Estimation of the mean degradation

Assume that the c.d.f. Fa of A and the function g are unknown. Fix the degradation measurement moments titi,... ,ti<mi of the i th individual (i = 1... ,n). If the death time Ti of this individual occurs in the interval [tiji ,ti,ji+i) (ji = 1,..., mi; tmi+i = m) then the values Zii,..., Ziji of the degradation process Zi of the ith individual are observed at the time moments tii,..., tiji.

The PAQUID data has the following properties:

1. number of measurements per individual is small;

2. the time of follow-up tiji — tii is short for each individual i;

3. there are important differences in intervals [tii; tiji] for different individuals, for example [60; 65] and [90, 95], etc.

In each short time interval \tii; tiji] we model the real degradation Zr (t) by the loglinear model g(t,Ai) = eAii (1+ t)Ai2, where Ai = (Aii,Aii), and Ai,...,An are n independent replicates of the random vector A.

To have stable estimators of the mean degradation attained at the moment t we use the degradation values of individuals with indices s such that s G D(t) = {i : t G [tii; tiji], ji > 2} .

Yij Zij , Yi (Yii , ..., Yiji ) . Then given Ai = ai, ji

Pi = (Pii,...,Pi,ji)T, Pij = Pij(ai) = lng(tij,ai), Si =\\ Sikl \\jixji,

Denote by bk the elements of the inverse matrix S^, and by N the number of individuals such that ji > 2.

The predictors A?, of the random vectors Ai for the ith individual are found minimizing with respect to ai,...,an the quadratic form

Denote by m = i=i ji the total number of measures over all of individuals. The estimator a2 of the parameter a2 is found maximizing with respect to a2 the conditional likelihood function

| |-1/2 exp - — ]T (Y - ^i(Ai))T S-1 (Yi - ^(Âi))\ .

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