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

-3.0 -2.5 -2.0 -1.5

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


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