6476) | ||
RUL(4) | 0.9060(2.7260) | 0.9530(2.0512) | 0.9145(2.7046) | 0.9625(2.0479) | ||
(10,8) | a | 0.9015(-0.7763) | 0.9525(-0.9249) | 0.9055(-0.7809) | 0.9560(-0.9278) | |
b | 0.8995(0.8159) | 0.9490(0.7041) | 0.8905(0.8152) | 0.9500(0.7039) | ||
c | 0.8955(-1.1031) | 0.9445(-1.1882) | 0.9065(-1.1142) | 0.9560(-1.2075) | ||
d | 0.9025(1.4501) | 0.9525(1.4087) | 0.9065(1.4532) | 0.9550(1.4119) | ||
μ0 | 0.9030(0.1066) | 0.9535(0.0640) | 0.9040(0.1079) | 0.9565(0.0659) | ||
σ20 | 0.8965(0.8010) | 0.9485(0.7494) | 0.9070(0.7934) | 0.9555(0.7366) | ||
R(5) | 0.9050(0.6699) | 0.9520(0.6012) | 0.8940(0.6735) | 0.9485(0.6061) | ||
E(T) | 0.9065(0.0268×104) | 0.9515(0.0111×104) | 0.8885(0.0353×104) | 0.9430(0.0162×104) | ||
X0(10) | 0.8975(-1.2025) | 0.9465(-2.5920) | 0.8960(-1.2022) | 0.9450(-2.5857) | ||
RUL(4) | 0.8930(2.8285) | 0.9525(2.1574) | 0.9120(2.8721) | 0.9590 (2.1891) |
(n,r) | parameter | UCL in GPQ method | UCL in bootstrap-p method | |||
0.9 | 0.95 | 0.9 | 0.95 | |||
(5,4) | a | 0.9045(0.8079) | 0.9490(1.1184) | 0.8925(0.7807) | 0.9470(1.0792) | |
b | 0.9030(2.0010) | 0.9525(2.2329) | 0.8950(2.0136) | 0.9480(2.2399) | ||
c | 0.9025(-0.1204) | 0.9540(0.0834) | 0.8925(-0.1762) | 0.9410(-0.0067) | ||
d | 0.8955(1.8914) | 0.9455(1.9784) | 0.9070(1.9054) | 0.9575(1.9922) | ||
μ0 | 0.9055(1.0601) | 0.9505(1.2647) | 0.8920(1.0396) | 0.9430(1.2327) | ||
σ20 | 0.9025(1.8569) | 0.9540(2.1869) | 0.8870(1.7710) | 0.9390(2.0194) | ||
R(5) | 0.8925(0.9535) | 0.9450(0.9735) | 0.8855(0.9539) | 0.9440(0.9743) | ||
E(T) | 0.9020(0.0000×104) | 0.9485(4.3930×104) | 0.9025(0.0000×104) | 0.9470(4.3245×104) | ||
X0(10) | 0.9425(0.0000) | 0.9580(14.4502) | 0.9330(0.0000) | 0.9705(14.1208) | ||
RUL(4) | 0.9030(256.4904) | 0.9545(984.8106) | 0.9035(274.7000) | 0.9550(1053.0000) | ||
(8,6) | a | 0.9020(0.4084) | 0.9540(0.6010) | 0.9150(0.4357) | 0.9570(0.6254) | |
b | 0.9025(1.7209) | 0.9450(1.8660) | 0.9045(1.6961) | 0.9445(1.8400) | ||
c | 0.8950(-0.3928) | 0.9445(-0.2703) | 0.8930(-0.4057) | 0.9425(-0.2964) | ||
d | 0.8925(1.7898) | 0.9515(1.8437) | 0.9135(1.7916) | 0.9615(1.8454) | ||
μ0 | 0.9030(0.7902) | 0.9565(0.9176) | 0.9135(0.8048) | 0.9570(0.9301) | ||
σ20 | 0.8940(1.4264) | 0.9445(1.5732) | 0.8930(1.4061) | 0.9410(1.5314) | ||
R(5) | 0.8980(0.9570) | 0.9455(0.9721) | 0.8970(0.9563) | 0.9450(0.9719) | ||
E(T) | 0.8975(0.0000×104) | 0.9480(3.8629×104) | 0.9035 (0.0000×104) | 0.9465(3.8199×104) | ||
X0(10) | 0.9095(0.0000) | 0.9580(11.3939) | 0.9315(0.0000) | 0.9665(11.4959) | ||
RUL(4) | 0.9040(148.1373) | 0.9540(532.5225) | 0.9050 (139.0714) | 0.9560(498.6037) | ||
(10,8) | a | 0.8990(0.2619) | 0.9520(0.4101) | 0.8915(0.2559) | 0.9400(0.4026) | |
b | 0.9020(1.5987) | 0.9515(1.7099) | 0.9045(1.5994) | 0.9525(1.7103) | ||
c | 0.9030(-0.4769) | 0.9540(-0.3841) | 0.8960(-0.4880) | 0.9450(-0.4033) | ||
d | 0.8975(1.7399) | 0.9485(1.7812) | 0.9035(1.7430) | 0.9515(1.7843) | ||
μ0 | 0.9060(0.6921) | 0.9515(0.7903) | 0.8885(0.6862) | 0.9425(0.7833) | ||
σ20 | 0.9045(1.3143) | 0.9540(1.4152) | 0.8950(1.3016) | 0.9430(1.3910) | ||
R(5) | 0.9035(0.9563) | 0.9590(0.9697) | 0.8965(0.9560) | 0.9485(0.9696) | ||
E(T) | 0.9020(0.0000×104) | 0.9540(3.3212×104) | 0.9025(0.0000×104) | 0.9545(3.3070×104) | ||
X0(10) | 0.9090(0.0000) | 0.9555(10.4619) | 0.9170(0.0000) | 0.9590(10.3875) | ||
RUL(4) | 0.9085(103.6819) | 0.9560(345.9644) | 0.9160(111.2404) | 0.9655(375.1649) |
Parameter | 90% | Length | 95% | Length |
a | (-0.0739, 0.4786) | 0.5525 | (-0.1286, 0.5323) | 0.6609 |
b | (-0.3882, 0.5149) | 0.9031 | (-0.4743, 0.6016) | 1.0759 |
c | (3.7885, 4.1246) | 0.3361 | (3.7577, 4.1592) | 0.4015 |
d | (0.0953, 0.4927) | 0.3974 | (0.0571, 0.5302) | 0.4731 |
μ0 | (0.0000, 0.4786) | 0.4786 | (0.0000, 0.5323) | 0.5323 |
σ20 | (44.1879, 61.8413) | 17.6534 | (42.8486, 64.0231) | 21.1745 |
R(100) | (0.4704, 0.6095) | 0.1391 | (0.4570, 0.6196) | 0.1626 |
T | (0.0008, 1.3891)×104 | 1.3883×104 | (0.0005, 4.3066)×104 | 4.3061×104 |
RUL(126) | (0.0004, 0.9336)×104 | 0.9332×104 | (0.0003, 2.9032)×104 | 2.9029×104 |
X0(150) | (-62.2527, 76.5406) | 138.7933 | (-76.1094, 90.2592) | 166.3686 |