The simple case f = c in found in [BN04] where f (t) = c(t) = ln(1 +1). In this case the calculations reduce to
UJ V(Y}* — YD/(f f ) — f(t\))) , and thus the estimate is not consistent in this case.
3.2 Nonlinear estimation
In the nonlinear case, whatever the hypothesis about the correlation of the noises is, the predictor of 0l is found by least square minimization
= argminoeRP Y - g(f, a))'S-1(Yi - g(t, a)) i = 1..n where g(ti,a) is the vector in Rfi of the values (g(tij,a))j=i..fi. Under Hl, [LM93] provide direct estimation of the distribution of the 0i via a parametric assumption 0i ~ N(fl,£), i = 1..n and estimate f3 and S in a nonlinear mixed effect model with maximum likelihood estimators. Here we do not make such assumption. Thus we have to construct some predictor 0i of the random coefficient 0i and shall plug these predictors in some non parametric estimate of Fg in the next section. Closed form for these predictors are not available but numerous numerical optimization procedures exist. We illustrate two well known algorithms on the Fatigue crack size propagation Alloy-A data ([LM93]). These data represent the growth of cracks in metal for 21 test units with a maximum of twelve equally spaced measurements. Testing was stopped if the crack length exceeded 1.60 inches which defines the threshold zo. No traumatic failure mode is defined in this example. The Paris growth curve
/, i__<\ (.-. .-■ 2-m 2 - m \ 2-m g(t,m,C) = [0.9 2 + Cjn tj with unit-to-unit coefficients 9i = (mi,Ci) is fitted on each item. The aim is thus to estimate the distribution function F(m C) with the noised measurements of degradation under hypothesis Hl.
Millions of cycles
Millions of cycles
Fig. 1. Fatigue Crack Size for Alloy-A and Nonlinear fitting of Paris curves with nls and gnls n-1-1-1-1-1-r
Millions of cycles
Millions of cycles
Fig. 1. Fatigue Crack Size for Alloy-A and Nonlinear fitting of Paris curves with nls and gnls
Under Hl, We used a Gauss-Newton algorithm implemented by Bates and DebRoy in nls function of Splus and R softwares ([BC92]). A similar algorithm minimizing generalized least squares and allowing correlated errors is obviously useful under H2 and H3. Thus we tested also the gnls function by Pinheiro and Bates in R. Both methods give a very well fit to the data but the predictions present some little difference as it is shown in figure 2.
In fact we could compare the methods by calculating the empirical mean square error. In this case, the mean square error of method nls in the lowest (MSEgnls = 0.0175, MSEnls = 0.0105).
3.3 Estimation of the reliability functions
If the Oi where known, an estimate of the distribution function Fg would be the classical empirical distribution function Fg. If the predictions O' of the random coefficients O' are consistent then a nonparametric estimation of Fg is
i=i and when a traumatic failure mode exists we can plug the 's in a NelsonAalen type estimate of the cumulative hazard function in the degradation space A(z) = f0 X(u)d,u to get
For further details we refer to [BN04]. The overall survival function S of the failure time U = min(T0,T) is estimated by
Example : As an illustration we consider three simulations of n=100 degradation curves Z(t,0i,02) = e61 (1 + t)°2, t e [0,12] with multiplicative noise and traumatic failure times with a hazard rate in the degradation space of Weibull-type X(t) = [3/a(x/a)/3-i, a = 5, ¡3 = 2.5 and (0i, 02) is a gaussian vector with mean (-2,2) and Var^ =Var6>2 = 0.12, Corr(9i,92) = -0.7. In this case the results of section 3.1 hold with Yj =lnZj and the additive errors e follow H1 or H2 with & = 0.9 or H3 with c(t)=ln(1+t) and a, = 0.05 for all i. The path i is censored by the minimum min(T0),T'1, 12) and the estimation of the coefficients are carried out according to section 3.1.
Fig. 3. 3 simulated path with CAR(1) and Wiener-type noises
The estimation behaves well under H1 and H2 but is less efficient under H3 (see the right hand side of figure 4). In fact, figure 5 shows that under H1 the distribution function of the random variable 0i is well estimated both by F61 and F61 but not under H3.
e.d.f or A, with known Ai or predictions e.d.f or A, with known Ai or predictions e.d.f or A, with known Ai or predictions
e.d.f or A, with known Ai or predictions
Fig. 4. Estimation of Fg1 under H1 and H3
Finally, we present an estimation of the cumulative hazard rate functuion A in the degradation space under H1 and a Monte Carlo simulation giving a 95% empirical confidence band under H2.
95% empirical confidence band, CAR model
95% empirical confidence band, CAR model
degradation space degradation space
Fig. 5. Estimation of A under H1 and 95 % empirical confidence band of A under H2
[AG01] Aalen, O.O., Gjessing, H.K.: Understanding the shape of the hazard rate: a process point of view. Statist. Sci. 16(1) 1-22 (2001) [BBK04] Bagdonavicius V, Bikelis A, Kazakevicius V.: Statistical analysis of linear degradation and failure time data with multiple failure modes. Lifetime Data Anal., 10(1), 65-81 (2004) [BC92] Bates, D. M., Chambers, J. M.: Nonlinear models, Chapter 10 of Statistical Models in S, eds J.M. Chambers and T.J. Hastie, Wadsworth & Brooks/Cole (1992) [BK85] Bogdanoff, J. L., Kozin F.: Probability Models of Cumulative
Damage. Wiley, New York (1985) [BN97] Bagdonavicius V., Nikulin M.S.: Transfer functionals and semi-
parametric regression models. Biometrika, 84 365-78 (1997) [BN01] Bagdonavicius V., Nikulin M.S.: Estimation in degradation models with explanatory variables. Lifetime Data Anal., 7(1), 85-103 (2001)
[BN02] Bagdonavicius V., Nikulin M.S.: Accelerated Life Models : Modeling and Statistical Analysis. Chapman & Hall / CRC, Boca Raton (2002)
[BN04] Bagdonavicius, V., Nikulin, M., Semiparametric analysis of degradation and failure times data with covariates, in Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life Series : Statistics for Industry and Technology Nikulin, M.S.; Balakrishnan, N.; Mesbah, M.; Limnios, N. (Eds.). Birkauser (2004) [C0U04] Couallier, V., Comparison of parametric and semiparametric estimates in a degradation model with covariates and traumatic censoring in Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life Series : Statistics for Industry and Technology Nikulin, M.S.; Balakrishnan, N.; Mesbah, M.; Limnios, N. (Eds.). Birkauser (2004) [C0X99] Cox DR.: Some remarks on failure-times, surrogate markers, degradation, wear, and the quality of life. Lifetime Data Anal., 5(4), 307-14 (1999) [DN95] Doksum K.A., Normand S.L.: Gaussian models for degradation processes-Part I: Methods for the analysis of biomarker data. Lifetime Data Anal., 1(2), 131-44 (1995) [DL00] Duchesne T, Lawless J.: Alternative time scales and failure time models. Lifetime Data Anal., 6(2),157-79 (2000) [DL02] Duchesne T, Lawless J.: Semiparametric inference methods for general time scale models. Lifetime Data Anal., 8(3),263-76 (2002) [FIN03] Finkelstein MS.: A model of aging and a shape of the observed force of mortality. Lifetime Data Anal., 9(1),93-109 (2003)
[GLJ03] Girish, T., Lam, S.W.: Jayaram, S.J.: Reliability Prediction Using Degradation Data - A Preliminary Study Using Neural Network-based Approach. Safety and Reliability - Bedford & van Gelder(eds): Proceedings of ESREL 2003, European Safety and Reliability Conference, 15-18 June, Maastricht, The Netherlands. 681-88. (c) Swets & Zeitlinger, Lisse (2003) [HA03] Huang, W., Askin, R.G.: Reliability analysis of electronic devices with multiple competing failure modes involving performance aging degradation. Quality and Reliability Engineering International 19(3), 241-54 (2003) [KW04] Kahle, W., Wendt, H.: On a cumulative damage process and resulting first passages times. Applied Stochastic Models in Business and Industry, 20(1) 17-26 (2004) [LC04] Lawless, J., Crowder, M.: Covariates and random effects in a gamma process model with application to degradation and failure. Lifetime Data Anal., 10(3), 213-27 (2004) [LM93] Lu, C. J. , Meeker, W. Q.: Using degradation measures to estimate a time-to-failure distribution. Technometrics, 35, 161-174 (1993) [ME98] Meeker, W.Q., Escobar, L.: Statistical Analysis for Reliability
Data. John Wiley and Sons, New York (1998) [0C04] Branco de Oliveira, V.R., Colosimo E.A.: Comparison of Methods to Estimate the Time-to-failure Distribution in Degradation Tests. Quality and Reliability Engineering International, 20(4), 363-73 (2004)
[PT04] Padgett, W.J., Tomlinson, M.A.: Inference from Accelerated Degradation and Failure Data Based on Gaussian Process Models. Lifetime Data Anal., 10(2) 191-206 (2004) [RC00] Robinson, M.E., Crowder, M.J.: Bayesian methods for a growth-curve degradation model with repeated measures. Lifetime Data Anal., 6(4), 357-74 (2000) [SW03] Seber, G.A.F., Wild, C.J.: Nonlinear regression. John Wiley &
Sons, New Jersey (2003). [SIN95] Singpurwalla, N.D.: Survival in dynamic environments. Statist.
Sci. 10(1), 86-103 (1995) [WK04] Wendt, H., Kahle, W.: On parameter stochastic poisson process, in Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life, Series : Statistics for Industry and Technology, Nikulin, M.S.; Balakrishnan, N.; Mesbah, M.; Limnios, N. (Eds.), Birkauser, 473-86 (2004) [WHI95] Whitmore, G.A.: Estimating degradation by a Wiener diffusion process subject to measurement error. Lifetime Data Anal., 1(3), 307-19 (1995)
[WS97] Whitmore, G.A., Schenkelberg F.: Modelling accelerated degradation data using Wiener diffusion with a time scale transformation. Lifetime Data Anal., 3(1), 27-45 (1997)
[WCL98] Whitmore, G.A., Crowder, M.J., Lawless, J.F.: Failure inference from a marker process based on a bivariate Wiener model. Lifetime Data Anal., 4(3), 229-51 (1998) [WT97] Wulfsohn M.S., Tsiatis A.A.: A joint model for survival and longitudinal data measured with error. Biometrics 53 330-39 (1997) [YM97] Yashin, A.I., Manton, K.G.: Effects of unobserved and partially observed covariate processes on system failure: a review of models and estimation strategies. Statist. Sci., 12(1), 20-34 (1997) [YU03] Yu, H.F.: Designing an accelerated degradation experiment by optimizing the estimation of the percentile. Quality and Reliability Engineering International, 19(3), 197-214 (2003) [YU04] Yu, H.F.: Designing an accelerated degradation experiment with a reciprocal Weibull degradation rate. Journal of Statistical Planning and Inference, In press, corrected proof (2004)
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