Jo

Hence the process M(z) is a martingale.

The proof is complete.

Note that we can not use the Nelson-Aalen estimator based on the obtained decomposition because the function Y(y) depends on the values of A in the point zo > y. On the other hand, the decomposition is useful for demonstration of asymptotic properties of estimators.

Let us consider another decomposition of the process N(z). Set

N*(t) = 1{T<t}, Y*(t) = 1{T>t}. Denote by F* the a-algebra generated by N*(s),Y*(s), 0 < s < t. Then

N*(t)= [ X(Z(u))Y*(u)du + M*(u), o where M*(u) is a martingale with respect to the filtration (F* | t > 0). Sett zo, if j < m; Z, if j = m.

Non-parametric estimation in degradation-renewal-failure models 27 Theorem 2. The process N(z) can be written as the sum

o where f r

Y**(y) = J_] Ajl{Zj>y}, M**(z) = l{z(u)<z}dM*(u). (12)

Proof.

N (z) = 1{Z (T)<z} = 1{Z(u)<z}dN*(u) = 1{Z(u)<z}A(Z (u))Y*(u)du oo p T ^ / _ S \

l{Z(u)<z}dM *(u) = J^ l{Z(u)<z}J2 A^T1) <u<s,+1}dn+M**

5 C '{u**)A (U—Sj) dU+C '{-AS-<z}A (^) dU+M" =

El J Am

/ l{v<z}A (v) Ajdv +i l{v<z}A (v) Af dv + M** =

V / l{v<z}AjdA(v) + M** = VAj l{v<Zj}dA(v) + M**. j=iJo j=i Jo

4 Estimation 4.1 The data

Suppose that n units are on test and, for ith unit, denote by Sij (j > 1) the moment of jth renewal, by Aij the inverse to the degradation rate in the interval (Sij; Sij+i], by Zi(t) the degradation process and by Ti the moment of its failure.

Set ASij = Sitj+i — Sij and define mi as in the case of one unit writing (for the ith unit) Sij and Ti instead of Sj and T. Denote Zi = Zi(Ti). Then the data can be defined as the following collection of vectors of a random length:

Define the processes Ni(z), Yi(y) and Mi(z) as in Theorem 1, with Z(T), Am, Sm replaced by Zi(Ti), Sf, Af. Set

N(z) = J2 Ni(z), Y(y) = YJ Yi(y), M(z) = £ Mi(z) i=1 i=1 i=1

and let Fz denote the a-algebra, generated by the events of the form given just before Theorem 1 with Aj, m, Z(T) replaced by Aij, m,, Zi(Ti) (i = l,...,n). Then Theorem 1 implies that

o and M(z) is a martingale with respect to the filtration (Fz). Theorem 2 implies another decomposition:

M(z) = J2 l{Zi(u)<z}dM* (u); i=iJo here M* is a martingale with respect to the filtration (F*t \ t > 0), where

F*t = a(N*(s),Y* (s), 0 < s < t), N*(t) = l{T<t},Y*(t) = 1{T>}-

4.2 Estimation of A

Consider the problem of non-parametric estimation of A. Note that we can not use the Nelson-Aalen estimator based on the decomposition (13) because the function Y(y) depends on the values of the function A in the interval

The decomposition (14) implies the estimator

We shall show that this estimator can be obtained by other way, considering the non-parametric model as the limit of a sequence of parametric models.

Consider some parametric family (Xg \ 9 G O C Rp) of intensity functions A and find the maximum likelihood estimators 9 of unknown parameter 9. Let O = (0; >x)p and for 9 = (91, ...,9p) set

here 0 = z(o) < z(i) < ■ ■■ < z(p-i) < zp = zo are fixed cut points. A natural container of the data vectors

is the space E = U°=i Ej, where Ej = (0; to)^1 and ]J stands for the direct sum of topological spaces. Define the Borel measure v on E by setting, for each Borel subset C C E, v(C) = Vj(C n Ej);

here dvj(ai,...,aj, z) = dm(ai) ■ ■ ■ dn(aj)dz. Then the probability density function p(u) of the random element (16) (with respect to the measure v) is given by the equalities p(u) = e-A(sj +aj a j X(sj + aj z) for u = (a1,..., aj, z) € Ej;

t here sj = (a1 +-----+ aj-1)zo, X(t) = X(Z(t)), A(t) = fo X(s)ds. Indeed, if

C n Ej = (0; a*1] x ■ ■ ■ x (0; a*] x (o; z*], then tt p{Ui € c} = J2 P{Ui € C n Ej}

= J2 P {mi = j, Aii < a*i,..., Aij < a* ,Zi(T) < z*} j=i tt

= J2 P{Aii < a*i,..., Aij < a*, Sj <T < Sj + Aijz*} j=i tt ra* rah

= ^ dn(a1) ■ ■ ■ dn(aj) [e-Ai(sj) - e-A(sj +aj j=i o o

V / dn(a1) ■ ■ ■ / dn(aj) e A(sj +ajz)ajX(sj + ajz)dz j=i o o o p( u) dv(u) .

The log-likelihood equals n

L(0) = ^2 \-A(Simi + AimZimi) + log(Aimi) + log X(Si„H + AimZimi) i=i

" r m r Sij +Aij Zij = 22 Xe((s - Sij)/Aij)ds + log Xe(Zimi) +const i=i j=i-J Sij n mi Zrt i3

+ const

Y.Aj 0k l{Zi3 >z}dz + ^Y.l°g 0k 1{z(k-l)<Zimi <Z(k)}+COnSt.

The maximum likelihood estimators satisfy the equations fz(k) 1 !{Zi3 >z}dz + J i=1 j=1 Jz(k-1) Jk i=1

Jk v^n T^mi A Fz, zn=1z m=1 Aij/zc-i) i{Zi3 >z}dz'

Now formally take p = n and z(k) = Z(k), where Z(1),...,Z(n) are the values Zimi in ascending order. Then 1{Zi3>z} = 1{Zi3>Z(h-)} for each z € (Z(k~1)', Z(k)] and therefore

The cumulative intensity function at point Z(k) then can be estimated by

We get the following estimator:

4.3 Large sample properties of A

Proposition 1. The process A is a semi-martingale with the characteristics (Bh,Ch,v), where f z

Ch(z)= i h2(Y-1(y))Y(y)dA(y), o o v(dy,du) = Y(y)dA(y)cy-i(y)(du), where eu denotes the Dirac measure concentrated at point u, h : R ^ R is a continuous function with compact support, which equals u for u in some neighborhood of 0.

Non-parametric estimation in degradation-renewal-failure models 31 Let us find the first characteristic, Bh, of the process A We have

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