Step Stress Model - IITKhome.iitk.ac.in/~kundu/seminar25.pdf · 2006-02-26 · Accelerated life...

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Department of Mathematics & Statistics, IIT Kanpur, INDIA

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Step Stress Model

Debasis Kundu

Department of Mathematics and Statistics

Indian Institute of Technology, Kanpur

India

Joint Work: N. Balakrishnan & N. Kannan & Tony Ng

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Outline of the Talk

• Accelerated Life Test

• Step Stress Test

• Acceleration Model

• Cumulative Exposure Model

• Parametric Inference

• Bayesian Approach

• Optimal Test Plan

• Open Problems

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What is an Accelerated Life Test

Accelerated life test is a popular experimental strategy to ob-

tain information on life distributions of highly reliable product.

The main idea is to submit materials to higher than usual envi-

ronmental conditions to ensure early failure. Data obtained

from such an experiment need to be extrapolated to estimate

lifetime distribution under normal conditions.

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Stress factors

• Single Stress

•Multiple Stress

The classical stresses are

• Temperature

• Voltage

• Current

• Pressure

• Load

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Advantages

(a) If you increase the stress, reasonable number of failures are

ensured

(b) Reduce the experimental time

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Disadvantages

(a) Need to know the exact relation between the stress and life-

time

(b) Model must take into account the effect of accumulation of

stress

(c) Model becomes more complex

(d) Even in simple case analysis becomes difficult

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What is a Step-Stress Life Test?A Step-Stress life test is a particular acceleratedlife-test. You observe the failure times of the ob-jects at a particular stress level, then change thestress level to a different level. Observe the failuretimes in the new stress level, and then the change thestress level again and so on.

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Several Variations are possible depending on the need

(a) The Stress can be changed at the pre specified tim-ing. It is known as Step-Stress with Type-I censor-ing. In this case number of failures is random.

(b) The Stress can be changed at the pre specified num-ber of failures. It is known as the Step-Stress withType-II censoring. In this case failure time is ran-dom.

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Example

It is from Nelson (1980). The test is regarding cable insulation.

The stress here is voltage (kilovolts). The aim is to estimate life

at certain design stress.

Step 1 2 3 4 5 6 7 8 9 10 11

Stress 5 10 15 20 26 28.5 31 33.4 36 38.5 41

Time exposure 10 10 10 10 15 15 15 15 15 15 15

(min) 10 10 10 10 60 60 60 60 60 60 60

10 10 10 10 240 240 240 240 240 240 240

10 10 10 10 960 960 960 960 960 960 960

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Acceleration Model

The main problem with the Step-Stress testing is to find the

relationship between the lifetime in test (stress condition) and

lifetime in used (normal condition). A relationship between the

lifetimes need to be established.

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Some Definitions

• θ(s): Stress Function at the stress level s

• X : The time to failure under stress

• Y : The time to failure under normal condition

• FX : The cumulative distribution function of X

• FY : The cumulative distribution function of Y

• φθ : The acceleration function

The acceleration function

φθ : [0,∞) → [0,∞).

and

Y = φθ(X)

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Relations Between X and Y

• The Distribution Functions

FY (y) = FX(φ−1θ (y))

• The Density Functions

fY (y) = fX(φ−1θ (y))

∣∣∣∣∣∣

d

dyφ−1θ (y))

∣∣∣∣∣∣

• Hazard Functions

λY (y) =

∣∣∣∣∣∣

d

dyφ−1θ (y))

∣∣∣∣∣∣λX(φ−1θ (y))

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Different Forms of φθ

• Linear acceleration function

φθ = θ(s)X ; θ(s) ≥ 1.

In this case

λY (y) =1

θ(s)λX(x).

This is a Proportional Hazard Model

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• Power Transform

Y = AXθ(s)

With such an acceleration function, a Weibull(α, β)is transformed to another Weibull(Aαθ, β

θ).

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Lifetime Distribution Under Step-Stress

• Cumulative Exposure Model

Classical assumption for the analysis of step-stress data. The

main idea is to assume that the remaining lifetime of the spec-

imens depends only on the current accumulated stress, re-

gardless of how it has accumulated.

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• F1 : The cumulative distribution function understress s1

• F2 : The cumulative distribution function understress s2

• G : The cumulative distribution function under astep-stress pattern

G can be obtained from F1 and F2 under the assump-tions that the lifetime F1 (under stress s1) at the timet1 has an equivalent time u2 of the distribution func-tion F2 (under stress s2).

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Graphical Representation

F

F2

1

u t2 1 0

0.2

0.4

0.6

0.8

1

0

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Step-Stress: Type-I Censoring

• [0, τ1) → Stress is s1

• [τ1, τ2) → Stress is s2

• [τ2, τ3) → Stress is s2

• ↓

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• Step 1: G(t) = F1(t); for 0 ≤ t ≤ τ1

By assumption there exists a u1 such that

F1(τ1) = F2(u1)

• Step 2: G(t) = F2(t− τ1 + u1); for τ1 ≤ t ≤ τ2

By assumption there exists a u2 such that

F2(τ2 − τ1 + u1) = F3(u2)

• Step 3: G(t) = F3(t− τ2 + u2); for τ2 ≤ t ≤ τ3

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Step Stress - Type II Censoring

• Put n items on test

• Continue with the stress s1 until, the first r1 itemsfail

• Change the stress level to s2

• Continue with the stress s2 until total r2 items fail

• Change the stress level of s3 and so on

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Step-Stress: Type-II Censoring

If t1:n < . . . < tr:n are the failure times

• [0, tr1:n) → Stress is s1

• [tr1:n, tr2:n) → Stress is s2

• [tr2:n, tr3:n) → Stress is s2

• ↓

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Distribution Function Under Step-Stress with Type-IICensoring

• G(t) = F1(t) if 0 ≤ t ≤ tr1:n.

• G(t) = F2(t− tr1:n + u1) if 0 ≤ tr1:n ≤ tr2:n.

• G(t) = F3(t− tr2:n + u2) if 0 ≤ tr2:n ≤ tr3:n.

Where ui Satisfies the following Relations

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• F1(tr1:n) = F2(u1)

• F2(tr2:n − tr1:n + u1) = F3(u2)

• F3(tr3:n − tr2:n + u2) = F4(u3)

• ↓.

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Exponential Distribution, Type-I Censoring

Let us consider simple step stress model only. Thestress changes at τ .

• F1 Follows exponential with mean θ1

• F2 Follows exponential with mean θ2

In this case u = θ2θ1τ

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The distribution function G(t) under step-stress con-dition

• G(t) = F1(t) = 1− e− t

θ1 if 0 < t < τ .

• G(t) = F2(θ2

θ1+ t− τ ) = 1− e

− τθ2

+τ−tθ2 if τ < t <∞.

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Suppose n1 is the number of failures before time τ andt1:n < . . . < tr:n are the observed failures.

Clearly, 0 ≤ n1 ≤ r. If 1 ≤ n1 ≤ r − 1 then the MLEs ofθ1 and θ2 exist

θ1 =

∑n1i=1 ti:n + (n− n1)τ

n1

θ2 =

∑ri=n1+1(ti:n − τ ) + (n− r)(tr:n − τ )

(r − n1)

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The problem occurs in computing the distribution ofθ1 and

θ1. Xiong (1998) computed the the order statis-tics and used to compute the distributions. He hasclaimed the following results:

• The minimum order statistics of G(t) has an exponen-tial distribution therefore, nt1:n has an exponentialdistribution

• He has used the spacings of t2:n − t1:n, . . . , tn1:n − tn1−1:n

have independent exponential distributions

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Both are not correct

• Note that for fixed τ , nt1:n has bounded support andit can not exceed nτ .

• t2:n − t1:n, . . . , tn1:n − tn1−1:n are order statistics from aright censored distribution

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The exact density functions of θ1 and

θ2 can be ob-tained using the conditional moment generating func-tion approach. They are quite complicated.

• The density function of θ1 can be written as a

double summation of weighted shifted gamma. Theweight functions may be negative also.

• The density function of θ1 is a mixture of gamma

distribution

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• All the moments can be calculated for both θ1 and

θ2

•θ1 is biased,

θ2 is unbiased.

• Confidence intervals of θ1 and θ2 cannot be ob-tained directly

• Under the assumptions: Pθ1(θ1 ≥ b) and Pθ2(

θ2 ≥ b)

are increasing functions of θ1 and θ2 respectively,confidence intervals of θ1 and θ2 can be obtained.

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Confidence Intervals

• Exact Confidence Intervals

• Asymptotic Confidence Intervals

• Bootstrap Confidence Intervals

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Step Stress Model Under Type II Censoring

The MLEs of θ1 and θ2 always exist.

θ1 =

∑r1i=1 ti:n + (n− r1)τ

r1

θ2 =

∑ri=r1+1(ti:n − tr1:n) + (n− r1)(tr:n − tn1:n)

(r − r1)

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• All the moments can be calculated for both θ1 and

θ2

•θ1 and

θ2 both are unbiased.

• Confidence intervals of θ1 and θ2 can be obtaineddirectly

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Optimal Design

• Type-I Censoring: Optimal choice of τ

• Type-II Censoring: Optimal Choice of n1

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Bayesian Analysis

• Type-I Censoring: Not much work has been done

• Type-II Censoring: Possible to do

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Open Problems

• Other Distributions

• Competing Risks Set Up

• Link function approach

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Refernces

• Guono and Balakrishnan (2001): Hand Book of Statistics

• Xiong (1998), IEEE Trans. Reliability

• Balakrishnan, Kundu, Ng, Kannan (2006), Jour. Qual. Tech.

• Nelson (1990), Accelerated Testing, Wiley

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Thank You

Step Stress Model - IITKhome.iitk.ac.in/~kundu/seminar25.pdf · 2006-02-26 · Accelerated life...

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