Z

Set ci = infz<zo b(z), C2 = supz<zo b(z) and suppose g(u) = 0 for I u I< c. Then, for n > 2/(cci),

= p{3z < zo : Y-i(z) > c, f ) g(Y-i(z))Y(z)dA(z) > e}

< P{3z < zo : Y-i(z) >c} < P{3z < zo : n-iY(z) < — } < P{3z < zo : n-iY(z) < ci/2} < P{3z < zo : n-iY(z) < ci/2 + b(z) -ci} < P{ sup I n-iY(z) - b(z) I> ci/2} 0, z<zo which gives 3).

Now suppose that h(u) = u for I u I< c. Similarly as above

PU h2(Y-\z))Y(z)l{y-1{z)>c}dA(z) >e} < P{3z < zo : Y-1(z) > c) ^ 0.

Moreover, for n sufficiently large,

P{JZ h2(Y-1(z))Y(z)l{y-1{z)<c}dA(z) >e} < p{£ Y-2(z)Y(z)dA(z) > e]

_inïz n-2Y2(z) n < P(supn-1 Y(z) > n1/2e)+P(inf n-2Y2(z) < n-1/2A(zo))

= P{3z < zo : n 1 Y(z) > n 1/2e} + P{3z < zo : n-2Y2(z) < n-1'2A(zo)}

< P{3z < zo : n-1 Y(z ) > 2c2) + P{3 z < zo : n 1 Y(z) < C1/2}} < P { sup \ n-1 Y(z) - b(z) \> c2} + P { sup \ n-1 Y(z) - b(z) \> cxj2} 0.

zkzq zkzq

This yields 2).

Relation 1) will be proved if we show that sup I h(Y-1 (z))Y(z) - 1 0.

z<zo

Similarly as above we get p{ sup I h(Y-\z))Y(z) - 1 I l{n-iy{z)<ci/2} > 4

z<z0

< p{3z < z0 : n-1Y(z) < c1j2\ 0. On the other hand, for n sufficiently large

SUp | h(Y-1(z))Y(z) - 1 I l{n-1y(z)>Cl/2} = z<z0

2 - ~ 2 < - sup I Y(z) - Y(z) I= — sup I n-1Y(z) - n-1Y(z) |

22 < — sup I n-1Y(z) - b(z) I +— sup I n-1 Y(z) - b(z) I~> 0.

The proof is complete.

Theorem 4. Suppose that the conditions of Theorem 3 are satisfied, E(A) < x. Then the random function vn,(A - a)

tends in distribution in the space D[0,zo] to the mean zero Gaussian process V with the covariance function

Jo Jo f=1 eA(Sk+i) - eAS+Ak(uAv)) b(u)b(v) Jo b(y) '

Proof. The asymptotic distribution of the estimator A can be found using the martingale decomposition of N, i.e. using the equality rs/2m - AM) = n'/2 f • (^ - l) dA(y) + n'/2 f'

=n-i/2 [ dA(y>+n-'" f (Yy — w>) ^^ "Ay

Let us find the mean and the covariance function of the first term. The equality EY(y) = EY(y) implies that

Let us find the covariance function: for zi < z2 < zo cov(Ai(zi),Ai (z2))

= n-i -rn TTT E(Y(u) — Y(u))(Y(v) — Y(v))dA(u)dA(v),

1{Sk + Aku<T<Sk+i} _ 1 1 _ e-Am(A(z0)-A(u)) 1{Ti>Sik + Aiku}

1{Sk+Akv<T <Sk + i} _ 1 . 1 _ e-Am(A(zo)-A(v)) 1{Ti>Sik+Aikv}

The second term converges in probability to zero uniformly on [0,zo]. The first and the third terms are asymptotically independent. The asymptotic distribution of the third term is obtained similarly as the asymptotic distribution of the Nelson-Aalen estimator.

It tends in distribution in the space D[0; zo] to the zero mean Gaussian process W with the covariance function

where

4.4 Estimation of the probability pj(z)

If ni = n then the probability

to attain the level of degradation z (0 < z < zo) before a failure occurs given that an unit had been renewed j — 1 times (j = 1, 2,...) is estimated by the statistic x

because

Otherwise, the estimator is

Pj(z) = J expj-aA(z)^ dnj(a) = —j ^ exp | -Aj ^ 1lY(Zk,mk)

where m(j)

, , i=1 '-{A-ij <a,mi>j} , .. V"^ nj (a) = -—j)-> m(j =2^ '{j<mi}.

References

[BN02] Bagdonavicius, V. and Nikulin, M. Accelerated Life Models. Chapman and Hall/CRC: Boca Raton(2002) [BN01] Bagdonavicius, V., Nikulin, M.: Estimation in Degradation Models with Explanatory variables. Lifetime Data Analysis, 7, 85-103 (2001) [BBK04] Bagdonavicius, V., Bikelis, A., Kazakevicius, V. : Statistical Analysis of Linear Degradation and Failure Time Data with Multiple Failure Modes. Lifetime Data Analysis, 10, 65-81(2004) [1] Jacod, J. and Shyriayev, A. N.: Limit theorems for stochastic processes.

Springer, New York (1987) [ME98] Meeker, W.Q. and Escobar L.: Statistical Methods for Reliability

Data. J.Wiley and Sons (1988) [WT97] Wulfsohn M. and Tsiatis A. A Joint Model for Survival and Longitudinal Data Measured with Error. Biometrics, 53, 330-339 (1997) [L04] Lehmann, A. On a Degradation-Failure Models for Repairable Items. In: Nikulin, M., Balakrishnan, N, Mesbah, M., Limnios, N. (ed) Parametric and Semiparametric Models with Applications to Reliability, Survival Analysis, and Quality of Life. Birkhauser, Boston, 65-80 (2004) [WK04] Wendt, H., Kahle, W. On Parametric Estimation for a Position-Dependent Marking of a Doubly Stochastic Poisson Process. In: Nikulin, M., Balakrishnan, N, Mesbah, M., Limnios, N. (ed) Parametric and Semi-parametric Models with Applications to Reliability, Survival Analysis, and Quality of Life. Birkhauser, Boston, 473-486 (2004)

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