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Accelerated life testing when a process of production is unstable
STATISTICS& PROBABILITY LETTERS ELSEVIER
Statistics & Probability Letters 35 (1997) 269-275
Accelerated life testing when a process of production is unstable Vilijandas Bagdonavi6ius
a,
Mikhail Nikulin b, *
a Departement of Mathematical Statistic, University of Vilnius, Lithuania b UFR MI2S, UniversitO Bordeaux-2, Bordeaux, France and Steklov Mathematical Institute, St. Petersbury, R~sia Received July 1996; revised December 1996
Abstract
The developement of the heredity principle in reliability is discussed. The possible invariants in the generalized additive and multiplicative models were proposed. Estimation of these invariants with application in accelerated life testing is considered. A test for the heredity hypothesis is proposed. @ 1997 Elsevier Science B.V.
AMS classification: 62F10; 62J05; 62G05; 62N05 Keywords." Heredity principle; Transfer functional; Accelerated life testing; Reliability function; Semiparametric models
I. Introduction
Suppose that a time to failure Tx under a constant stress x is a non-negative random variable with the reliability function S,(t) = P{Tx > t}. A process of production is unstable if the reliability of items can change from one group of produced items to another. The models of accelereted life testing are some hypotheses about the change of the expressions of the reliability function or other reliability characteristics in dependence of the applied stress. For the concrete models the reliability characteristics under the "normal" stress Xl can be often written as some functions of the reliability characteristics under the stress x2, higher than "normal" and some function p(xl,x2) of the stresses Xl and x2. If the process of production is unstable and the function p is estimated using the preliminary experiments, then estimation of the reliability under the "normal" stress xl, using only the experiment under the higher stress x2 and an estimator ~ of p is possible if the function p is invariant going from one group of items to another. In this paper we consider semiparametric models of accelerated life, proposed by Bagdonavirius and Nikulin (1994-1997), the possible invariants in these models, estimation of the function p and goodness-of-fit tests for the considered models. * Correspondence address: UFR MI2S, Universit6 Bordeaux-2, 146 rue Leo Saignat, B.P. 69, 33076 Bordeaux Cedex, France. 0167-7152/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PH S01 67-7 1 5 2 ( 9 7 ) 0 0 0 2 2 - 9
270
V. Baodonavi~ius, M. Nikulin / &at&tics & Probability Letters 35 (1997) 269-275
2. Semiparametric multiplicative and additive models Consider the class fq = {G} of reliability functions, strictly decreasing and differentiable on their supports. For all G E f# denote H = G -1. Denote fxa = H o Sx. In this case G(fxa(t)) =Sx(t) and the moment t under the reliability function Sx is equivalent to the moment fxa(t) under the reliability function G. So f ~ ( t ) is called the G-resource used till the moment t under the stress x, the function fx(t) being the transfer function. We will say that the G-multiplicative (G-additive) model (see Bagdonavi6ius and Nikulin, 1995a) is satisfied on some set of stresses E if for all x E E dfxa (t)dt = r(x)df-~t(t)
(1)
dfxa(t) _ dfoa(t) - dt dt
(2)
or + a(x)
respectively. Here r : E --+ ~+, a : E --~ ~ are some functions of the stress and foa = H o So is some "baseline" transfer function. So for the G-multiplicative model the rate of resource use at the moment t is proportional to some "baseline" rate and the constant of proportionality depends only on the stress. In the case of G-additive model the stress influences additively the rate of resource use. The particular cases of these models when G ( t ) = e -t,
t>~0,
are the proportional and additive hazards models. Really, in this case
dfxG(t)=-S~(t)/Sx(t)=~x(t), where C~x(t) is the hazard rate under the stress x, and a0 is the basel&e hazard rate. In this case the equalities (1) and (2) are
~x(t) =r(x)~o(t),
~x(t) = a 0 ( t ) + a(x).
We obtain the proportional hazards or Cox model and additive hazards model. Taking
G(t) = e x p { - exp t},
tE~,
we have two models
~x(t) , , ~o(t) Ax(t) = rtx)A-
and
• x(t) Ax(t)
~o(t) - Ao(t)
+a(x),
where t
Ax(t) = fOt Ctx(U)du, Taking
1 G(t)= l + t '
t>~O,
Ao(t) =
fO
~o(u) du.
I~ Baodonavi6ius, M. Nikulin I Statistics & Probability Letters 35 (1997) 269-275
271
we have two models Otx(t) _ . ,~0(t) Sx(t) - r [ X ) s - ~ ) '
~x(t) Sx(t)
~0(t) + a(x), So(t~)
where the first model is a generalization of the well-known logistic regression model; it is near to the proportional hazards model when t is small. Taking 1
G(t) -- 1 + e t'
t E R,
we obtain Ctx(t) O~o(t) 1 - Sx(t) : r(x) 1 - S0(t)'
O~x(t) _ ~o(t) + a ( x ) , 1 - Sx(t) 1 - So(t--~)
where the first model is near to the proportional hazards model when t is large. The models (1) and (2) imply that for xl,x2 E E Sx2(t) = G{p(xl,x2)H o Sxl(t)},
(3)
Sx:(t) = G { H o Sx.(t) + b(xl,x2)t},
(4)
where p(xl,x2)=r(xz)/r(xl)
and
b ( x l , x z ) = a ( x 2 ) - a(xl).
In the particular case, taking G = Sx,, we obtain the accelerated life model: Sx2(t) = Sx, (p(xl ,x2)t).
(5)
If the reliability functions Sx, and Sx2, the functions p and b are completely unknown and G is specified in the models (3) and (4); these models are semiparametric.
3. The heredity hypothesis
Suppose that the G-multiplicative or G-additive model is true, and the process of production is unstable. We will say that the heredity hypothesis is satisfied on E, if for all Xl,X2 E E the function p(xl,x2) or b(Xl,X2) is correspondingly invariant going from one group of items to the another one. If the heredity hypothesis is satisfied on E and sufficiently large "normal" and accelerated data is accumulated during a long period of observations, then good estimators of the functions p(xl,x2) or b(Xl,X2) can be obtained. The reliability of newly produced items under the "normal" stress Xl can be estimated from accelerated life data under the stress x2 >xl, using the estimators t~(xl,x2) or/~(Xl,X2) and without using the experiment under the normal stress. We call the formulated hypothesis the heredity hypothesis as it has some associations with the "heredity principle" of Kartashov and Perrote (1968) and is motivated by it. The heredity principle is formulated as follows. Suppose items of ith group are observed under the stress x and are characterized by some multivariate technical parameter W/(x). The parameter IV//is good if c<~ Wi(x)<~d,
c, d E R k.
The parameter Wg is some function of the inferior physical parameters vi of items:
Wii(x) = fi(vi(x) ).
V. Bagdonavirius, M. Nikulin I Statistics & Probability Letters 35 (1997) 269-275
272
It is supposed that there exist a function ~b such that v2(x) = ¢(v~(x),x).
The heredity principle states that the distribution of the random v e c t o r s Yi(X) can change going from one group of items to another but the functions j] and ~b are invariant. For more details about the heredity principle see, for example, Bagdonavirius and Nikulin (1995a), Kartashov and Perrote (1968), Singpurwalla (1987), and Rukhin and Hsieh (1987).
4. Estimation of p and b
Suppose that ni items are tested under the stress xi (i = 1,2). Denote No(t ) the numbers of observed failures and Yiy(t) the numbers of items "at risk" (non-censored and non-failed) just prior to t for each item on test. Consider the filtration o~t, generated by the processes Nij and Yij:
o~t =a{Nij(s), Y/)(s): O<,s<~t, j = 1..... ni, i= 1,2}. We suppose that the times to failure are absolutely continuous random variables and the compensators Aij of the counting processes Nij are absolutely continuous:
Aij(t) = foot •iy(S) ds. We assume that the random intensities 2~j satisfy the multiplicative intensity model
~At) = ~(t)r~At), where a i ( t ) = ~x,(t) are the hazard rates in the ith group. Denote ni
ni
Ni(t): ZNij(t)'
ni
Ai(t): ZAij(t)'
j=l
Yi(t): Z
j=l
GI c~=-~-,
Mi(t)=N,(t)-Ai(t),
Yij(t),
j=l
ff=~oH.
The Doob-Meyer decomposition 5/2 =M2 + A2; the equalities (3) and (4) imply M2(t) = Nz(t) - p(xl,x:)
/o'
O(S2(u))Y2(u)dH
0
SI(U)
(6)
or
M2(t) = N2(t) -
f0'
~(S2(u))Yz(u)dH
o
Sl(U)
-
b(xl,x2)
f0'
~,(S2(u))Yz(u)du
for the G-multiplicative and G-addditive models respectively. Note that EMz(t) :- O. Denote Ji(t)=I{Yi(t)>O} the indicator of the event {Yi(t)>0}, zn•' = s u p { t : Y i ( t ) > 0 } ,
zn = z ~ ' A z 2n 2.
(7)
V. Bagdonavidius, M. Nikulin I Statistics & Probability Letters 35 (1997) 269-275
273
We propose to estimate the functions p and b by solving the estimating equations U(p,z")=O
or
U(b,z")=O,
respectively, where g ( p , t ) = N2(t) - p(xl,x2)
~O(S2(u-))Y2(u)dH 0 Sl(U),
U(b, t) = N2(t) - ~ t ~/'(S2(u-))Y2(u) dH o S, (u) - b(Xl, x2 ) ~o' ~l,(S2(u-))Y2(u) du and $1, $2 are the Kaplan-Meier estimators of the reliability functions $1 and $2. So we obtain the estimators Nz(z") ^ , ,a(Xl,X2) = fo" ff(S2(u-))Y2(u) dH o S1 (u)
(8)
~9(Xl,X2 ) = Ng('cn) -- JO" ~l(S2(u-))Y2(u)dH o Sl(u)
(9)
fo" ~l(S2(u- ) )Y2(u) du Suppose that the estimators ~ or D are obtained from the preliminary experiments and the heredity principle is satisfied. If the newly produced items are tested only under the accelerated stress Xl and the Kaplan-Meier estimator Sx, is obtained, then the reliability function Sx2 under the "normal" stress x2 can be estimated as follows: Sx2(t) = G{p(xl,X2)H o Sx,(t)}
or
,~x2(t)= G{H o Sx,(t) + b(x,,x2)t}.
5. Asymptotical properties of the estimators ~ and Denote n = nl + nz and suppose that ni/n-+ liE]O,l[,
zn e , z,
sup IY,.(u)/ni - yi(u)l *" >0 uC[0,t°]
as n --+ cx~.
Denote Bn =
O o S2(u)Y2(u)dH o SI(U),
Dn =
O o S2(u)Y2(u)du.
Using the results for the Kaplan-Meier estimators, Rebolledo's central theorem for martingales (see Fleming and Harrington, 1991 ), we obtained YG(fi-P)
~>N(0, a~),
vG(b-b)
~ , N ( 0 , a2),
the variances a2 and a~ can be consistently estimated by the statistics n
¢~1(U) --
Yl(u)
[
Ip o S2(z")Y2(zn)H ' o SI(z")SI(z n) -
/u
H' o Sl(v)S1(v) d(¢ o S2(v)Y2(v))
]
274
K Bagdonavi~ius, M. Nikulin / Statistics & Probability Letters 35 (1997) 269-275
Y2(v)O'oS2(v)S2(v)dHoSl(v) ,
~2(u)= 1 + ~
nff
4 = ~
(a~2(u) dN, (u) + a~'E(u) dN2(u)),
a *(u)= " l~" [~b° S2(zn)r2(zn)H' ° S'(zn)s'(zn) - ~ H '
a2(u)= l+r--~
° S'(v)S'(v)d(~k ° S2(v)Y2(v))]
Y2(v)O'o~2(v)~2(v)(dnO~l(V)+~,dv).
Taking into account that p is positive, the rate of convergence to the normal distribution in the G_ multiplicative model can be ameliorated considering the estimator ¢* = In ¢. If we denote p* = In p, then O-2 2 __ P v'~(¢* - p*) ~, N(0, @ ) where O-o* - )-~"
The variance 4 can be consistently estimated by ~o2. = ~/¢2. If during m different periods the estimators ~*(l)=ln¢(1),...,¢*(m)=ln¢(m)
and
~O),...,~(m)
are obtained and the heredity principle is satisfied, we can determine
¢.= f-: ¢*(i) / m E" i=l ( p * )
1
^
^(i)'2'
¢ = e p"
i=l (O-p')
and
where
/~(i) /
m
i=1 [O-b ) l
i=l
1
(~i2)2 and (e~i))2 are the
estimators of the variances of ¢*(i) and ~(i) respectively.
6. Test for the heredity principle Suppose that two groups of n{1) and n~l) items, produced during period (Sl, tl ), are tested under the stresses Xl and x2 respectively and other two groups of n{2) and n~2) items, produced during some another period (s2, t2), are tested also under the stresses Xl and x2, respectively. If the heredity principle is true, the estimators ¢(1) and ¢(2) (or the estimators ~(l) and ~(2)) from the first and the second period, respectively, are estimating the same parameter p (or b). Denote n(1) =n~0 + n~2) n(2) =nl-(2) 7"---r~2-(2) and
N = n (0 + n (2).
Suppose that n(J) i ~
oo,
_ (i J ) /t~ /_(j) ~ r~
I-i( j ) ,
n(J)/N
k (j).
V. Bagdonavi~ius, M. Nikulin I Statistics & Probability Letters 35 (1997) 269-275
275
In the case of G-multiplicative model the test statistic
ln f(1) 2 ,(',,G),2 + n(1)
9 x2(1), -
-
n(2)
and in the case of G-additive model the test statistic XJ =
(bO) - b(2))2 n(1)
~-~ Z2(1),
n(2)
see, for example, Greenwood and Nikulin (1996). So the critical regions with the approximative significance level ~ are determined by inequality X2 > X2_~(1) or by inequality Xff > ;~_~(1), respectively, where ;~_~(1) is the (1 -a)-quantile of the chi-square distribution with one degree of freedom. References Bagdonavi~ius, V. and M. Nikulin (1994), Stochastic models of accelerated life, in: R. Guti&rez and M.J. Valderrama, eds., Selected Topics on Stochastic Modelling (World Scientific, Singapore) pp. 73-87. Bagdonavi~ius, V. and M. Nikulin (1995a), Semiparametric models in accelerated life testing, Queen's Papers in Pure and Applied Mathematics, Vol. 98 (Queen's University, Kingston, Ontario, Canada) p. 70. Bagdonavi~ius, V. and M. Nikulin (1995b), On accelerated testing of system, European J. Diagnosis Safety Automation 5, 307-316. Bagdonavi~ius, V. and M. Nikulin (1996), Analyses of generalized additive semiparametric models, Comptes Rendus Academie Sci. Paris S6r. I, 323, 1079-1084. Bagdonavi~ius, V. and M. Nikulin (1997), Transfer functionals and semiparametric regression models, Biometrika, 84, N2. To appear. Fleming, T.R. and D.P. Harrington (1991), Counting Processes and Survival Analysis (Wiley, New York). Greenwood, P.E. and M.S. Nikulin (1996), A Guide to Chi-squared Testing (Wiley, New York). Kartashov, G.D. and A.I. Perrote (1968), On the principle of "heredity" in reliability theory, Eng. Cybernet. 9, 231-245. Lin, D.Y. and Z. Ying (1996), Semiparametric analysis of the general additive-multiplicative hazard models for counting processes, Ann. Statist. 23, 1712-1734. Nelson, W. (1990), Accelerated Testing. Statistical Models, Test Plans and Data Analysis (Wiley, New York). Rukhin, A.L. and H.K. Hsieh (1987), Survey of Soviet work in reliability, Statist. Sci. 294, 484-503. Singpurwalla, N.D. (1987), Comment on "Survey of Soviet Work in Reliability" by A.L. Rukhin and H.K. Hsieh, Statist. Sci. 2, 497-498.
Statistics & Probability Letters 35 (1997) 269-275
Accelerated life testing when a process of production is unstable Vilijandas Bagdonavi6ius
a,
Mikhail Nikulin b, *
a Departement of Mathematical Statistic, University of Vilnius, Lithuania b UFR MI2S, UniversitO Bordeaux-2, Bordeaux, France and Steklov Mathematical Institute, St. Petersbury, R~sia Received July 1996; revised December 1996
Abstract
The developement of the heredity principle in reliability is discussed. The possible invariants in the generalized additive and multiplicative models were proposed. Estimation of these invariants with application in accelerated life testing is considered. A test for the heredity hypothesis is proposed. @ 1997 Elsevier Science B.V.
AMS classification: 62F10; 62J05; 62G05; 62N05 Keywords." Heredity principle; Transfer functional; Accelerated life testing; Reliability function; Semiparametric models
I. Introduction
Suppose that a time to failure Tx under a constant stress x is a non-negative random variable with the reliability function S,(t) = P{Tx > t}. A process of production is unstable if the reliability of items can change from one group of produced items to another. The models of accelereted life testing are some hypotheses about the change of the expressions of the reliability function or other reliability characteristics in dependence of the applied stress. For the concrete models the reliability characteristics under the "normal" stress Xl can be often written as some functions of the reliability characteristics under the stress x2, higher than "normal" and some function p(xl,x2) of the stresses Xl and x2. If the process of production is unstable and the function p is estimated using the preliminary experiments, then estimation of the reliability under the "normal" stress xl, using only the experiment under the higher stress x2 and an estimator ~ of p is possible if the function p is invariant going from one group of items to another. In this paper we consider semiparametric models of accelerated life, proposed by Bagdonavirius and Nikulin (1994-1997), the possible invariants in these models, estimation of the function p and goodness-of-fit tests for the considered models. * Correspondence address: UFR MI2S, Universit6 Bordeaux-2, 146 rue Leo Saignat, B.P. 69, 33076 Bordeaux Cedex, France. 0167-7152/97/$17.00 (~) 1997 Elsevier Science B.V. All rights reserved PH S01 67-7 1 5 2 ( 9 7 ) 0 0 0 2 2 - 9
270
V. Baodonavi~ius, M. Nikulin / &at&tics & Probability Letters 35 (1997) 269-275
2. Semiparametric multiplicative and additive models Consider the class fq = {G} of reliability functions, strictly decreasing and differentiable on their supports. For all G E f# denote H = G -1. Denote fxa = H o Sx. In this case G(fxa(t)) =Sx(t) and the moment t under the reliability function Sx is equivalent to the moment fxa(t) under the reliability function G. So f ~ ( t ) is called the G-resource used till the moment t under the stress x, the function fx(t) being the transfer function. We will say that the G-multiplicative (G-additive) model (see Bagdonavi6ius and Nikulin, 1995a) is satisfied on some set of stresses E if for all x E E dfxa (t)dt = r(x)df-~t(t)
(1)
dfxa(t) _ dfoa(t) - dt dt
(2)
or + a(x)
respectively. Here r : E --+ ~+, a : E --~ ~ are some functions of the stress and foa = H o So is some "baseline" transfer function. So for the G-multiplicative model the rate of resource use at the moment t is proportional to some "baseline" rate and the constant of proportionality depends only on the stress. In the case of G-additive model the stress influences additively the rate of resource use. The particular cases of these models when G ( t ) = e -t,
t>~0,
are the proportional and additive hazards models. Really, in this case
dfxG(t)=-S~(t)/Sx(t)=~x(t), where C~x(t) is the hazard rate under the stress x, and a0 is the basel&e hazard rate. In this case the equalities (1) and (2) are
~x(t) =r(x)~o(t),
~x(t) = a 0 ( t ) + a(x).
We obtain the proportional hazards or Cox model and additive hazards model. Taking
G(t) = e x p { - exp t},
tE~,
we have two models
~x(t) , , ~o(t) Ax(t) = rtx)A-
and
• x(t) Ax(t)
~o(t) - Ao(t)
+a(x),
where t
Ax(t) = fOt Ctx(U)du, Taking
1 G(t)= l + t '
t>~O,
Ao(t) =
fO
~o(u) du.
I~ Baodonavi6ius, M. Nikulin I Statistics & Probability Letters 35 (1997) 269-275
271
we have two models Otx(t) _ . ,~0(t) Sx(t) - r [ X ) s - ~ ) '
~x(t) Sx(t)
~0(t) + a(x), So(t~)
where the first model is a generalization of the well-known logistic regression model; it is near to the proportional hazards model when t is small. Taking 1
G(t) -- 1 + e t'
t E R,
we obtain Ctx(t) O~o(t) 1 - Sx(t) : r(x) 1 - S0(t)'
O~x(t) _ ~o(t) + a ( x ) , 1 - Sx(t) 1 - So(t--~)
where the first model is near to the proportional hazards model when t is large. The models (1) and (2) imply that for xl,x2 E E Sx2(t) = G{p(xl,x2)H o Sxl(t)},
(3)
Sx:(t) = G { H o Sx.(t) + b(xl,x2)t},
(4)
where p(xl,x2)=r(xz)/r(xl)
and
b ( x l , x z ) = a ( x 2 ) - a(xl).
In the particular case, taking G = Sx,, we obtain the accelerated life model: Sx2(t) = Sx, (p(xl ,x2)t).
(5)
If the reliability functions Sx, and Sx2, the functions p and b are completely unknown and G is specified in the models (3) and (4); these models are semiparametric.
3. The heredity hypothesis
Suppose that the G-multiplicative or G-additive model is true, and the process of production is unstable. We will say that the heredity hypothesis is satisfied on E, if for all Xl,X2 E E the function p(xl,x2) or b(Xl,X2) is correspondingly invariant going from one group of items to the another one. If the heredity hypothesis is satisfied on E and sufficiently large "normal" and accelerated data is accumulated during a long period of observations, then good estimators of the functions p(xl,x2) or b(Xl,X2) can be obtained. The reliability of newly produced items under the "normal" stress Xl can be estimated from accelerated life data under the stress x2 >xl, using the estimators t~(xl,x2) or/~(Xl,X2) and without using the experiment under the normal stress. We call the formulated hypothesis the heredity hypothesis as it has some associations with the "heredity principle" of Kartashov and Perrote (1968) and is motivated by it. The heredity principle is formulated as follows. Suppose items of ith group are observed under the stress x and are characterized by some multivariate technical parameter W/(x). The parameter IV//is good if c<~ Wi(x)<~d,
c, d E R k.
The parameter Wg is some function of the inferior physical parameters vi of items:
Wii(x) = fi(vi(x) ).
V. Bagdonavirius, M. Nikulin I Statistics & Probability Letters 35 (1997) 269-275
272
It is supposed that there exist a function ~b such that v2(x) = ¢(v~(x),x).
The heredity principle states that the distribution of the random v e c t o r s Yi(X) can change going from one group of items to another but the functions j] and ~b are invariant. For more details about the heredity principle see, for example, Bagdonavirius and Nikulin (1995a), Kartashov and Perrote (1968), Singpurwalla (1987), and Rukhin and Hsieh (1987).
4. Estimation of p and b
Suppose that ni items are tested under the stress xi (i = 1,2). Denote No(t ) the numbers of observed failures and Yiy(t) the numbers of items "at risk" (non-censored and non-failed) just prior to t for each item on test. Consider the filtration o~t, generated by the processes Nij and Yij:
o~t =a{Nij(s), Y/)(s): O<,s<~t, j = 1..... ni, i= 1,2}. We suppose that the times to failure are absolutely continuous random variables and the compensators Aij of the counting processes Nij are absolutely continuous:
Aij(t) = foot •iy(S) ds. We assume that the random intensities 2~j satisfy the multiplicative intensity model
~At) = ~(t)r~At), where a i ( t ) = ~x,(t) are the hazard rates in the ith group. Denote ni
ni
Ni(t): ZNij(t)'
ni
Ai(t): ZAij(t)'
j=l
Yi(t): Z
j=l
GI c~=-~-,
Mi(t)=N,(t)-Ai(t),
Yij(t),
j=l
ff=~oH.
The Doob-Meyer decomposition 5/2 =M2 + A2; the equalities (3) and (4) imply M2(t) = Nz(t) - p(xl,x:)
/o'
O(S2(u))Y2(u)dH
0
SI(U)
(6)
or
M2(t) = N2(t) -
f0'
~(S2(u))Yz(u)dH
o
Sl(U)
-
b(xl,x2)
f0'
~,(S2(u))Yz(u)du
for the G-multiplicative and G-addditive models respectively. Note that EMz(t) :- O. Denote Ji(t)=I{Yi(t)>O} the indicator of the event {Yi(t)>0}, zn•' = s u p { t : Y i ( t ) > 0 } ,
zn = z ~ ' A z 2n 2.
(7)
V. Bagdonavidius, M. Nikulin I Statistics & Probability Letters 35 (1997) 269-275
273
We propose to estimate the functions p and b by solving the estimating equations U(p,z")=O
or
U(b,z")=O,
respectively, where g ( p , t ) = N2(t) - p(xl,x2)
~O(S2(u-))Y2(u)dH 0 Sl(U),
U(b, t) = N2(t) - ~ t ~/'(S2(u-))Y2(u) dH o S, (u) - b(Xl, x2 ) ~o' ~l,(S2(u-))Y2(u) du and $1, $2 are the Kaplan-Meier estimators of the reliability functions $1 and $2. So we obtain the estimators Nz(z") ^ , ,a(Xl,X2) = fo" ff(S2(u-))Y2(u) dH o S1 (u)
(8)
~9(Xl,X2 ) = Ng('cn) -- JO" ~l(S2(u-))Y2(u)dH o Sl(u)
(9)
fo" ~l(S2(u- ) )Y2(u) du Suppose that the estimators ~ or D are obtained from the preliminary experiments and the heredity principle is satisfied. If the newly produced items are tested only under the accelerated stress Xl and the Kaplan-Meier estimator Sx, is obtained, then the reliability function Sx2 under the "normal" stress x2 can be estimated as follows: Sx2(t) = G{p(xl,X2)H o Sx,(t)}
or
,~x2(t)= G{H o Sx,(t) + b(x,,x2)t}.
5. Asymptotical properties of the estimators ~ and Denote n = nl + nz and suppose that ni/n-+ liE]O,l[,
zn e , z,
sup IY,.(u)/ni - yi(u)l *" >0 uC[0,t°]
as n --+ cx~.
Denote Bn =
O o S2(u)Y2(u)dH o SI(U),
Dn =
O o S2(u)Y2(u)du.
Using the results for the Kaplan-Meier estimators, Rebolledo's central theorem for martingales (see Fleming and Harrington, 1991 ), we obtained YG(fi-P)
~>N(0, a~),
vG(b-b)
~ , N ( 0 , a2),
the variances a2 and a~ can be consistently estimated by the statistics n
¢~1(U) --
Yl(u)
[
Ip o S2(z")Y2(zn)H ' o SI(z")SI(z n) -
/u
H' o Sl(v)S1(v) d(¢ o S2(v)Y2(v))
]
274
K Bagdonavi~ius, M. Nikulin / Statistics & Probability Letters 35 (1997) 269-275
Y2(v)O'oS2(v)S2(v)dHoSl(v) ,
~2(u)= 1 + ~
nff
4 = ~
(a~2(u) dN, (u) + a~'E(u) dN2(u)),
a *(u)= " l~" [~b° S2(zn)r2(zn)H' ° S'(zn)s'(zn) - ~ H '
a2(u)= l+r--~
° S'(v)S'(v)d(~k ° S2(v)Y2(v))]
Y2(v)O'o~2(v)~2(v)(dnO~l(V)+~,dv).
Taking into account that p is positive, the rate of convergence to the normal distribution in the G_ multiplicative model can be ameliorated considering the estimator ¢* = In ¢. If we denote p* = In p, then O-2 2 __ P v'~(¢* - p*) ~, N(0, @ ) where O-o* - )-~"
The variance 4 can be consistently estimated by ~o2. = ~/¢2. If during m different periods the estimators ~*(l)=ln¢(1),...,¢*(m)=ln¢(m)
and
~O),...,~(m)
are obtained and the heredity principle is satisfied, we can determine
¢.= f-: ¢*(i) / m E" i=l ( p * )
1
^
^(i)'2'
¢ = e p"
i=l (O-p')
and
where
/~(i) /
m
i=1 [O-b ) l
i=l
1
(~i2)2 and (e~i))2 are the
estimators of the variances of ¢*(i) and ~(i) respectively.
6. Test for the heredity principle Suppose that two groups of n{1) and n~l) items, produced during period (Sl, tl ), are tested under the stresses Xl and x2 respectively and other two groups of n{2) and n~2) items, produced during some another period (s2, t2), are tested also under the stresses Xl and x2, respectively. If the heredity principle is true, the estimators ¢(1) and ¢(2) (or the estimators ~(l) and ~(2)) from the first and the second period, respectively, are estimating the same parameter p (or b). Denote n(1) =n~0 + n~2) n(2) =nl-(2) 7"---r~2-(2) and
N = n (0 + n (2).
Suppose that n(J) i ~
oo,
_ (i J ) /t~ /_(j) ~ r~
I-i( j ) ,
n(J)/N
k (j).
V. Bagdonavi~ius, M. Nikulin I Statistics & Probability Letters 35 (1997) 269-275
275
In the case of G-multiplicative model the test statistic
ln f(1) 2 ,(',,G),2 + n(1)
9 x2(1), -
-
n(2)
and in the case of G-additive model the test statistic XJ =
(bO) - b(2))2 n(1)
~-~ Z2(1),
n(2)
see, for example, Greenwood and Nikulin (1996). So the critical regions with the approximative significance level ~ are determined by inequality X2 > X2_~(1) or by inequality Xff > ;~_~(1), respectively, where ;~_~(1) is the (1 -a)-quantile of the chi-square distribution with one degree of freedom. References Bagdonavi~ius, V. and M. Nikulin (1994), Stochastic models of accelerated life, in: R. Guti&rez and M.J. Valderrama, eds., Selected Topics on Stochastic Modelling (World Scientific, Singapore) pp. 73-87. Bagdonavi~ius, V. and M. Nikulin (1995a), Semiparametric models in accelerated life testing, Queen's Papers in Pure and Applied Mathematics, Vol. 98 (Queen's University, Kingston, Ontario, Canada) p. 70. Bagdonavi~ius, V. and M. Nikulin (1995b), On accelerated testing of system, European J. Diagnosis Safety Automation 5, 307-316. Bagdonavi~ius, V. and M. Nikulin (1996), Analyses of generalized additive semiparametric models, Comptes Rendus Academie Sci. Paris S6r. I, 323, 1079-1084. Bagdonavi~ius, V. and M. Nikulin (1997), Transfer functionals and semiparametric regression models, Biometrika, 84, N2. To appear. Fleming, T.R. and D.P. Harrington (1991), Counting Processes and Survival Analysis (Wiley, New York). Greenwood, P.E. and M.S. Nikulin (1996), A Guide to Chi-squared Testing (Wiley, New York). Kartashov, G.D. and A.I. Perrote (1968), On the principle of "heredity" in reliability theory, Eng. Cybernet. 9, 231-245. Lin, D.Y. and Z. Ying (1996), Semiparametric analysis of the general additive-multiplicative hazard models for counting processes, Ann. Statist. 23, 1712-1734. Nelson, W. (1990), Accelerated Testing. Statistical Models, Test Plans and Data Analysis (Wiley, New York). Rukhin, A.L. and H.K. Hsieh (1987), Survey of Soviet work in reliability, Statist. Sci. 294, 484-503. Singpurwalla, N.D. (1987), Comment on "Survey of Soviet Work in Reliability" by A.L. Rukhin and H.K. Hsieh, Statist. Sci. 2, 497-498.