j. appl. prob. vol. 37, no. 2 (june 2000

J. Appl. Prob. 37, 521–533 (2000)Printed in Israel

Applied Probability Trust 2000

MULTICOMPONENT LIFETIME DISTRIBUTIONSIN THE PRESENCE OF AGEING

JUANJUAN FAN,∗ University of California

S. G. GHURYE,∗∗ University of Alberta

RICHARD A. LEVINE,∗ ∗∗∗ University of California

Abstract

Lifetime distributions for multicomponent systems are developed through the interplayof ageing and stress shocks to the system. The ageing process is explicitly modeled by anexponential function with rate affected by the magnitude of stresses from a compoundPoisson process shock model. Applications of these life distributions and associatedfailure rates towards the study of multicomponent system survival are discussed. Inparticular, we illustrate the behavior of these survival functions in relevant subsets of theparameter space.

Keywords: Poisson shock process; ageing process; survival function; failure rate

AMS 1991 Subject Classification: Primary 60K10Secondary 62N05

1. Introduction

A common problem faced in the engineering and medical disciplines is that of survival ofa machine, or individual, under stresses to the system. A natural assumption one may makein constructing models of survival is that of ageing: the longer the system has been in use,the greater the effect of the stress on the system and thus the larger the likelihood of failure.We wish to make it clear that when we refer to ‘ageing’, we do not have in mind either themechanism of or the intrinsic process of ageing, but merely the differential effect of the passageof time on different systems.

We assume the system of interest is exposed to stresses according to the Poisson shockmodel as developed in Marshall and Olkin (1967), Esary et al. (1973), and Ghurye and Mar-shall (1984). For more recent applications of and references to the shock model see Sheu andGriffith (1996).

The incorporation of the ageing process in Poisson shock models is not a trivial problem.For a simple example, consider a system subject to shocks of unit magnitude from a simplePoisson process P(λ) where a total damage of two units results in system failure. Let T denotea random variable representing the time at which the component fails for the first time. Thesurvival function may be written as

G(t) = Pr(T > t) = e−λt (1 + λt).

Received 25 March 1998; revision received 31 August 1999.∗ Postal address: Division of Statistics, University of California, Davis, CA 95616, USA.∗∗ Postal address: University of Alberta, Edmonton, Canada.∗∗∗ Email address: [email protected]

521

522 J. FAN ET AL.

The hazard rate for this function,

h(t) = lim�t↓0

Pr{t ≤ T < t + �t | T ≥ t}�t

= λ2t

(1 + λt),

is increasing in t , which may be presumed indicative of an ageing process. However, thismodel does not show any symptoms of ageing as it assumes that the system waits passively forthe second shock to cause failure.

Ageing is often modeled by accumulated damage from non-fatal shocks (see, for example,Fagiuoli and Pellerey (1994)). However, these models do not explicitly represent the physicalprocess of ageing and/or present unwieldy mathematical expressions for the ageing factor(Ghurye (1987)).

In this paper, we consider models based on compound Poisson process shocks incorporatingintuitive representations of ageing. In particular, we explicitly represent the ageing processthrough an exponential decay function. The exponential distribution is widely used to modelthe increase of human mortality with age in the medical literature (Horiuchi and Wilmoth(1997)) and plays a central role in life testing and reliability models (see Balakrishnan andBasu (1995)). Intuitively, exponential ageing arises from the idea of a system consisting ofseveral identical components in parallel. As some components fail, the stress on the survivorsincreases, consequently increasing their failure probability. The exponential function thusapplies well for the ageing process since the failure probability of the surviving componentsincreases as the number of surviving components of the system decreases. The combination ofthe Poisson shock models and exponential ageing, as we will see, explicitly models the ageingprocess while retaining fairly simple survival functions.

Our ultimate goal is the development of optimal repair/replacement policies in the presenceof realistic maintenance cost functions. Such a cost analysis requires an understanding of thelifetime distributions underlying the survival of the system. Consequently, in this paper, westudy survival functions and associated failure rates under the shock model proposed, in aneffort to understand their behavior in various ranges of the relevant parameter space. As thisstudy is a task unto itself, we will hold off the application of our survival models to the optimalrepair scheme problem to future work.

The paper unfolds as follows. In Section 2, we introduce lifetime distributions in thepresence of ageing for a one-component system. We also compare the survival functions of ourmodel in this situation with one in which the ageing process is absent. The necessary inclusionof an ageing process in survival models is apparent from this explication. In Sections 3 and 4,we extend our model to a two-component and then general K -component system. In Section 5,we study the behavior of the two-component survival probabilities in different domains of theparameter space. We are particularly interested in the interplay between ageing and the systemshocks with respect to the failure probabilities. Throughout the work, we take the generic viewof studying the probability of the survival of system components subject to stresses whichcause damage and eventual failure. Finally, in Section 6, we fit our model to some medicaldata.

2. One-component shock model

We consider an extension of the shock model of Marshall and Olkin (1967) as suggested byGhurye (1987). Suppose we are interested in studying a system consisting of one component.The system is affected by a compound Poisson process shock of magnitude x and rate λ,denoted by P(λ; x). The system is further assumed to age exponentially, independently of

Multicomponent lifetime distributions in the presence of ageing 523

the shocks, at a rate δ, denoted by the Exp(δ) function. Under these suppositions, for a one-component system of age a ≥ 0 at time zero, a shock of magnitude x ≥ 0 arriving at timeu ≥ 0 is fatal to the component with probability

1 − exp[−δ(a + u) − x]. (1)

Note that this model assumes failure to be due to a fatal shock provided by the P(λ; x)

process. Failure can occur only at shock arrival moments irrespective of the magnitude x .Susceptibility to the shocks increases with age according to the Exp(δ). Thus, as exhibited in(1), the ageing process essentially increases the magnitude of the shock by adding a constantδ×(age) to the exponential term. An inherent consequence of this assumption is that the shocksdo not affect ageing in that the internal clock of the system does not accelerate. Furthermore,we assume for the present that the damage is not self-propagating after infliction. We haveconsidered the case of self-propagating damage, but since this case is even more complex, itsdetailed discussion is saved for future work.

Under this shock model, we can exploit the properties of the Poisson processes to calculatethe survival function of the component, i.e., the probability that the component survives totime t , given it is of age a at time zero. We will detail this calculation here, since the ideas andmanipulations may be utilized for all lifetime distributions shown in subsequent sections.

Let T ≥ 0 denote a random variable representing the time at which the component fails forthe first time. The survival function in the one component system may be derived as

G11(t; a) = Pr{T > t; age of component is a at time 0}

=∞∑

n=0

Pr {n shocks occur in (0, t) and component survives all}

=∞∑

n=0

EX

[e−λt

∫0<u1<···<un <t

n∏i=1

λ e−δ(a+ui )−Xi dui

]

=∞∑

n=0

e−λt (λξ e−δa)n∫

0<u1<···<un <t

n∏i=1

e−δui dui

=∞∑

n=0

e−λt(

λξ e−δa

δ

)n(1 − e−δt)n

n!

= exp[−λt + λξ e−δa

(1 − e−δt

δ

)], (2)

where the magnitudes of each shock are independent and identically distributed with ξ =E(e−X ). The subscript ‘11’ denotes the one-component nature of the system with shock modelcharacterized by one parameter, namely λ. The use of this notation delineates this survivalprobability from others derived in subsequent sections.

One may argue that the assumption of one P(λ; x) shock process is unrealistic. If wegeneralize to the more realistic countable and mutually independent class � = {P(λi ; xi) :i = 1, 2, . . . } of Poisson processes, the survival function is equivalent. In particular, for� = ∑∞

i=1 λi < ∞ and �θ = ∑∞i=1 λiξi , where θ ≤ 1 and ξi = E[exp(−Xi )], the survival

524 J. FAN ET AL.

FIGURE 1: G11 against λ for a = 0, t = 1, and x = 0. The symbols denote •: δ = 0; v: δ = 0.5;×: δ = 1; +: δ = 2; ◦: δ = ∞. The survival probability is 1 for δ = 0 without an ageing factor since the

system does not incur any damage from shocks x = 0. Ageing causes a sharper descent for larger δ.

function for a one component system under the shock process � is

G1K (t; a) =∞∏

i=1

exp[−λi t + λiξi e−δa

(1 − e−δt

δ

)]

= exp

[−�t + �θ e−δa

(1 − e−δt

δ

)]. (3)

G1K is identical to G11(t; a) with λ replaced by � and λξ replaced by �θ . Restriction to thesingle shock process P(λ; x) is thus reasonable under the given framework.

It is interesting to note that the survival function without an ageing factor (δ = 0) is of theform

G(t) = exp[−λt + λξ t ] (4)

independent of the age a of the system at time zero. In comparing (2) and (4) we see that theageing process enhances the magnitude of the shock x by a linear term δa. Furthermore, theageing of the system from time zero to time t is incorporated through an exponential decayfunction {1 − exp(−δt)}. This exponential function contrasts with the linear time piece in thesecond term of (4). Consequently, the ageing process lowers the probability of survival at timet in that G11(t; a) < G(t) for all t > 0.

Figures 1 and 2 display the survival functions (2) and (4) for a new system (a = 0) andolder system (a = 1) for several values of δ.

3. Two-component shock model

In many applications, the system of interest contains more than one component. The modeldescribed in Section 2 may be easily extended to such a situation. We will introduce the two-component shock model in this section and generalize to a K -component system in Section 4.


FIGURE 2: G11 against λ for a = 1, t = 1, and x = 0. The symbols denote •: δ = 0; v: δ = 0.5;×: δ = 1; ◦: δ = ∞. An older system (a = 1) under the same shock magnitude x = 0 is moresusceptible to ageing in that the survival function approaches the limiting curve represented by the symbol◦ faster than a new system as shown in Figure 1. A similar behavior occurs as the magnitude x of the

shock process increases.

We assume that the two-component system is affected by three different compound Poissonshock processes as follows:

(i) P1(α; x) shock of magnitude x and rate α affects component I only;

(ii) P2(β; y) shock of magnitude y and rate β affects component II only;

(iii) P0(γ ; z) shock of magnitude z = (z1, z2) and rate γ affects components I and IIsimultaneously.

The two components are also assumed to age independently of the shocks and of each other,according to simple exponential functions with coefficients δ and ε respectively, denoted byExp(δ) and Exp(ε).

Assume that at time zero, component I is of age a and component II of age b. A shock ofmagnitude x from P1 arriving at time u is fatal to component I with probability {1−exp[−δ(a+u) − x]}. Similarly, a shock of magnitude y from P2 arriving at time v is fatal to component IIwith probability

{1 − exp[−ε(b + v) − y]}.A shock of magnitude z = (z1, z2) from P0 arriving at time w is fatal to component I withprobability {1 − exp[−δ(a +w)− z1]} and to component II with probability {1 − exp[−ε(b +w) − z2]}.

This model, as with the model leading to (1) for a one-component system, assumes failureto either component to be due to a fatal shock. Ageing affects the magnitude of the effectof shock: the older the component, the more susceptible to failure from a shock of a specificmagnitude.

526 J. FAN ET AL.

In a multi-component system, the typical question raised is how the components interact.In particular, are the components arranged in parallel or series? For this paper, we will restrictourselves to the simplest situation of physically unrelated components; that is, the operationalcharacteristics of one component do not affect those of the second component. The P0(γ, z)shock process can affect both components, but the effect on one component is independent ofthe effect on the other component. Considerations and modeling of the interaction betweencomponents is left for future work.

The survival probability under this two-component model may be obtained in an analogousmanner to the one-component case in Section 2. The details are thus left to the reader. Inparticular, since the shock processes are independent compound Poisson,

G23(u, v; a, b) = Pr{components (I, II) survive to (u, v) given initial ages (a, b)}= exp[−αu − βv − γ (u ∨ v) + αξ e−δa D(u) + βη e−εb E(v)

+ γ ζ0 e−δa−εb F(u ∧ v) + H ] (5)

where ξ = E(e−X ), η = E(e−Y ), ζ0 = E[e−(Z1+Z2)], ζi = E(e−Zi ), i = 1, 2, D(u) =[1 − exp(−δu)]/δ, E(v) = [1 − exp(−εv)]/ε, F(w) = {1 − exp[−(δ + ε)w]}/(δ + ε),

H ={

γ ζ2 e−εb[E(v) − E(u)], if u < v

γ ζ1 e−δa[D(u) − D(v)], if u > v,

and ∧ denotes minimum and ∨ denotes maximum. The subscript ‘23’ denotes the two-component nature of the system with a three-parameter shock model (α, β, γ ).

Notice, as in the one-component case, the ageing processes parameterized by δ and ε forcomponents I and II respectively enhance the magnitude of the three shocks to the systemby linear terms. Furthermore, the ageing of the system from time zero to time t affects thesurvival probabilities through the exponential functions D(t), E(t), and F(t). Ageing of thelonger-lived component from the time of the first failure of one of the components affects thesurvival probabilities through the term H .

The three mutually independent compound Poisson process shock model might seem to beunnecessarily complex. Alternatively, one may choose to describe the shock process through asingle bivariate compound Poisson process with rate λ and magnitude z = (z1, z2). We denotethis process by P(λ, z), affecting both components simultaneously. In this situation, zi is theeffect of the shock on component i(= 1, 2).

Assume that at time zero, the system is of age (a, b). Thus a shock of magnitude z =(z1, z2) arriving at time t is fatal to component I with probability {1 − exp[−δ(t + a) − zi ]},and component II with probability {1−exp[−ε(t+b)−zi ]}, under the same simple independentexponential ageing processes Exp(δ) and Exp(ε) as described earlier.

Under this single bivariate compound Poisson shock process, the survival probabilities maybe shown, in an analogous derivation to (5), to be

G21(u, v; a, b) = exp[−λ(u ∨ v) + λζ0 e−δa−εb F(u ∧ v) + H ], (6)

where

H ={

λζ2 e−εb[E(v) − E(u)], if u < v

λζ1 e−δa[D(u) − D(v)], if u > v,

and all other terms are defined as in (5). The subscript ‘21’ denotes the two-component natureof the system with a shock model characterized by one parameter, namely λ.


The survival function G21 in (6) is simpler than G32, in that the function is characterizedby fewer parameters while retaining the essential complexity of the failure probabilities. Thisreduction in parameters, of course, decreases model adaptability. However, as seen in Sec-tion 5, the lower-dimensional parameter space allows for easier study of the behavior of thesurvival probabilities. From a statistical point of view, the simpler parametric model lendsmore easily to parameter estimation. Hence, we will pursue analysis of only the latter single-parameter bivariate compound Poisson process in the remainder of this paper.

A number of failure probabilities associated with G21 are of interest in studying the two-component system. First, the following three probability density functions provide relevantinformation about the system:

g212(u, v; a, b) = ∂2G21(u, v; a, b)

∂u∂v

= λ2G21(u, v; a, b) e−δ(a+v)(1 − ζ1 e−δ(a+u))

× (ζ1 − ζ0 e−ε(b+v)), u > v, (7)

g211(u, v; a, b) = λ2G21(u, v; a, b) e−ε(b+u)(1 − ζ2 e−ε(b+v))

× (ζ2 − ζ0 e−δ(a+u)), u < v, (8)

g210(t; a, b) = ∂2G21(u, v; a, b)

∂u∂v

∣∣∣∣u=v=t

= λG21(t, t; a, b)(1 − ζ1 e−δ(a+t) − ζ2 e−ε(b+t) + ζ0 e−δ(a+t)−ε(b+t)) (9)

where all terms are defined as in (5). The densities g211 and g212 are the probabilities thatcomponent I fails at time u, and component II fails at time v, with component I failing first inthe former density (u < v) and component II failing first in the latter density (u > v). Thefunction g210 is the probability that components I and II fail at the same time u = v = t .The function g210 may also be interpreted as the density of the survival times along the timebisector u = v on which both components fail simultaneously.

Second, for the purpose of designing repair policies, one is interested in first failure prob-abilities. In particular, it is easy to show

f211(t; a, b) = Pr{component I fails at time t and II survives beyond t}= λG21(t, t; a, b)(ζ2 e−ε(b+t) − ζ0 e−δ(a+t)−ε(b+t)) (10)

and

f212(t; a, b) = Pr{component II fails at time t and I survives beyond t}= λG21(t, t; a, b)(ζ1 e−δ(a+t) − ζ0 e−δ(a+t)−ε(b+t)). (11)

Third, the survival probabilities of one component, given the relationship in survival timesbetween the two components, is relevant for optimal repair decisions. For example, if T1 andT2 denote random variables representing the time at which the two components fail for the firsttime respectively,

Pr{T1 > t | T1 < T2} = G21I (t; a) − ET2 [G21I (T2; a)]1 − ET2 [G21I (T2; a)]

528 J. FAN ET AL.

where G21I (t; a) is the marginal survival function for component I, and ET2 [·] is the expectedvalue with respect to the random variable T2. Pr{T2 > t | T1 < T2} is derived analogously.

All these probabilities and failure rates are relevant in predicting the lifetime of the systemand type of the first failure, in addition to optimal repair policies. These developments will bepursued in future work. We will study the behavior of these lifetime distributions over theirrespective parameter spaces in Section 5.

4. K-component shock model

Generalization of the two-component shock model of Section 3 to K > 2 components isan extension of the Poisson process mathematics. Suppose a K -component system experi-ences shocks according to a multivariate compound Poisson process P(λ, z) with rate λ andmagnitude z = (z1, . . . , zK ). The K components are affected simultaneously where zi is theeffect of the shock on components i = 1, . . . , K respectively. We assume that, if at time zerothe system is of age a = (a1, . . . , aK ), a shock of magnitude z arriving at time t is fatal tocomponent i with probability {1 − exp[−δi(t + ai) − zi ]}. As in Section 3, we presume thecomponents to be physically unrelated. Considerations of interacting components are left forfuture work.

Let Ti denote a random variable representing the time at which component i fails for thefirst time, i = 1, 2, . . . , K . Under the assumption of temporal ordering u1 < u2 < · · · < uK ,the survival function of the system is obtained in an analogous manner to the work of previoussections as

GK (u1, . . . , uK ; a) = Pr{T1 > u1, . . . , TK > uk; a}= Pr{all K components survive to u1}

× Pr{components 2, . . . , K − 1 survive through (u1, u2)}. . . Pr{component K survives through (uK −1, uK )}

= P1 · P2 · · · · · PK ,

where

P1 =∞∑

n=0

Pr{components 1, . . . , K survive all n shocks in (0, u1)}

= exp(−λu1) exp{

λ exp(−∑Kj=1 δ j a j )∑K

j=1 δ j

× EZ

[exp

(−

K∑j=1

Z j

)][1 − exp

(− u1

K∑j=1

δ j

)]},

P2 =∞∑

n=0

Pr{components 2, . . . , K survive all n shocks in (u1, u2)}

= exp[−λ(u2 − u1)] exp{

λ exp(− ∑Kj=2 δ j a j )∑K

j=2 δ j· EZ

[exp

(−

K∑j=2

Z j

)]

×[

exp(

−u1

K∑j=2

δ j

)− exp

(−u2

K∑j=2

δ j

)]},


FIGURE 3: g210 against λ and t for a = b = 0, z1 = z2 = 0, and δ = ε = 1. The probabilitythe components fail increases as the system ages on top of the shock process. A time t∗ > 0 is reached,however, at which the probability the system has failed at any time prior to t∗ is larger than the probabilityof failure subsequent to t∗ . As time increases beyond t∗ , the two components are less and less likely tofail together as such an event most probably occurred at an earlier time. For large values of λ, the systemhas high probability of failure early in its existence as the shocks are too frequent. As λ decreases, the

system has a better chance for an extended lifetime.

PK =∞∑

n=0

Pr{component K survives all n shocks in (uK −1, uK )}

= exp[−λ(uK − uK −1)] exp{(λ/δK ) exp(−δK aK )EZ[exp(−Z K )]× [exp(−uK −1δK ) − exp(−uK δK )]},

and P3, . . . , PK −1 follow similarly to the derivation of P2. After some simplification,

GK (u1, . . . , uK ; a) = exp(−λuK )

K∏l=1

exp{

λ exp(− ∑Kj=l δ j a j)∑K

j=l δ j· EZ

[exp

(−

K∑j=l

Z j

)]

×[

exp(

−ul−1

K∑j=l

δ j

)− exp

(−ul

K∑j=l

δ j

)]}(12)

where u0 = 0 and u1 < · · · < uK . The formulation is analogous for other time orderingpermutations of u1, . . . , uK .

5. Graphical analysis of lifetime distributions

In this section, we graphically present some of the relevant features of the lifetime dis-tributions developed in Section 3. Figures 3 and 4 show the density (9) as a function ofλ over two sets of shock magnitudes. The sensitivity of the lifetime distribution to shockmagnitude is similar if we consider survival probabilities as a function of δ or ε for a fixed λ

as well. Furthermore, the behavior is analogous if the initial age of the components, a or b,

530 J. FAN ET AL.

FIGURE 4: g210 against λ and t for a = b = 0, z1 = z2 = 1, and δ = ε = 1. An increase in themagnitude (z1, z2) of the shock process causes the concave ridge of the curve in Figure 3 to collapseagainst the t = 0 boundary. In particular, the ridge compresses into an exponential decay function for

large values of λ and flattens out for small shock rates λ.

FIGURE 5: G21 against δ(= ε) and λ for a = 0, b = 0.1, z1 = 0.1, z2 = 0, and u = v = 1.

increases. This latter characteristic is a consequence of the effect of ageing and magnitude onthe survival density function in the form {exp[−δ(a + b) − (z1 + z2)]} for fixed δ = ε. Thedensity functions (7) and (8) exhibit similar behavior to g210.

The importance of understanding the behavior of the survival functions in subsets of theparameter space becomes apparent when developing repair policies and system maintenanceprocedures. Optimal repair strategies depend on the sensitivity of the system to ageing andshocks in addition to the cost of repair/replacement. For example, we must study graphics suchas Figure 5 to decide if it is economically more feasible to reduce the frequency or magnitude


FIGURE 6: Empirical distribution function (solid line) and one-component shock model (dotted line) fitto appendectomy data from one twin of the pair.

of the shocks or the effect of ageing through repairs. Which is more costly? How will repairimpact the life of the system and need for future costly repairs? The survival functions derivedin this paper set the ground work for answering such questions.

6. Data analysis

We fit the survival models presented in the previous sections to the Australian twin dataset (Duffy et al. (1990)). This data consists of responses by twins to a survey concerningtheir past history of a number of diseases and operations. One of the operations of interest isappendectomy, in particular, toward the genetics of acute appendicitis. We consider a subsetof the data: 287 twins who retrospectively reported an appendectomy as a consequence ofappendicitis. We wish to estimate the probability of such an occurrence over the age of theindividuals.

We first fit the one-component shock model from equation (2) on data from one twin ofthe pair. Recall the parameters in the model are the shock process rate λ, shock magnitudeξ , exponential ageing parameter δ, and initial age a. Maximum likelihood estimates of theseparameters are found to be λ = 1.75, ξ = 1.00, δ = 0.0030, and a = −1.95. Figure 6displays the Kaplan–Meier (empirical distribution function) estimate of the probability ofappendectomy as a function of age as well as our model fit. The shock model appears to fitquite well. In fact, a Kolmogorov goodness-of-fit test is not able to reject the null hypothesisof the data arising from the shock process probability model (p-value � 0.20).

A couple of remarks are in order. First, a shock magnitude parameter of ξ = 1.0 indicatesthe shocks affect the subjects only through the ageing process. Second, the initial age isestimated to be less than zero. Recall, the initial age is not constrained to the positive realline in the model formulations of Sections 2 and 3. Though a negative initial age is seeminglyunusual, in this data set it can be explained by the negligible probability of an infant undergoingan appendectomy as a consequence of appendicitis. According to our model, infants are

532 J. FAN ET AL.

FIGURE 7: Bivariate empirical distribution function and 2-component shock model fit to appendectomydata from both twins.

born with an ‘immunity’ to appendectomies as a consequence of appendicitis. Ageing, andconsequently greater susceptibility to appendicitis, does not begin until the subject is of agetwo.

An initial age less than zero can be described as the effect of a retail warranty. A full-servicewarranty for any mechanical device often ensures the owner a brand new piece of equipmentif the appliance fails during a certain period of time. Upon the conclusion of the warrantyperiod, any ageing, subsequent damage, and failure of the device proceeds as usual at theowner’s expense. An initial age of negative two suggests a warranty period of two years. Ofcourse, in our shock model, the machine does not age during the warranty period (we maythink of this also as the owner receiving a brand new device at the conclusion of the warrantyperiod).

As ageing typically occurs throughout a warranty period, an alternative explanation of anegative initial age may be illustrated through fuel supply use in military aircraft. Airplanesmay have a full fuel tank in addition to a supplemental fuel supply. The plane will most likelyuse up the supplemental supply prior to tapping into the fuel tank. Once the supplementalfuel is depleted, the supplemental fuel compartment is dropped from the plane, and the regularfuel tank is used. This process is explained by a negative initial supply of fuel. For example,if the plane carries fifty gallons of supplemental fuel, the initial fuel supply is negative fiftygallons. The standard complement of fuel is not touched until the supply reaches zero gallons.Analogously, in the twin data, once a subject is of two years of age, the initial protection fromthe appendectomy is over and the individual ages as expected. The negative initial age thusincorporates a time lag into the ageing process.

From a mathematical perspective, the probability of an appendectomy upon the arrival of ashock at age t is {

1 − ξ exp[−δ(a + t)] t ≥ −a

1 t < −a.


This probability generalizes Equation (1) to allow a to range anywhere on the real line.The fit discussed above and displayed in Figure 6 is essentially the same for each twin. Of

course, the twin responses are paired. A bivariate analysis utilizing the two-component shockmodel in Equation (6) is more appropriate. This model is characterized by eight parameters:shock process rate λ, shock magnitudes ζ0, ζ1, and ζ2, ageing parameters δ and ε, and initialages a and b. Since the data contains responses from identical twins and the univariate prob-ability model fit on each half of the twin pairs is the same, we assume the ageing parametersand initial ages are equal in that δ = ε and a = b.

The maximum likelihood estimates of the six remaining parameters are found to be λ =39.93, ζ0 = ζ1 = ζ2 = 1.00, δ = 0.000 15, and a = −2.00. Figure 7 displays the bivariateempirical distribution function and our model fit. Though more difficult to see in this three-dimensional plot, the two curves again fall on top of each other. Our shock model thus seemsto fit the probability of appendectomy as a consequence of appendicitis well in the Australiantwin data.

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