Asymmetric distributions such as the skew-normal (SN) distribution have been used for modeling the unit-to-unit variation of the degradation rate of products, and its superiority for model fitting and lifetime estimation has been demonstrated in literature. However, there still exist two issues that need to be further studied. That is, how to simply estimate the parameters of the degradation model with asymmetric random effects and whether the asymmetry or skewness affects the remaining useful life (RUL) prediction. Driven by these two issues, the parameter estimation and RUL prediction of the degradation model with general random effects are first introduced and a two-step maximum likelihood estimation (MLE) method for parameter estimation is proposed, which could greatly simplify the parameter estimation of the degradation model with asymmetric random effects. Then, the SN distribution is used as an example to analyze the second issue regarding the effect of model mis-specification on RUL prediction based on regression models and Wiener processes with different asymmetric random effects in theory. Finally, the parameter estimation and mis-specification analysis of degradation models with other asymmetric random effects are conducted through numerical examples and two case studies. The results demonstrate that the proposed two-step MLE method is satisfactory and the effect of asymmetry on RUL prediction is negligible. Therefore, there is no need to consider the asymmetry of random effects for RUL prediction.
XiXZhouD. Prognostics of fractional degradation processes with state-dependent delay. Proc IMechE, Part O: J Risk and Reliability2021; 236: 114–124.
2.
PechtM. Prognostics and health management of electronics. New Jersey: John Wiley & Sons; 2008.
3.
XiaTDongYXiaoL, et al. Recent advances in prognostics and health management for advanced manufacturing paradigms. Reliab Eng Syst Saf2018; 178: 255–268.
4.
van NoortwijkJM. A survey of the application of gamma processes in maintenance. Reliab Eng Syst Saf2009; 94: 2–21.
5.
SiXWangWHuC, et al. Remaining useful life estimation – a review on the statistical data driven approaches. Eur J Operat Res2011; 213: 1–14.
6.
YeZ-SXieM. Stochastic modelling and analysis of degradation for highly reliable products. Appl Stoch Models Bus Ind2014; 31: 16–32.
7.
ZhangZSiXHuC, et al. Degradation data analysis and remaining useful life estimation: a review on Wiener-process-based methods. Eur J Operat Res2018; 271: 775–796.
8.
LuCJMeekerWO. Using degradation measures to estimate a time-to-failure distribution. Technometrics1993; 35: 161–174.
9.
LiaoGYinHChenM, et al. Remaining useful life prediction for multi-phase deteriorating process based on Wiener process. Reliab Eng Syst Saf2021; 207: 107361.
10.
YangJTangSFangP, et al. Remaining useful life prediction of implicit linear Wiener degradation process based on multi-source information. Proc IMechE, Part O: J Risk and Reliability. Epub ahead of print 19November2022. DOI: 10.1177/1748006X22113260.
11.
CuiJCaoLZhangT. A two-stage Gaussian process regression model for remaining useful prediction of bearings. Proc IMechE, Part O: J Risk and Reliability. Epub ahead of print 19November2023. DOI: 10.1177/1748006X221141744.
12.
SunLZhaoFBalakrishnanN, et al. A nonlinear Wiener degradation model integrating degradation data under accelerated stresses and real operating environment. Proc IMechE, Part O: J Risk and Reliability2020; 235: 356–373.
13.
CholetteMEYuHBorghesaniP, et al. Degradation modeling and condition-based maintenance of boiler heat exchangers using gamma processes. Reliab Eng Syst Saf2019; 183: 184–196.
14.
YeZ-SChenN. The inverse Gaussian process as a degradation model. Technometrics2013; 56: 302–311.
15.
PengC-Y. Inverse gaussian processes with random effects and explanatory variables for degradation data. Technometrics2015; 57: 100–111.
16.
PengCTsengS. Mis-specification analysis of linear degradation models. IEEE Trans Reliab2009; 58: 444–455.
17.
TangSYuCWangX, et al. Remaining useful life prediction of Lithium-ion batteries based on the Wiener process with measurement error. Energies2014; 7: 520–547.
18.
TangSGuoXYuC, et al. Real time remaining useful life prediction based on nonlinear Wiener based degradation processes with measurement errors. J Cent South Univ2014; 21: 4509–4517.
19.
SiX-SWangWHuC-H, et al. Estimating remaining useful life with three-source variability in degradation modeling. IEEE Trans Reliab2014; 63: 167–190.
20.
ZhengJFSiXSHuCH, et al. A nonlinear prognostic model for degrading systems with three-source variability. IEEE Trans Reliab2016; 65: 736–750.
21.
GebraeelNZLawleyMALiR, et al. Residual-life distributions from component degradation signals: a Bayesian approach. IIE Trans2005; 37: 543–557.
22.
GebraeelN. Sensory-updated residual life distributions for components with exponential degradation patterns. IEEE Trans Automat Sci Eng2006; 3: 382–393.
23.
GebraeelNPanJ. Prognostic degradation models for computing and updating residual life distributions in a time-varying environment. IEEE Trans Reliab2008; 57: 539–550.
24.
GebraeelNElwanyAJingP. Residual life predictions in the absence of prior degradation knowledge. IEEE Trans Reliab2009; 58: 106–117.
25.
LiNLeiYLinJ, et al. An improved exponential model for predicting remaining useful life of rolling element bearings. IEEE Trans Indu Electron2015; 62: 7762–7773.
26.
SiXWangWChenM, et al. A degradation path-dependent approach for remaining useful life estimation with an exact and closed-form solution. Eur J Operat Res2013; 226: 53–66.
27.
SiXWangWHuC, et al. A Wiener-process-based degradation model with a recursive filter algorithm for remaining useful life estimation. Mech Syst Signal Process2013; 35: 219–237.
28.
SiX-S. An adaptive prognostic approach via nonlinear degradation modeling: application to battery data. IEEE Trans Indu Electron2015; 62: 5082–5096.
29.
WangZHuCFanH. Real-time remaining useful life prediction for a nonlinear degrading system in service: application to bearing data. IEEE ASME Trans Mechatron2018; 23: 211–222.
30.
WangXBalakrishnanNGuoB. Residual life estimation based on a generalized Wiener degradation process. Reliab Eng Syst Saf2014; 124: 13–23.
31.
WangZ-QHuC-HWangW, et al. An additive Wiener process-based prognostic model for hybrid deteriorating systems. IEEE Trans Reliab2014; 63: 208–222.
32.
ChakrabortySGebraeelNLawleyM, et al. Residual-life estimation for components with non-symmetric priors. IIE Trans2009; 41: 372–387.
33.
ChenXSunX. Reliability assessment for products with two performance characteristics based on marginal stochastic processes and copulas. Commun Stat Simul Comput2022; 51: 3621–3644.
34.
PanDLiuJ-BHuangF, et al. A Wiener process model with truncated normal distribution for reliability analysis. Appl Math Modell2017; 50: 333–346.
35.
YeZ-SChenNShenY. A new class of Wiener process models for degradation analysis. Reliab Eng Syst Saf2015; 139: 58–67.
36.
ZhaiQChenPHongL, et al. A random-effects Wiener degradation model based on accelerated failure time. Reliab Eng Syst Saf2018; 180: 94–103.
37.
YanBWangHMaX. Modeling left-truncated degradation data using random drift-diffusion Wiener processes. Qual Technol Quantitat Manage2023; 21(2): 200–223.
38.
WangZZhaiQChenP. Degradation modeling considering unit-to-unit heterogeneity-a general model and comparative study. Reliab Eng Syst Saf2021; 216: 107897.
39.
PengC-YTsengS-T. Statistical lifetime inference with Skew-Wiener linear degradation models. IEEE Trans Reliab2013; 62: 338–350.
40.
HuangZXuZKeX, et al. Remaining useful life prediction for an adaptive skew-Wiener process model. Mech Syst Signal Process2017; 87: 294–306.
41.
WuDJiaMCaoY, et al. Remaining useful life estimation based on a nonlinear Wiener process model with CSN random effects. Measurement2022; 205: 112232.
42.
TsaiC-CTsengS-TBalakrishnanN. Mis-specification analyses of gamma and Wiener degradation processes. J Statist Plann Inference2011; 141: 3725–3735.
43.
TsengS-TYaoY-C. Mis-specification analysis of Gamma with inverse Gaussian degradation processes. In: ChenD-GLioYNgHKT, et al. (eds) Statistical modeling for degradation data. Singapore: Springer Singapore; 2017, pp.193–208.
44.
SunLBalakrishnanNZhaoF-C, et al. Mis-specification analysis of the impact of covariates on the diffusion coefficient in Wiener degradation process. Commun Stat Simul Comput2022; 51(6): 3204–3222.
45.
ZhangFNgHKTShiY. Mis-specification analysis of Wiener degradation models by using f-divergence with outliers. Reliab Eng Syst Saf2020; 195: 106751.
46.
WangXBalakrishnanNGuoB. Mis-specification analyses of nonlinear wiener process-based degradation models. Commun Stat Simul Comput2015; 45: 814–832.
47.
TangSGuoXZhouZ. Mis-specification analysis of linear Wiener process–based degradation models for the remaining useful life estimation. Proc IMechE, Part O: J Risk and Reliability2014; 228: 478–487.
48.
TangSWangFSunX, et al. Unbiased parameters estimation and mis-specification analysis of Wiener process-based degradation model with random effects. Appl Math Modell2022; 109: 134–160.
49.
AzzaliniA. A class of distributions which includes the normal ones. Scand J Statist1985; 12: 171–178.
50.
AzzaliniAValleAD. The multivariate skew-normal distribution. Biometrika1996; 83: 715–726.
51.
AzzaliniACapitanioA. Statistical applications of the multivariate skew normal distribution. JR Stat Soc Series B: Stat Methodol1999; 61: 579–602.
52.
WangXBalakrishnanNGuoB. Residual life estimation based on a generalized Wiener process with skew-normal random effects. Commun Stat Simul Comput2016; 45: 2158–2181.
53.
PanDLiuJ-BYangW. A new result on lifetime estimation based on skew-Wiener degradation model. Statist Probab Lett2018; 138: 157–164.
54.
HaoSYangJBerenguerC. Degradation analysis based on an extended inverse Gaussian process model with skew-normal random effects and measurement errors. Reliab Eng Syst Saf2019; 189: 261–270.
55.
ChenXSunXDingX, et al. The inverse Gaussian process with a skew-normal distribution as a degradation model. Commun Stat Simul Comput2020; 49: 2827–2843.
56.
HanYMaCTangS, et al. Residual life estimation of lithium-ion batteries based on nonlinear Wiener process with measurement error. Proc IMechE, Part O: J Risk and Reliability2022; 237: 133–151.
57.
XuXYuCTangS, et al. Remaining useful life prediction of lithium-ion batteries based on Wiener processes with considering the relaxation effect. Energies2019; 12: 1685.
58.
PewseyA. Problems of inference for Azzalini’s skewnormal distribution. J Appl Stat2000; 27: 859–870.
59.
SartoriN. Bias prevention of maximum likelihood estimates for scalar skew normal and skew t distributions. J Stati Plann Inference2006; 136: 4259–4275.
60.
TangSYuCSunX, et al. A note on parameters estimation for nonlinear wiener processes with measurement errors. IEEE Access2019; 7: 176756–176766.
61.
KeatsJB. Statistical methods for reliability data. J Qual Technol2018; 31: 466–468.
62.
HenzeN. A probabilistic representation of the ‘Skew-Normal’ distribution. Scand J Statist1986; 13: 271–275.
