- Email: [email protected]
Inference for step-stress accelerated life tests
Journal of Statistical Planning and Inference 7 (1983) 295-306 North-Holland
INFERENCE
FOR STEP-STRESS
ACCELERATED
295
LIFE TESTS
Moshe SHAKED* University of Arizona, Tucson, AZ 85721, USA
Nozer D. SINGPURWALLA** George Washington University, Washington, DC 20052, USA Received 17 November 1981 Revised manuscript received 3 May 1982 Recommended by S. Zacks
Abstract: In this paper we consider the more realistic aspect of accelerated life testing wherein the stress on an unfailed item is allowed to increase at a preassigned test time. Such tests are known as step-stress tests. Our approach is nonparametric in that we do not make any assumptions about the underlying distribution of life lengths. We introduce a model for step-stress testing which is based on the ideas of shock models and of wear processes. This model unifies and generalizes two previously proposed models for step-stress testing. We propose an estimator for the life distribution under use conditions stress and show that this estimator is strongly consistent. AMS Subject Classification: Primary 62N05; Secondary 62FlO. Key words: Accelerated life tests; Step-stress tests; Nonparametric estimation; Shock models; Damage accumulation; Inverse power law; Test patterns.
Consistency;
1. Introduction Suppose that it is desired to estimate the probability distribution of the life lengths of a population of devices operating under normal conditions. With modern high reliability devices, the observed life lengths tend to be very long, and thus the time necessary to test a sample of such devices tends to be excessive. Such circumstances call for testing the sample under conditions which are more severe than the operating conditions, with the result that failure data can be obtained in a short period of test time. Such life tests are called accelerated or overstress tests, and by now several papers pertaining to the statistical issues which emanate from such tests have appeared in the literature. It is appropriate to mention here that accelerated life tests are so universally used by government and industry throughout the world that they have been codified by the U.S. Department of Defense in their handbook, * Research supported by NSF Grant MCS-79-27150. ** Research supported by the Office of Naval Research Contract NOOO14-77-C-0263,and the U.S. Army Research Office Grant, DAAG-29-80-C-0067 with the George Washington University. 0378-3758/83/$03.00
0 1983 Elsevier Science Publishers B.V. (North-Holland)
296
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
MIL-STD-781C, under the name ‘environmental tests’. The main problem with accelerated life tests is how to extrapolate from the data, obtained under conditions of severe stress, to statements about the failure behavior of the device under conditions of normal stress. Most of the published work pertaining to inference from accelerated tests has been restricted in the following two ways: First, it is required that every item which is subjected to an overstress is observed until it fails or is removed (censored) under the constant application of the large stress. The stresses are allowed to vary from one item to the other, but changing the stress level for an item before it fails or is withdrawn is not permitted. Such tests are also referred to as constant stress tests. The two references which drop this requirement are De Groot and Goel (1979), and Nelson (1980). Second, the life distribution of the items under every accelerated stress level is assumed to be known, except for the underlying parameters. This assumption may not be practical in many applications. Exceptions to the above are the recent papers by Shaked, Zimmer, and Ball (1979), Proschan and Singpurwalla (1979,198O) and Sethuraman and Singpurwaha (1982) which allow for the underlying life distributions to be unknown. However, the above references do require that the testing be conducted under constant stresses. The purpose of this paper is to consider the more realistic situation of accelerated life testing [cf. MIL-STD-781C, p. 71 wherein the stress on an unfailed item is allowed to change (increase) at a preassigned test time. Such testing is referred to as step-stress testing. Furthermore, our approach here is nonparametric in the sense that no assumptions are made about the form of the underlying distribution. To illustrate a particular application of step-stress testing suppose that cable insulation normally operates under a voltage stress of 20 kilovolts (kv). In order to make some statements about the reliability of the insulation under the normal operational voltage, m samples, the i-th one consisting of ni items, are subjected to a life test under the constant application of a voltage stress, say I$, where vr20kv, i=l,..., m. After tl hours of testing some of the items in each sample would have failed and the others would still be alive (i.e. not failed). The times to failure of the failed items would be recorded and so will the values of the associated stress I$, i=l,..., m. In order to terminate the test procedure in a short amount of time the stress on all the surviving items will now be increased from vi, i = 1,. . . , m, to a value V,, 1, where V,+ 1 is a large value, say 50 kv. The value V, + , has been chosen because it may be known to the engineer that under this stress the life of the cable insulation is quite short, so that all the remaining items will soon fail. We shall show that the additional information gained by increasing the stress levels to V,, , and inducing early failures will enable us to obtain better estimates of the life characteristics under the normal operating conditions. In Section 2 we introduce our model for step-stress testing and show its relationship to the models of De Groot and Goel (1979), Nelson (1980), and Endicott and Zoellner (1961). We shall show that by utilizing the very ideas that lead to ‘shock
M. Shaked,
N.D.
Singpurwaila
/ Step-stress
accelerated
291
life tests
models and wear processes’ [cf. Esary, Marshall and Proschan (1973)], we are able to interpret the otherwise unmotivated models of the previous authors. In Section 3 we describe our inference procedures, and in Section 4 we prove the strong consistency of our estimators. Throughout this paper, corresponding to any life length distribution function F, the survival function F is defined as 1 -F.
2. A model for step-stress testing Let Vi, . . . . V, be a collection of accelerated stress levels and let Vc and denote the use condition stress; assume that V, < Vi < ... < V1. An item is initially put on test under stress I$, where ii E { 1,2, . . . , I}. At a preassigned time ti the stress level on the item, provided that it is surviving at t, , is changed to vi,, where iz E { 1,2, . . . , I}. In general, the index iz is greater than the index i,, but this need not be so. The above strategy is continued so that at time tj, if the item is still surviving, the stress is changed from I$ to Vi,+,. Eventually, at time tJ, if the item is still surviving, its stress is changed to I$+, and it stays so until the item fails. Note that we need not assume that vj # y,+, , j= 1, . . . , J, nor do we require that K,l &<***I I$+,; however, in most applications this will indeed be the case. The ordered collection (vi,, t,, vi,, tZ, . . . , I$, fJ, vi,+,) will be called the stress pattern under which the test item runs and it will be denoted by Yet, where V=(~,,..., I$,+,) and t=(t,, . . . . tJ). Let Fi denote the distribution of the life length of the device when the underlying stress is Vi, i= 1, . . . ,I. Assume that Fi(t)=F(Fa’t),
tr0,
(2.1)
where F is some unknown distribution and (Yis an unknown constant. Notice that assuming (2.1) is equivalent to assuming that F,(t) =A(Cvat),
to 0,
(2.1’)
where A is some unknown distribution and C> 0 and czare unknown constants. This is so because one can identify F(a) of (2.1) with A(C.) of (2.1’). Assumption (2.1) is the familiar time transformation known as the ‘inverse power law’. The analysis given below can apply to other time transformations as well, but in order to save space these will not be discussed by us. Let T denote the life length of an item which is subjected to the test pattern (K,, *a*, vi,+,)o(tr, ***,fJ) and let H denote its life distribution. Thus, by (2.1), if t~[O,ti).
H(t)=P(T(t)=F;:,(t)=F(~~t)
We shall assume for now, with a justification H(t) = F( V;:piI+ vr(f - tl))
to be given in Section 2.1, that
if t E [t,, t2),
298
M. Shaked, N.D. Singpurwalla-/ Step-stress accelerated life tests
so that in general we have as our model for step-stress testing H(t)=F
(
i
y,i”(tj_tj_1)+
vE,(t-tl)
j=l
>if
TV VI,tt+ 11,
(2.2)
where i=O, 1, . . . . J, with te=O, and tJ+i =QO. A special case of the model given above is the model of De Groot and Goel(1979); in their case J= 1 and F is taken to be the exponential distribution. Under certain circumstances, which will be described soon, the above model is equivalent to that considered by Nelson (1980). These authors provide little motivation to justify their choice of the model. Thus, a contribution of this paper lies in the discussion below, which sheds some light on the models of De Groot and Goel and of Nelson and thereby provides the necessary motivation. A referee has observed that (2.2) can be written as H(I)=F(/;V’(u)du)
where J+1
VW =
c ~jx[fj_,,fJo;
j=l
here XD is the indicator function of the set D. This representation succinctly summarizes the model. It should be emphasized that this model assumes that the distribution of the life length is a function of the total accumulated Va and does not depend on the order of testing. The tester could run a test first to check that the life distribution is indeed independent of the order of the stress applied. 2.1. Motivation for the model The model (2.2) can be justified via two conceptually different scenarios which are plausible in practice. To describe the first scenario suppose that under the constant application of a stress, say I$, an item is subjected to shocks which occur in time according to the postulates of a Poisson process with a constant rate li= I$: i=l,..., I. If the probability that the item survives m shocks is Fm then the life length of the item has the survival function Ri(t)=
ODecAWit)m ~~, C m!
t20
(2.3)
m=O
This is the general shock model considered by Esary, Marshall, and Proschan (1973). Now consider an item which has been subjected to step-stress testing via a stess v; during the time interval [tj_ , , tj) for j = 1, . . . , J+ 1. Note that a change of stress corresponds to a change in the rate of the underlying Poisson process. Thus, following our convention that to= 0 and tJ+, = 00, the survival function of the item for
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
any tE[tl,tl+l), Ij(f)=
299
l=O,I ,..., J, is i WI=0
'
exp
[
j$,
1[ -
f: A;,(fj-tj-l)+Aj,+,(f-t,) j=l
~i,(~j-fj-l)+~i,+,(f-fl)
m Pm 1 * 11
z
(2.4)
Thus (2.2) must hold with m
F(t)=
e-lp
c ---ym. *=o
m.
(2.5)
Note that both (2.2) and (2.4) convey the notion that when the stress level changes the item is not ‘like new’ anymore; it retains its present state of damage from the old stresses to the new environment. Typically, the Ij, of (2.3) and (2.4), for m=O, 1, . . . . are unknown, and consequently the F of (2.5) is also not known. Thus, the nonparametric assumption underlying the model equation (2.2) is quite germane. Note that in many contexts is,,, is taken to be P(“)(x), where x is a fixed threshold, P is the distribution of the random variable representing the damage inflicted on the item by a shcok and pCrn) denotes the m-fold convolution of P with itself; see Esary, Marshall, and Proschan (1973). When P and x are both unknown F is also unknown and so the nonparametric assumption of (2.2) is again appropriate. The second setting which provides us with another motivation for our model is based on the ‘damage accumulation’ concept of Endicott and Zoellner (1961). Here it is assumed that under the influence of a constant stress the damage to an item accumulates at a constant rate. For example, we may assume that under stress level vi, the rate at which damage is accumulated as I.i = va, where Q is an unknown constant. Let X be the threshold of the item, assumed to be a random variable with an unknown distribution function F. Assume that the item fails when the accumulated damage reaches the threshold X, so that under stress 6 the life length of the item has distribution function Fi, where
This is precisely equation (2.1). Thus, under the stress pattern Vat, the distribution of the life length of the item is given by (2.2). Endicott and Zoellner (1961) have used this model for objectives which are different from those considered by us. When Vat = (Ve, V,) o tt and F is taken to be an exponential distribution, then (2.2) specializes to the model of De Groot and Goel (1979). Furthermore, it is not difficult to verify that the life distribution of items under the step-stress model of Nelson (1980) reduces to the model given by equation (2.2) if the life distribution of an item under a constant stress v is described by (2.1) for some life distribution F.
300
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
3. Estimation
of F,
In industrial step-stress life testing experiments it is a common practice to put several items on the test under various stress patterns. It will be helpful for us to recall from Section 2 that given the time points t = (I,, . . . , tJ and the corresponding stress levels V=( V;:,,Fz,, . . . , y,, V,,,,), with J$E {V,, V,, . . . , VI}, j= 1,2, . . . , J+ 1, the stress pattern is denoted by Vet. Assume that, in a step-stress life test, the collection of all the stress patterns is { Wet(‘) and that nk items are run under the stress ,-**, V(K)Ot(K)} pattern V(@o 0, k = 1, . . . , K. More specifically, for every k~ { 1,2, . . . , K}, t(k) = (t(k) Y(k)= ( v,‘“),. . . , v!” 9*--s vCk) 1 , . . . , ty), . . . , tz)) and ), where Vj(” E Jk+l J {v,, v,,..., VI}, j=l,2 ,..., J,+l. Some simplification in our notation will be achieved by pointing out that without loss of generality we can assume that t(l) = tc2)= *-- = tck)= t where t = (t,, . . . , tJ) for some fixed J. This is so because we can take the epochs at which changes in stress occur as the union of the ty)‘s, since we do not require Tjk)# J$t!. The stress patterns, then, are completely determined by the vectors Y(l), . . . , VcK), where y(k) = ( v/W, vz’k’,***, Vck)) J+, , k=l ,***, K Let F0 be the distribution of life lengths at use conditions stress V,, where, per our model assumption (2.1), &j(t) =F( vo”t>.
Our aim is to obtain an estimate of FO, say & using the failure data from stepstress testing as discussed above. Our estimate f10 is obtained in two steps. First an estimate of cz, say 6, is obtained, and then C?is used to transform all the observed life times to correspond to those that would have been obtained under V,. The transformed observations are then used to obtain the empirical distribution function of F,, & and so PO is our estimator of the distribution of lifelengths at V,. Let cl, /=1,2 )...) nk, kc&2 )..., K, denote the observed lifetimes under the stress pattern Vck)Ot; recall that nk is the total number of items which are tested under this pattern and that 6’) = t(‘)= -me= tcK)= t. Let Sk, 5 &2< -essSknr be the order statistics in the k-th sample; that is, Sk, is the l-th smallest observation associated with the group of items which are run under the test pattern Vck)ot, k=l,2 ,..., K. 3.1. Estimation of (r In order to estimate (Ywe need at least one time interval, say [tj_ 1, tj), during which items are tested under at least two different stress levels; otherwise, F is unidentifiable. This requirement is met whenever, for some k1 # k2, V-(kl)# Vck2). First let us assume that the first such interval is [0, tl), and that K= 2, so that, for t c [0, tl), P{Tl,st}=F((V/‘))at)
,
I=1 ,-**, n, ,
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
301
and P{T,,st}
=F((Vj2))V),
I= 1, . ..) n2,
where Y(r) # Yt2). Let G:(t) anrd G,(t) be the empirical distribution functions of the first and the second samples respectively. That is, I Gk(t) = ni’(number of Tkrs t), k= 1,2. Let G,’ be the right continuous G,‘(u)=sup{t:
inverse of Gk; that is,
G&)
UE [O,l),
(3.1)
k = 1,2, and let ~z=min(G,(t,),~~(f~)}.
We recommend that the reader graphically verify the meaning of these quantities. Note, for example, that if V,(r)< Vt@’then it is likely that Gr(tr)
*
Our proposed estimator of t9r2is 6r2, where u2
G;‘(U) du 42=
i
"u,
(3.2)
* e;'(u)
du
s0
and an estimator of (Ythen is .m a =
log
42
(3.3)
log( V;“/V,“‘) *
In Section 4 we will show that 6 is strongly consistent. Note that when t, = oo (3.2) reduces to the estimator (2.7) of Shaked, Zimmer, and Ball (1979), with (i,j) being replaced by (1,2). We shall now show that 8,, can also be written more explicitly as a function of the lifetimes Sk,, I= 1, . . . , nk; k= l,2. Let &=max{/: &(tr},
k= 1,2.
Then u2 is either II/n, or 12/n2 depending upon whether G,(t, c e2(t2)or otherwise. Suppose that u2 = Ir/nr ; let & = [(/,n2)/nr] where [a] denotes the largest integer which is less than or equal to a. In this case we can verify that
=ny’C;=, CT,,-T, ,I_, )(f,-~+l)+tt,-T,IJ~
g 12
n;’ Cf=,
(T2/- T2,1_,)(1;--I+ l)+(tr-
T2l;)l; ’
When u2 = 12/n2 the formula for 6t2 is of an analogous form.
(3.4)
302
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
Note that the estimator of a given by (3.3) is based upon failure data obtained under only two different stress levels V/” and Vy’. In practice however, it is very likely that the number of distinct stress levels under which life test data is obtained for the time interval [O,rr], is more than two. Thus, we would have several 0,‘s, each of which would lead us to an estimator of a. Consequently, what we need to consider next is a procedure for obtaining a pooled estimator of a. To this end, let the number of distinct (different) stress levels under which the life tests are conducted over the time interval [0, tr] be M, where 2rMsK. That is, M is the number of different VF)‘s among V/r’, . . . , VcK) , . Let us denote these distinct Vy)‘s by CI;‘),Ur@’ , **-, ujM , and let fi, be the number of items that are run under stress Ui”’ (that is, ii, is a sum of some Q’S), m = 1,2, . . . , M. For every pair of indices m and m’ with m},
U2 =
min{
Gl(ij),
62}.
As before, we define Br2= ( l$12)/V/1))a and propose as an estimator of 8r2, u2 G;‘(U) du Q*2= s :I”’ (3.6) &r(u) du ’ s “=“I and as an estimator of a, a = log l$2/(log( $Vl$Ji))).
(3.7)
We shall show in Section 4, that this estimator of a is also strongly consistent.
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
303
We note that the estimator (3.7) reduces to the estimator (3.3), when j= 1 (with obvious change of notation). A formula similar to (3.4) can also be obtained, and when the number of distinct stress levels is M, for M<2, the +M(M- 1) estimates of a can be pooled as in (3.5). 3.2. Estimation of FO Once we have an estimate of a, 6, we transform
the observed life-times Tk/,
I= 1, . . . . nk, k= 1, . . . . K, as indicated below to obtain Fkj, where
Here, as was assumed before, to= 0 and tJ+ 1= oo. The empirical distribution function based on the N= C:=, nk transformed variables Tk:,,is our estimate of PO and is denoted by FO. The mean, the median, the various percentiles and the other parameters of F, can be estimated from their counterparts using Fe.
4. Properties of the estimator EO In this section we shall show that the estimator p0 discused in Section 3 is such that ~,,(x)-+F(x) uniformly in x with probability 1. For this, we shall assume that for each k, k=l,2 ,..., K, nk + 03 in such a way that nk/N-‘Pk> 0, where N=C:=, nk, and so c;=, Pk= 1. We shall first show that 8i2% 0i2 and, in general, gm,,,,% B,,,,. This result is then used to argue that &+a with probability 1. To formalize our development we will assume that the random variables Tkl, I= 1,2, . . . , nk; nk= 1,2, . . . ; k= 1, . . . , K, are defined on the same probability space and let P denoe the induced probability measure on this space. In considering 6iZ of (3.2), let us assume, without loss of generality, that V/i’= Vi < V, = Vi@). Then, by the Glivenko-Cantelli lemma, C?,*F, and 6, -*F2 uniformly on [O,t) with P-probability 1. Since F,(t,)
min{F,(t,), F2(tl)) =F,(t,)
M. Shaked. N.D. Singpurwalla / Step-stress accelerated life tests
304
and so
F,-‘(u) du
11
I
0
= 0
I
0
s w c-9
Fi- ’ (4 (4) W (4
0
I
0
=
s w (4
I’(v’/v*)asdF’(s)
=(V*/V,)“=4*.
I
0
Thus 0i2 of (3.2) is strongly consistent. Consider now 0i2 of (3.6). Here again, without loss of generality, assume that V!” = Vi < V2= y!“. Then G, -% B, and G2 s B2, where, because of (2.2), J j-l
B&)=F
1 (V~‘)a(f,-tm_,)+ m=l
V-;(M~_~) I
for tE [tj_i,ti), k= 1,2. ’ the first time interval Since I’(‘)= Y(*) for m= 1, . . ..j- 1 (recall that [t.J_1, ti) IS for whichyhe styess levels are different), if follows that ‘i’ (Vi))‘(f,-f,_i)=l$l
(Vz’)“(t,-t,_i)=a,
m=l
say.
Thus Bk(f)=F(a+
Vkol(t_tj_l)),fE[tj_1,ti),
It follows that, except on a set of P-probability
k=l,2. measure zero,
u+Bl(a)=B2(a) and ~2%
B,(o+ V~a(tj-tj_~))~
Thus
El@+
Vp(rj-(j-1))
B
a.s. 5
Bi ‘(u) du
In view of the above we conclude that 6-o. In order to show that &,+Fo uniformly with probability one, let rk/ be Fkj as defined in (3.8) with B replaced by CT.The distribution of each rk[ is easily seen to
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
305
be F of (2.1). From the strong consistency of G and the Gilvenko-Cantelli lemma, the almost sure convergence of p,, is therefore established. We close this section by remarking that without having data from a time interval in which stresses are applied at two or more different levels, it is not possible to estimate Fo. This follows from the fact that if H of (2.2) is explicitly given then F and a are not identifiable. That is, the same H can be obtained from different choices of F and a even if Vi, . . . , V,, 1 are given constants.
5. Summary and conclusions In this paper we have proposed a model and have developed a procedure for the analysis of data from accelerated life tests wherein the underlying stresses are allowed to vary during the test. This type of testing is more typical than the one in which items are tested under a constant stress, an assumption which has been commonly made in much of the previous literature. A prime example of the kind of testing considered here is described under what is known as ‘environmental testing’ of MIL-STD-781C. The model that we have proposed here has been motivated from two different points of view and it encompasses the models considered by other researchers who have also considered this type of accelerated testing. Our procedure is nonparametric in that we are not required to make an assumption about the underlying distribution of lifetimes. However, we do require the assumption of a time transformation function and also require that there exists at least one interval of time during which life tests have been conducted under at least two distinct stress levels. As of now, an investigator who has data from step-stress accelerated tests has a choice of the following three procedures for estimating the life distribution under the use conditions stress: The approach of DeGroot and Goel (1979) is Bayesian and requires some data obtained under the use conditions stress. In addition, it requires that all the items be switched from the use conditions stress to a particular prespecified higher stress level. The authors mention the possibility of applying their method to more general sets of data, but do not give details. For general sets of data from step-stress testing, the researcher may use the procedure of Nelson (1980) or the one of this paper. Nelson’s procedure should be used whenever the F of (2.1) is specified or can be assumed known. Note that whereas the procedure of this paper requires at least one time interval during which there be testing under at least two distinct stress levels, no such requirement is made of Nelson’s procedure. Also, we have not considered here the case of censored data, whereas Nelson has. Thus it appears necessary to impose a few requirements on the data in order to be able to conduct a nonparametric analysis.
306
M. Shaked. N.D. Singpurwalla / Step-stress accelerated life tests
Acknowledgment
We thank a referee for a careful reading of a draft of this paper and for helpful comments.
References [l] DeGroot, M. and Goel, P.K. (1979). Bayesian estimation and optimal designs in partially accelerated life testing. Naval Res. Logistic Quart. 26, 223-235. [2] Endicott, H.S. and Zoellner, J.A. (1961). A preliminary investigation of the steady and progressive stress testing of MICA capacitors. Proc. 7th National Symposium on Reliability and Quality Control, 229-235. [3] Esary, J.D., Marshall, A.W. and Proschan, F. (1973). Shock models and wear processes. Ann. Probab. 1, 627-649. [4] Military Standard-781-C (1977). Reliability design qualification and production acceptance tests: exponential distribution. Department of Defense, Washington, DC. [.5] Nelson, W. (1980). Accelerated life testing - step stress models and data analyses. IEEE Trans. Reliability 29, 103-108. [6] Proschan, F. and Singpurwalla, N.D. (1979). Accelerated life testing - A pragmatic Bayesian approach. Optimization Methods in Statistics (ed: Rustagi, J.), Academic Press, New York. [7] Proschan, F. and Singpurwalla, N.D. (1980). A new approach to inference from accelerated life tests. IEEE Trans. Reliability 29, 98-102. [8] Sethuraman, J. and Singpurwalla, N.D. (1982). Testing of hypotheses for distributions in accelerated life tests. J. Amer. Statist. Assoc. 77, 204-208. [9] Shaked, M., Zimmer, W.J. and Ball, C.A. (1979). A nonparametric approach to accelerated life testing. J. Amer. Statist. Assoc. 74, 694-699.
INFERENCE
FOR STEP-STRESS
ACCELERATED
295
LIFE TESTS
Moshe SHAKED* University of Arizona, Tucson, AZ 85721, USA
Nozer D. SINGPURWALLA** George Washington University, Washington, DC 20052, USA Received 17 November 1981 Revised manuscript received 3 May 1982 Recommended by S. Zacks
Abstract: In this paper we consider the more realistic aspect of accelerated life testing wherein the stress on an unfailed item is allowed to increase at a preassigned test time. Such tests are known as step-stress tests. Our approach is nonparametric in that we do not make any assumptions about the underlying distribution of life lengths. We introduce a model for step-stress testing which is based on the ideas of shock models and of wear processes. This model unifies and generalizes two previously proposed models for step-stress testing. We propose an estimator for the life distribution under use conditions stress and show that this estimator is strongly consistent. AMS Subject Classification: Primary 62N05; Secondary 62FlO. Key words: Accelerated life tests; Step-stress tests; Nonparametric estimation; Shock models; Damage accumulation; Inverse power law; Test patterns.
Consistency;
1. Introduction Suppose that it is desired to estimate the probability distribution of the life lengths of a population of devices operating under normal conditions. With modern high reliability devices, the observed life lengths tend to be very long, and thus the time necessary to test a sample of such devices tends to be excessive. Such circumstances call for testing the sample under conditions which are more severe than the operating conditions, with the result that failure data can be obtained in a short period of test time. Such life tests are called accelerated or overstress tests, and by now several papers pertaining to the statistical issues which emanate from such tests have appeared in the literature. It is appropriate to mention here that accelerated life tests are so universally used by government and industry throughout the world that they have been codified by the U.S. Department of Defense in their handbook, * Research supported by NSF Grant MCS-79-27150. ** Research supported by the Office of Naval Research Contract NOOO14-77-C-0263,and the U.S. Army Research Office Grant, DAAG-29-80-C-0067 with the George Washington University. 0378-3758/83/$03.00
0 1983 Elsevier Science Publishers B.V. (North-Holland)
296
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
MIL-STD-781C, under the name ‘environmental tests’. The main problem with accelerated life tests is how to extrapolate from the data, obtained under conditions of severe stress, to statements about the failure behavior of the device under conditions of normal stress. Most of the published work pertaining to inference from accelerated tests has been restricted in the following two ways: First, it is required that every item which is subjected to an overstress is observed until it fails or is removed (censored) under the constant application of the large stress. The stresses are allowed to vary from one item to the other, but changing the stress level for an item before it fails or is withdrawn is not permitted. Such tests are also referred to as constant stress tests. The two references which drop this requirement are De Groot and Goel (1979), and Nelson (1980). Second, the life distribution of the items under every accelerated stress level is assumed to be known, except for the underlying parameters. This assumption may not be practical in many applications. Exceptions to the above are the recent papers by Shaked, Zimmer, and Ball (1979), Proschan and Singpurwalla (1979,198O) and Sethuraman and Singpurwaha (1982) which allow for the underlying life distributions to be unknown. However, the above references do require that the testing be conducted under constant stresses. The purpose of this paper is to consider the more realistic situation of accelerated life testing [cf. MIL-STD-781C, p. 71 wherein the stress on an unfailed item is allowed to change (increase) at a preassigned test time. Such testing is referred to as step-stress testing. Furthermore, our approach here is nonparametric in the sense that no assumptions are made about the form of the underlying distribution. To illustrate a particular application of step-stress testing suppose that cable insulation normally operates under a voltage stress of 20 kilovolts (kv). In order to make some statements about the reliability of the insulation under the normal operational voltage, m samples, the i-th one consisting of ni items, are subjected to a life test under the constant application of a voltage stress, say I$, where vr20kv, i=l,..., m. After tl hours of testing some of the items in each sample would have failed and the others would still be alive (i.e. not failed). The times to failure of the failed items would be recorded and so will the values of the associated stress I$, i=l,..., m. In order to terminate the test procedure in a short amount of time the stress on all the surviving items will now be increased from vi, i = 1,. . . , m, to a value V,, 1, where V,+ 1 is a large value, say 50 kv. The value V, + , has been chosen because it may be known to the engineer that under this stress the life of the cable insulation is quite short, so that all the remaining items will soon fail. We shall show that the additional information gained by increasing the stress levels to V,, , and inducing early failures will enable us to obtain better estimates of the life characteristics under the normal operating conditions. In Section 2 we introduce our model for step-stress testing and show its relationship to the models of De Groot and Goel (1979), Nelson (1980), and Endicott and Zoellner (1961). We shall show that by utilizing the very ideas that lead to ‘shock
M. Shaked,
N.D.
Singpurwaila
/ Step-stress
accelerated
291
life tests
models and wear processes’ [cf. Esary, Marshall and Proschan (1973)], we are able to interpret the otherwise unmotivated models of the previous authors. In Section 3 we describe our inference procedures, and in Section 4 we prove the strong consistency of our estimators. Throughout this paper, corresponding to any life length distribution function F, the survival function F is defined as 1 -F.
2. A model for step-stress testing Let Vi, . . . . V, be a collection of accelerated stress levels and let Vc and denote the use condition stress; assume that V, < Vi < ... < V1. An item is initially put on test under stress I$, where ii E { 1,2, . . . , I}. At a preassigned time ti the stress level on the item, provided that it is surviving at t, , is changed to vi,, where iz E { 1,2, . . . , I}. In general, the index iz is greater than the index i,, but this need not be so. The above strategy is continued so that at time tj, if the item is still surviving, the stress is changed from I$ to Vi,+,. Eventually, at time tJ, if the item is still surviving, its stress is changed to I$+, and it stays so until the item fails. Note that we need not assume that vj # y,+, , j= 1, . . . , J, nor do we require that K,l &<***I I$+,; however, in most applications this will indeed be the case. The ordered collection (vi,, t,, vi,, tZ, . . . , I$, fJ, vi,+,) will be called the stress pattern under which the test item runs and it will be denoted by Yet, where V=(~,,..., I$,+,) and t=(t,, . . . . tJ). Let Fi denote the distribution of the life length of the device when the underlying stress is Vi, i= 1, . . . ,I. Assume that Fi(t)=F(Fa’t),
tr0,
(2.1)
where F is some unknown distribution and (Yis an unknown constant. Notice that assuming (2.1) is equivalent to assuming that F,(t) =A(Cvat),
to 0,
(2.1’)
where A is some unknown distribution and C> 0 and czare unknown constants. This is so because one can identify F(a) of (2.1) with A(C.) of (2.1’). Assumption (2.1) is the familiar time transformation known as the ‘inverse power law’. The analysis given below can apply to other time transformations as well, but in order to save space these will not be discussed by us. Let T denote the life length of an item which is subjected to the test pattern (K,, *a*, vi,+,)o(tr, ***,fJ) and let H denote its life distribution. Thus, by (2.1), if t~[O,ti).
H(t)=P(T(t)=F;:,(t)=F(~~t)
We shall assume for now, with a justification H(t) = F( V;:piI+ vr(f - tl))
to be given in Section 2.1, that
if t E [t,, t2),
298
M. Shaked, N.D. Singpurwalla-/ Step-stress accelerated life tests
so that in general we have as our model for step-stress testing H(t)=F
(
i
y,i”(tj_tj_1)+
vE,(t-tl)
j=l
>if
TV VI,tt+ 11,
(2.2)
where i=O, 1, . . . . J, with te=O, and tJ+i =QO. A special case of the model given above is the model of De Groot and Goel(1979); in their case J= 1 and F is taken to be the exponential distribution. Under certain circumstances, which will be described soon, the above model is equivalent to that considered by Nelson (1980). These authors provide little motivation to justify their choice of the model. Thus, a contribution of this paper lies in the discussion below, which sheds some light on the models of De Groot and Goel and of Nelson and thereby provides the necessary motivation. A referee has observed that (2.2) can be written as H(I)=F(/;V’(u)du)
where J+1
VW =
c ~jx[fj_,,fJo;
j=l
here XD is the indicator function of the set D. This representation succinctly summarizes the model. It should be emphasized that this model assumes that the distribution of the life length is a function of the total accumulated Va and does not depend on the order of testing. The tester could run a test first to check that the life distribution is indeed independent of the order of the stress applied. 2.1. Motivation for the model The model (2.2) can be justified via two conceptually different scenarios which are plausible in practice. To describe the first scenario suppose that under the constant application of a stress, say I$, an item is subjected to shocks which occur in time according to the postulates of a Poisson process with a constant rate li= I$: i=l,..., I. If the probability that the item survives m shocks is Fm then the life length of the item has the survival function Ri(t)=
ODecAWit)m ~~, C m!
t20
(2.3)
m=O
This is the general shock model considered by Esary, Marshall, and Proschan (1973). Now consider an item which has been subjected to step-stress testing via a stess v; during the time interval [tj_ , , tj) for j = 1, . . . , J+ 1. Note that a change of stress corresponds to a change in the rate of the underlying Poisson process. Thus, following our convention that to= 0 and tJ+, = 00, the survival function of the item for
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
any tE[tl,tl+l), Ij(f)=
299
l=O,I ,..., J, is i WI=0
'
exp
[
j$,
1[ -
f: A;,(fj-tj-l)+Aj,+,(f-t,) j=l
~i,(~j-fj-l)+~i,+,(f-fl)
m Pm 1 * 11
z
(2.4)
Thus (2.2) must hold with m
F(t)=
e-lp
c ---ym. *=o
m.
(2.5)
Note that both (2.2) and (2.4) convey the notion that when the stress level changes the item is not ‘like new’ anymore; it retains its present state of damage from the old stresses to the new environment. Typically, the Ij, of (2.3) and (2.4), for m=O, 1, . . . . are unknown, and consequently the F of (2.5) is also not known. Thus, the nonparametric assumption underlying the model equation (2.2) is quite germane. Note that in many contexts is,,, is taken to be P(“)(x), where x is a fixed threshold, P is the distribution of the random variable representing the damage inflicted on the item by a shcok and pCrn) denotes the m-fold convolution of P with itself; see Esary, Marshall, and Proschan (1973). When P and x are both unknown F is also unknown and so the nonparametric assumption of (2.2) is again appropriate. The second setting which provides us with another motivation for our model is based on the ‘damage accumulation’ concept of Endicott and Zoellner (1961). Here it is assumed that under the influence of a constant stress the damage to an item accumulates at a constant rate. For example, we may assume that under stress level vi, the rate at which damage is accumulated as I.i = va, where Q is an unknown constant. Let X be the threshold of the item, assumed to be a random variable with an unknown distribution function F. Assume that the item fails when the accumulated damage reaches the threshold X, so that under stress 6 the life length of the item has distribution function Fi, where
This is precisely equation (2.1). Thus, under the stress pattern Vat, the distribution of the life length of the item is given by (2.2). Endicott and Zoellner (1961) have used this model for objectives which are different from those considered by us. When Vat = (Ve, V,) o tt and F is taken to be an exponential distribution, then (2.2) specializes to the model of De Groot and Goel (1979). Furthermore, it is not difficult to verify that the life distribution of items under the step-stress model of Nelson (1980) reduces to the model given by equation (2.2) if the life distribution of an item under a constant stress v is described by (2.1) for some life distribution F.
300
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
3. Estimation
of F,
In industrial step-stress life testing experiments it is a common practice to put several items on the test under various stress patterns. It will be helpful for us to recall from Section 2 that given the time points t = (I,, . . . , tJ and the corresponding stress levels V=( V;:,,Fz,, . . . , y,, V,,,,), with J$E {V,, V,, . . . , VI}, j= 1,2, . . . , J+ 1, the stress pattern is denoted by Vet. Assume that, in a step-stress life test, the collection of all the stress patterns is { Wet(‘) and that nk items are run under the stress ,-**, V(K)Ot(K)} pattern V(@o 0, k = 1, . . . , K. More specifically, for every k~ { 1,2, . . . , K}, t(k) = (t(k) Y(k)= ( v,‘“),. . . , v!” 9*--s vCk) 1 , . . . , ty), . . . , tz)) and ), where Vj(” E Jk+l J {v,, v,,..., VI}, j=l,2 ,..., J,+l. Some simplification in our notation will be achieved by pointing out that without loss of generality we can assume that t(l) = tc2)= *-- = tck)= t where t = (t,, . . . , tJ) for some fixed J. This is so because we can take the epochs at which changes in stress occur as the union of the ty)‘s, since we do not require Tjk)# J$t!. The stress patterns, then, are completely determined by the vectors Y(l), . . . , VcK), where y(k) = ( v/W, vz’k’,***, Vck)) J+, , k=l ,***, K Let F0 be the distribution of life lengths at use conditions stress V,, where, per our model assumption (2.1), &j(t) =F( vo”t>.
Our aim is to obtain an estimate of FO, say & using the failure data from stepstress testing as discussed above. Our estimate f10 is obtained in two steps. First an estimate of cz, say 6, is obtained, and then C?is used to transform all the observed life times to correspond to those that would have been obtained under V,. The transformed observations are then used to obtain the empirical distribution function of F,, & and so PO is our estimator of the distribution of lifelengths at V,. Let cl, /=1,2 )...) nk, kc&2 )..., K, denote the observed lifetimes under the stress pattern Vck)Ot; recall that nk is the total number of items which are tested under this pattern and that 6’) = t(‘)= -me= tcK)= t. Let Sk, 5 &2< -essSknr be the order statistics in the k-th sample; that is, Sk, is the l-th smallest observation associated with the group of items which are run under the test pattern Vck)ot, k=l,2 ,..., K. 3.1. Estimation of (r In order to estimate (Ywe need at least one time interval, say [tj_ 1, tj), during which items are tested under at least two different stress levels; otherwise, F is unidentifiable. This requirement is met whenever, for some k1 # k2, V-(kl)# Vck2). First let us assume that the first such interval is [0, tl), and that K= 2, so that, for t c [0, tl), P{Tl,st}=F((V/‘))at)
,
I=1 ,-**, n, ,
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
301
and P{T,,st}
=F((Vj2))V),
I= 1, . ..) n2,
where Y(r) # Yt2). Let G:(t) anrd G,(t) be the empirical distribution functions of the first and the second samples respectively. That is, I Gk(t) = ni’(number of Tkrs t), k= 1,2. Let G,’ be the right continuous G,‘(u)=sup{t:
inverse of Gk; that is,
G&)
UE [O,l),
(3.1)
k = 1,2, and let ~z=min(G,(t,),~~(f~)}.
We recommend that the reader graphically verify the meaning of these quantities. Note, for example, that if V,(r)< Vt@’then it is likely that Gr(tr)
*
Our proposed estimator of t9r2is 6r2, where u2
G;‘(U) du 42=
i
"u,
(3.2)
* e;'(u)
du
s0
and an estimator of (Ythen is .m a =
log
42
(3.3)
log( V;“/V,“‘) *
In Section 4 we will show that 6 is strongly consistent. Note that when t, = oo (3.2) reduces to the estimator (2.7) of Shaked, Zimmer, and Ball (1979), with (i,j) being replaced by (1,2). We shall now show that 8,, can also be written more explicitly as a function of the lifetimes Sk,, I= 1, . . . , nk; k= l,2. Let &=max{/: &(tr},
k= 1,2.
Then u2 is either II/n, or 12/n2 depending upon whether G,(t, c e2(t2)or otherwise. Suppose that u2 = Ir/nr ; let & = [(/,n2)/nr] where [a] denotes the largest integer which is less than or equal to a. In this case we can verify that
=ny’C;=, CT,,-T, ,I_, )(f,-~+l)+tt,-T,IJ~
g 12
n;’ Cf=,
(T2/- T2,1_,)(1;--I+ l)+(tr-
T2l;)l; ’
When u2 = 12/n2 the formula for 6t2 is of an analogous form.
(3.4)
302
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
Note that the estimator of a given by (3.3) is based upon failure data obtained under only two different stress levels V/” and Vy’. In practice however, it is very likely that the number of distinct stress levels under which life test data is obtained for the time interval [O,rr], is more than two. Thus, we would have several 0,‘s, each of which would lead us to an estimator of a. Consequently, what we need to consider next is a procedure for obtaining a pooled estimator of a. To this end, let the number of distinct (different) stress levels under which the life tests are conducted over the time interval [0, tr] be M, where 2rMsK. That is, M is the number of different VF)‘s among V/r’, . . . , VcK) , . Let us denote these distinct Vy)‘s by CI;‘),Ur@’ , **-, ujM , and let fi, be the number of items that are run under stress Ui”’ (that is, ii, is a sum of some Q’S), m = 1,2, . . . , M. For every pair of indices m and m’ with m
U2 =
min{
Gl(ij),
62
As before, we define Br2= ( l$12)/V/1))a and propose as an estimator of 8r2, u2 G;‘(U) du Q*2= s :I”’ (3.6) &r(u) du ’ s “=“I and as an estimator of a, a = log l$2/(log( $Vl$Ji))).
(3.7)
We shall show in Section 4, that this estimator of a is also strongly consistent.
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
303
We note that the estimator (3.7) reduces to the estimator (3.3), when j= 1 (with obvious change of notation). A formula similar to (3.4) can also be obtained, and when the number of distinct stress levels is M, for M<2, the +M(M- 1) estimates of a can be pooled as in (3.5). 3.2. Estimation of FO Once we have an estimate of a, 6, we transform
the observed life-times Tk/,
I= 1, . . . . nk, k= 1, . . . . K, as indicated below to obtain Fkj, where
Here, as was assumed before, to= 0 and tJ+ 1= oo. The empirical distribution function based on the N= C:=, nk transformed variables Tk:,,is our estimate of PO and is denoted by FO. The mean, the median, the various percentiles and the other parameters of F, can be estimated from their counterparts using Fe.
4. Properties of the estimator EO In this section we shall show that the estimator p0 discused in Section 3 is such that ~,,(x)-+F(x) uniformly in x with probability 1. For this, we shall assume that for each k, k=l,2 ,..., K, nk + 03 in such a way that nk/N-‘Pk> 0, where N=C:=, nk, and so c;=, Pk= 1. We shall first show that 8i2% 0i2 and, in general, gm,,,,% B,,,,. This result is then used to argue that &+a with probability 1. To formalize our development we will assume that the random variables Tkl, I= 1,2, . . . , nk; nk= 1,2, . . . ; k= 1, . . . , K, are defined on the same probability space and let P denoe the induced probability measure on this space. In considering 6iZ of (3.2), let us assume, without loss of generality, that V/i’= Vi < V, = Vi@). Then, by the Glivenko-Cantelli lemma, C?,*F, and 6, -*F2 uniformly on [O,t) with P-probability 1. Since F,(t,)
min{F,(t,), F2(tl)) =F,(t,)
M. Shaked. N.D. Singpurwalla / Step-stress accelerated life tests
304
and so
F,-‘(u) du
11
I
0
= 0
I
0
s w c-9
Fi- ’ (4 (4) W (4
0
I
0
=
s w (4
I’(v’/v*)asdF’(s)
=(V*/V,)“=4*.
I
0
Thus 0i2 of (3.2) is strongly consistent. Consider now 0i2 of (3.6). Here again, without loss of generality, assume that V!” = Vi < V2= y!“. Then G, -% B, and G2 s B2, where, because of (2.2), J j-l
B&)=F
1 (V~‘)a(f,-tm_,)+ m=l
V-;(M~_~) I
for tE [tj_i,ti), k= 1,2. ’ the first time interval Since I’(‘)= Y(*) for m= 1, . . ..j- 1 (recall that [t.J_1, ti) IS for whichyhe styess levels are different), if follows that ‘i’ (Vi))‘(f,-f,_i)=l$l
(Vz’)“(t,-t,_i)=a,
m=l
say.
Thus Bk(f)=F(a+
Vkol(t_tj_l)),fE[tj_1,ti),
It follows that, except on a set of P-probability
k=l,2. measure zero,
u+Bl(a)=B2(a) and ~2%
B,(o+ V~a(tj-tj_~))~
Thus
El@+
Vp(rj-(j-1))
B
a.s. 5
Bi ‘(u) du
In view of the above we conclude that 6-o. In order to show that &,+Fo uniformly with probability one, let rk/ be Fkj as defined in (3.8) with B replaced by CT.The distribution of each rk[ is easily seen to
M. Shaked, N.D. Singpurwalla / Step-stress accelerated life tests
305
be F of (2.1). From the strong consistency of G and the Gilvenko-Cantelli lemma, the almost sure convergence of p,, is therefore established. We close this section by remarking that without having data from a time interval in which stresses are applied at two or more different levels, it is not possible to estimate Fo. This follows from the fact that if H of (2.2) is explicitly given then F and a are not identifiable. That is, the same H can be obtained from different choices of F and a even if Vi, . . . , V,, 1 are given constants.
5. Summary and conclusions In this paper we have proposed a model and have developed a procedure for the analysis of data from accelerated life tests wherein the underlying stresses are allowed to vary during the test. This type of testing is more typical than the one in which items are tested under a constant stress, an assumption which has been commonly made in much of the previous literature. A prime example of the kind of testing considered here is described under what is known as ‘environmental testing’ of MIL-STD-781C. The model that we have proposed here has been motivated from two different points of view and it encompasses the models considered by other researchers who have also considered this type of accelerated testing. Our procedure is nonparametric in that we are not required to make an assumption about the underlying distribution of lifetimes. However, we do require the assumption of a time transformation function and also require that there exists at least one interval of time during which life tests have been conducted under at least two distinct stress levels. As of now, an investigator who has data from step-stress accelerated tests has a choice of the following three procedures for estimating the life distribution under the use conditions stress: The approach of DeGroot and Goel (1979) is Bayesian and requires some data obtained under the use conditions stress. In addition, it requires that all the items be switched from the use conditions stress to a particular prespecified higher stress level. The authors mention the possibility of applying their method to more general sets of data, but do not give details. For general sets of data from step-stress testing, the researcher may use the procedure of Nelson (1980) or the one of this paper. Nelson’s procedure should be used whenever the F of (2.1) is specified or can be assumed known. Note that whereas the procedure of this paper requires at least one time interval during which there be testing under at least two distinct stress levels, no such requirement is made of Nelson’s procedure. Also, we have not considered here the case of censored data, whereas Nelson has. Thus it appears necessary to impose a few requirements on the data in order to be able to conduct a nonparametric analysis.
306
M. Shaked. N.D. Singpurwalla / Step-stress accelerated life tests
Acknowledgment
We thank a referee for a careful reading of a draft of this paper and for helpful comments.
References [l] DeGroot, M. and Goel, P.K. (1979). Bayesian estimation and optimal designs in partially accelerated life testing. Naval Res. Logistic Quart. 26, 223-235. [2] Endicott, H.S. and Zoellner, J.A. (1961). A preliminary investigation of the steady and progressive stress testing of MICA capacitors. Proc. 7th National Symposium on Reliability and Quality Control, 229-235. [3] Esary, J.D., Marshall, A.W. and Proschan, F. (1973). Shock models and wear processes. Ann. Probab. 1, 627-649. [4] Military Standard-781-C (1977). Reliability design qualification and production acceptance tests: exponential distribution. Department of Defense, Washington, DC. [.5] Nelson, W. (1980). Accelerated life testing - step stress models and data analyses. IEEE Trans. Reliability 29, 103-108. [6] Proschan, F. and Singpurwalla, N.D. (1979). Accelerated life testing - A pragmatic Bayesian approach. Optimization Methods in Statistics (ed: Rustagi, J.), Academic Press, New York. [7] Proschan, F. and Singpurwalla, N.D. (1980). A new approach to inference from accelerated life tests. IEEE Trans. Reliability 29, 98-102. [8] Sethuraman, J. and Singpurwalla, N.D. (1982). Testing of hypotheses for distributions in accelerated life tests. J. Amer. Statist. Assoc. 77, 204-208. [9] Shaked, M., Zimmer, W.J. and Ball, C.A. (1979). A nonparametric approach to accelerated life testing. J. Amer. Statist. Assoc. 74, 694-699.