E. L. CRUZ ET AL.

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187

12

exp

ss

0,s

(11)

with is a function of temperature and 1

and 2

are constants characteristic of the product fail-

ure mechanism and test conditions [9 ].

3.3. Relationship to Cox Proportional Hazards

Model and Arrhenius Model

In the next result we show the equivalence of the

GMALT model and the Cox regression model or the pro-

portional hazards model. The reader is referred to Flem-

ing and Harrington [4] for an exposition. This result

shows that the dual of the GMALT model is the Cox

proportional hazards in the counting process approach to

survival analysis, the GMALT being the lifetime ap-

proach to survival analysis. The Arhennius life model is

also presented as special case.

Proposition 6. In the GMALT model with accelera-

tion function

s

1

with stress space S, the hazard func-

tion at any stress s S is given by

,

0

sAs

0,x

01

x

for all (13)

If

exp

sX

xp

where X is the covariate,

then 01

0

,e

sx

1

X

which is the Cox

proportional hazards model with one covariate. In par-

ticular, if

s, 01

and 12

where s is

temperature, and 1

and 2

are constants, we get the

well-known Arhennius model [9].

Proof of Propositio n 6. Let . Then

SS

PXy F

,

S

X

PAsX yPX

FyAs

As y

,

where X

is the distribution of X. Also

X

1

X

yAs As

fyAs where fX is the

density of X. By definition the hazard function (at a given

stress level s) is

1

1

0

,11

,

S

SX

X

sfyAs

FyAs

fy

xs Fy

As x

by letting

yAs

,

.

3.4. Equivalence to Model of Cox and Oakes

Finally we show the equivalence of the proposed model

(4) to the one given by Cox and Oakes [5] and described

in Section 2.

Let X have distribution function F. Since

S

As X

we have

, .

S

PXtPAsX tPX

FtAs

,(

,tAs

Taking

,1,

ss

;,

tsFs t

gives

and the two are equivalent. Thus, the approach of Cox and

Oakes may be also taken as a special case of GMALT.

4. Concluding Remarks

In this paper, we considered a general model for acceler-

ated life testing and derive some of its properties. This

model is expressed in terms of the random variables

which is simpler, instead of distribution functions. An

important consideration is the choice of stress level,

where the GMALT model may fail. A threshold level

(stress) at which the scaling is no longer applicable must

be sought by the scientist. Future work may consider

where a test of hypothesis 0 applies. Rejec-

tion of this hypothesis means we can proceed to the test

or increase stress at a certain level. For the simple null

hypothesis

:1HAs

0:1HAs

, rejection in favor of the alter-

native

:1HAs

0

indicates that we can start doing

accelerated testing at stress level s where

1As

. For

products where

s is not known, the search for

s is the subject of many inquiries.

5. Acknowledgements

E. L. C. thanks the Office of the Vice President for

Academic Affairs of the University of the Philippines

where he was recipient of Doctoral Stude nt Grant.

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