Nonparametric estimation in degradationrenewalfailure models

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V. Bagdonavicius1, A. Bikelis1, V. Kazakevicius1 and M. Nikulin2

1 Vilnius University, Naugarduko 24, Vilnius, Lithuania [email protected] [email protected] Vytautas.kazakevicius.maf.vu.lt

2 Bordeaux Victor Segalen University, France [email protected]

1 Introduction

Classical reliability theory and survival analysis parts give methods for analysis of failure time data.

An important part of modern reliability theory and survival analysis is modelling and statistical analysis of ageing , wearing, damage accumulation, degradation processes of technical units or systems, living organisms ([ME98], [WK04], [BN02]).

Lately, methods for simultaneous degradation-failure time data analysis are being developed ([BN01], [WT97], [BBK04]).

Some degradation processes are not non-reversible and degradation processes may be renewed. For example, the degradation of pancreas and thyroid can be defined by the quantities of secreted insulin and thyroidal hormone, respectively. By injection of insulin the degradation process of pancreas is (indirectly) renewed. By injection of thyroxine (case of hyperthyroid) or car-bimazole (case of hypothyroid) the degradation process of thyroid is (indirectly) renewed. If the value of hormone (for example, insulin) approaches a critical value, the risk of failure increases quickly.

In reliability, a simple example is a tire with renewable protector. The risk of failure depends on the level of protector wear .

We consider relatively simple linear (or loglinear) degradation process. On the other side, we consider rather complicated situation of non-parametric estimation when units are renewable and the failure intensities depend on degradation level of the unit.

We consider nonparametric estimation of degradation and failure process characteristics using degradation and failure time data with renewals. See also the paper [L04] who considers parametric estimation in similar context.

2 Model

For j > 1, let Sj denote the moment of the jth renewal (we assume Si = 0) and Aj be the inverse to the degradation rate in the interval (Sj; Sj+i].

Assume that the random variables Ai,A2,... are independent and identically distributed according to some cumulative distribution function n (or only independent with the cumulative distribution functions ni,n2,...). Denote by Z(t) the value of the degradation process at the moment t. Degradation process model:

Let T denote the moment of a traumatic failure. Failure model:

P(T > t | Z(s), 0 < s < t) = exp j - A(Z(s))ds j , (2)

A being a positive function. Denote m(t)= j, if t e (Sj; Sj+i] (j > 1), m = m(T). (3)

The failure occurs in the interval (Sm,Sm+i].

The data (for one unit) can be defined as the following vector of a random length:

Remark 1. The conditional distribution of T (with respect to the a-algebra A generated by the random variables Ai,A2,...) can be defined in another way, which is more convenient for computer simulations.

Firstly, define recursively conditionally independent random variables ATi, AT2,... such that

(here Pa denotes the conditional probability with respect to A). Secondly, set f=\ Si + ATi, if ATi < Aizo;

Then conditional distribution of T coincides with that of T. Indeed, if t e (Sj, Sj+i] then

Pa{T >t} = Pa{ATi > Aizo,..., ATj-i > Aj-izo, ATj > t - Sj}

= Pa {ATi > Aizo} • • • PA{ATj-i > Aj-z}PA{ATj > t - Sj}

= e- foAlZ° X(s/Ai)ds . ,.e- J0Aj-1Z° X(s/Aj-1)dse- J0t-Sj X(s/A.)ds = e- JSS? X((s-Si)/Ai)ds _ _ _ e- Jsj^ X((s-Sj-i)/Aj-i)dse- JS. X((s-Sj )/Aj )ds

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