Using ^u+v (y — yo)k-u+b-1e-[c+n(t)Y(v-y0)dy = 1 we get k
c + n(t)_
c + n(t)_
c + n(t) = cn-i(t) +i and then we get yo[c + n(t)] = yon(t)[cn-i(t) + i]
Hence, pk (t) are the probabilities of the Delaport distribution with parameters nCt), b and yo • r/(t). From (5) the random number of shocks @(t) can be interpreted as a sum of two independent random variables Wi(t) and W2(t) where W^t) is Poisson distributed with expectation y0r/(t) and W2(t) is negative binomial distributed with parameters q(t) e (0,1) and b > 0. In the special case of y0 = 0 $(t) = W2(t) is negative binomial distributed and if Y is exponential distributed (b = 1) we get the geometrical distribution for W2(t) in $(t) = Wi(t) + W2(t).
3. Let Y be inverse Gaussian distributed with pdf fY(y) = I(y > 0)
k! ^(1 + 2r1(t)^2/p)k The moments of order k of the inverse Gaussian distribution are given by
Finally we get Pk(t) = exp (— ^(y/1 + 2n(t)n2/p — 1)^
2.2 Marking the sequence (Tn)
Next we consider a marking of the sequence (Tn). At every time point Tn a shock causes a random degradation. We describe the degradation increment at Tn by the mark Xn. <P = ((Tn,Xn)) is said to be a position-dependent G-marking of (Tn) if Xi, X2,... are conditionally independent given (Tn):
Moreover, we assume that each mark Xn and Y are conditionally independent given (Tn), i.e. P(Xn G B \ (Tn) ,Y) = P(Xn G B \ (Tn)). Note that the distribution of the n — th degradation increment Xn depends on the random time of the n-th shock. With a position-dependent-marking it is possible to describe degradation processes where the degradation becomes faster (or slower) with increasing time. We want to give two simple examples.
1. Let t0 > 0 be a fixed time and let (Un), (Vn) be two sequences of iid random variables with cdf Fu and Fv , respectively. The sequence of degradation increments (Xn) is defined by
That means that at time to the distribution of degradation increments is changing. For to = 0 we get the independent marking.
2. Let (Un) be a sequence of non-negative iid random variables with the density fu and let S be a real number. We assume that the sequence (Un) is independent on (Tn). The sequence (Xn) is defined by Xn = Un ■ eSTn. That means we get damage increments which tend to be increasing (S > 0) or decreasing (S < 0). The stochastic kernel G is given by
Again, for S = 0 we have the special case of independent marking. In this case G defines a probability measure which is independent on the time t.
Last we have to specify the distribution of the marks. This can be any distribution law with nonnegative realizations, such as the exponential, gamma, Weibull, lognormal or Pareto.
For practical applications it is necessary to estimate the parameters of all considered distributions. That can be done or by likelihood theory or by the method of moments. The likelihood function for such degradation models and parameter estimates are given in detail in [WeK04]. Some characteristics of
Xn := I (Tn < to) Un + I (Tn > to) Vn and G(t, [0,x]) is given by
G(t, [0, x]) = I(t < to) Fu(x) +1(t > to) Fy(x) .
the process such as the cumulative degradation at any time t, the moments of the counting process, and others, are developed. Here we will restrict us to the estimation of the distribution parameters of Y by maximum likelihood and moment methods. We can have different levels of information observing the degradation process:
1. All random variables, Y, (Tn), and (Xn) are observable. Then the likelihood function is a product of three densities and the parameters can be estimated independently for each random variable. This case is not very realistic.
2. More interesting is the the assumption that we can observe each time point of a shock and each increment of degradation but cannot observe the random variable Y. This is a more realistic assumption because Y is a variable which describes the individual shock intensity for each item, a frailty variable.
3. In many situations it might be possible that a failure is the result of an degradation process but we cannot observe the underlying degradation. By the maximum likelihood method it is possible to estimate all parameters in the model because the distribution of the the first passage time contains all these parameters.
Let 0 be given as 0 = (0Y, 0T, dX) G Rp with p = u+v+w. Here, 0Y G Ru is a parameter of the distribution function Fy of Y, 0T G Rv denotes a parameter of the deterministic terms n and its derivative £, respectively. And 0X G Rw represents a parameter of the distribution of degradation increments. Under the assumptions of section 2 we get the following stochastic intensity of the marked point process & = ((Tn, Xn))
X(t, B; 0) = Y • £(t; 0T) • G(t, B; 0X) , B gB+ .
If we have the full information about the degradation process then it can be shown that the likelihood function consists of three independent parts, each of them contains the full information about 0Y, 0T, and 0X, respectively. If we want to estimate 0Y, then we have the classical problem of estimating parameters from a sample of m iid observations [Wen99]. In [WeK04] it is shown that in the second case (Y is not observable) the intensity A is given by oo fy^(t-)+1e-yv(t;0T)Fy(dy; 0Y)
The essential part for estimating the parameter iY is the last term
which can be interpreted as the conditional expectation of Y given the history of observation. It is easy to see that this term depends only on 0Y and 0T. Our aim is to determine an estimator of 0Y based on m > 1 independent copies of the process Let ^i(t) be the observed number of shocks in the i—th copy (i = 1,..., m). For the three special distributions of Y introduced in section 2 we get the following essential parts of the process intensity and resulting maximum likelihood estimates:
1. If Y is rectangular distributed in [a, b]:
Ht— + 1 f \bn(t;9T)]" e-bv(t;eT) - \an(teT)]" e-aV(t;eT )\
n(t; eT) *(t-) f \bn(t;eT )]„ bn(t;eT)_ \an(t;eT)]- aV(t;eT
For this distribution we get two likelihood equations which are linear dependent and which both leads to
Consequently, it is not possible to estimate both parameters a and b. 2. If (Y — yo) is gamma distributed:
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