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Statistical Science

2006, Vol. 21, No. 4, 532–551DOI: 10.1214/088342306000000448c© Institute of Mathematical Statistics, 2006

On the Statistical Modeling and Analysisof Repairable SystemsBo Henry Lindqvist

Abstract. We review basic modeling approaches for failure and main-tenance data from repairable systems. In particular we consider im-perfect repair models, defined in terms of virtual age processes, andthe trend-renewal process which extends the nonhomogeneous Poissonprocess and the renewal process. In the case where several systems ofthe same kind are observed, we show how observed covariates and un-observed heterogeneity can be included in the models. We also considervarious approaches to trend testing. Modern reliability data bases usu-ally contain information on the type of failure, the type of maintenanceand so forth in addition to the failure times themselves. Basing our workon recent literature we present a framework where the observed eventsare modeled as marked point processes, with marks labeling the typesof events. Throughout the paper the emphasis is more on modelingthan on statistical inference.

Key words and phrases: Repairable system, preventive maintenance,nonhomogeneous Poisson process, renewal process, marked point pro-cess, virtual age process, trend-renewal process, heterogeneity, trend,competing risks.

1. INTRODUCTION

According to a commonly used definition of a re-pairable system [5], this is a system which, afterfailing to perform one or more of its functions satis-factorily, can be restored to fully satisfactory perfor-mance by a method other than replacement of theentire system. For the present paper and followingrecent literature on the subject, we suggest extend-ing this definition to include the possibility of ad-ditional maintenance actions which aim at servicingthe system for better performance. We shall referto this as preventive maintenance (PM), where one

Bo H. Lindqvist is Professor, Department of

Mathematical Sciences, Norwegian University of

Science and Technology, Trondheim, Norway e-mail:

bo@math.ntnu.no.

This is an electronic reprint of the original articlepublished by the Institute of Mathematical Statistics inStatistical Science, 2006, Vol. 21, No. 4, 532–551. Thisreprint differs from the original in pagination andtypographic detail.

may further distinguish between condition-based PMand planned PM. The former type of maintenance isdue when the system exhibits inferior performance,while the latter is performed at predetermined pointsin time. In this presentation we will consider someaspects of condition-based PM, while the plannedPM will be briefly touched on in the concluding re-marks.

Traditionally, the literature on repairable systemsis concerned with modeling failure times, with pointprocess theory being the main tool. The most com-monly used models for the failure process of a re-pairable system are renewal processes (RP), includ-ing the homogeneous Poisson processes (HPP) andnonhomogeneous Poisson processes (NHPP). Whilesuch models often are sufficient for simple reliabilitystudies, the need for more complex models has ofcourse emerged.

There is currently a rapidly increasing literatureconcerning modeling and analysis of recurrent events,with a wide range of applications, including relia-bility analysis of repairable systems, which is the

1

2 B. H. LINDQVIST

present topic. In a recent review paper, Cook andLawless [14] presented several examples from medi-cal studies where models and methods for recurrentevents are appropriate. The review paper by Pena[55] gave examples from both medical and reliabil-ity studies. The scope of our paper is biased towardreliability applications, although most of the mod-els considered have a wider applicability. We will,in particular, consider models which incorporate ef-fects of different kinds of repair and maintenance,and with the possibility of handling several failurecauses, for example.

In a review paper like this, it is of course impossi-ble to cover all models or methods which have beensuggested in the literature. Our aim is rather to em-phasize some important ideas, and in this respectthere will be a clear bias toward work in the direc-tion of our own interests and in work by ourselvesand collaborators. Throughout the paper the em-phasis will be more on modeling than on statisti-cal inference. In addition we will try to give somehistorical perspectives on the theory and practicerelated to repairable systems, again not necessarilycomplete and possibly biased by our own views.

One of the first comprehensive treatments of sta-tistical methods for recurrent events with reliabilityemphasis is the talk by David R. Cox, read beforethe Royal Statistical Society in London in March1955 and published in [17]. Cox touched a largenumber of topics, most of them motivated from theclothing industry. Topics of particular importancefor reliability applications were trend testing, test-ing whether a failure process is a Poisson process,autocorrelated time gaps, doubly stochastic Poissonprocesses, heterogeneity between systems, correla-tions between different types of events, mean repairtimes, availability of service and so forth. Many re-sults from the paper are contained in the subsequentbook by Cox and Lewis [19], which still is a very use-ful and much cited source on the subject.

Another early contribution to the study of re-pairable systems is the heavily cited 1963 paper byProschan [58], “Theoretical explanation of observeddecreasing failure rate.” This paper is particularlyimportant since it led to the awareness that properanalysis of recurrent events is an important partof reliability theory. In particular it is one of thefirst treatments of heterogeneity in the theory of re-pairable systems.

What seems to be the first book devoted solely torepairable systems reliability was published by As-cher and Feingold [5] in 1984. For a long time this

was the main reference for repairable systems andit is still a major source. The subtitle of the book isModeling, Inference, Misconceptions and Their Causes.The authors were complaining that reliability re-searchers and practitioners did not recognize thecrucial difference between the statistical treatmentof repairable systems and nonrepairable components.They demonstrated by simple examples how conclu-sions from data may be very wrong if times betweenfailures are treated as i.i.d. if there is a trend inthem.

Data from repairable systems are usually givenas ordered failure times T1, T2, . . . with data comingfrom a single system or from several systems of thesame kind. The implicit assumption is usually thatthe system is repaired and put into new operationimmediately after the failure. This restriction, dis-regarding repair times, is not serious if one is inter-ested in modeling and estimation of the probabilitymechanisms behind failure occurrences. It is, more-over, justified if the time scale is taken to be oper-ation time, number of cycles, number of kilometersrun and so forth. We will impose this restriction inthis paper, and we will therefore not cover impor-tant topics such as availability and unavailability ofsystems, where the standard tool is to use alternat-ing renewal processes with operation periods alter-nating with repair periods (see, e.g., [59], Chapter7).

A common recipe for analysis of data from a re-pairable system is as follows. First, apply a testfor trend in the interfailure times Xi = Ti − Ti−1.If no significant trend is found, then use a RP asa model, in which case the well established statis-tical tools for analysis of i.i.d. observations can beused. Otherwise, use a NHPP model, which handlestrend through specification of an intensity functionλ(t). For example, a deteriorating system will thencorrespond to an increasing function λ(t), while animproving system will correspond to a decreasingλ(t). A homogeneous Poisson process, HPP(λ), cor-responds to a constant intensity λ(t) ≡ λ and is atthe same time a renewal process with exponentiallydistributed interfailure times.

A RP model is also called a perfect repair model,since the system is as good as new after a failure. Onthe other hand, a NHPP model corresponds to whatis called minimal repairs, meaning that the systemafter repair is only as good as it was immediatelybefore the failure. Lindqvist, Elvebakk and Hegg-land [48] represent the problem of distinguishing

ANALYSIS OF REPAIRABLE SYSTEMS 3

between the two “extreme” kinds of repair as cor-responding to the first “dimension” of a repairablesystem description in the form of a so-called modelcube (Figure 3). The second dimension is the ap-pearance of trend or no trend in interfailure times.This particular aspect of system behavior has tra-ditionally received much attention in reliability the-ory and is resolved by considering trend tests. Fi-nally, the third dimension corresponds to the exis-tence of unobserved heterogeneity between systems.This problem is of course relevant only when sev-eral systems of the same kind are observed. There iscurrently a large and increasing interest in the mod-eling of heterogeneity, usually known as frailties inthe survival analysis literature. To some extent, het-erogeneity may have been much overlooked in relia-bility studies, but there are important exceptions inthe literature.

Several classes of models have in turn been sug-gested for cases not covered by the “extreme” mod-els RP and NHPP. These include the so-called im-perfect repair models. The idea is that after a re-pair the “virtual” age of the unit is not necessarilyreduced to 0, such as for a perfect repair, nor is itthe same as before the repair, such as for a mini-mal repair. Instead, the virtual age is reduced by acertain amount that depends on the type of repair.We review the basic properties of such models andwe will see how the concept of virtual age can begeneralized to more than one dimension.

Another class of alternatives to NHPP and RPmodels, which includes these models, is the so-calledtrend-renewal process (TRP). This model is a gen-eralization of Berman’s modulated gamma process[9] and has been extensively studied in [48]. In thepresent paper we will use TRP models and their ex-tensions as our basic framework to illustrate somemain issues on modeling and statistical analysis ofdata from repairable systems. The TRP is partic-ularly suitable to illustrate the already mentionedthree dimensions of repairable systems.

Modern reliability data bases usually contain moreinformation than just the failure times. For example,there may be information on the times of preventivemaintenance (PM), identity of a failed component,type of failure, type of repair, cost of replacementand so forth. Thus we shall more generally assumethat observations from repairable systems are repre-sented as marked point processes where the markslabel the types of events. For example, the marksmay be of two kinds, corresponding to the type of

maintenance, repair or PM. We review some recentliterature in this direction with the aim of extend-ing the theory of repairable systems to a competingrisks setting.

In addition to information on types of events, thedata bases may contain covariates that represent en-vironmental conditions, measures of various forms ofload and usage stress, and so forth. Such covariatescould be constant or are possibly varying with time.Regression models for repairable systems are usefulfor obtaining better understanding of the underly-ing failure and PM mechanisms, or for predictingthe behavior of new items.

The outline of the paper is as follows. The basicnotation and definitions used are given in Section 2,including the introduction of the marked point pro-cess setup. Section 3 reviews models for the case offailure data with a single type of events, with em-phasis on virtual age models and trend-renewal pro-cesses. Section 4 is devoted to a discussion of unob-served heterogeneity in repairable systems data. Themodel cube for heterogeneous trend-renewal processesis considered in particular. In Section 5 we considervarious approaches to trend testing, both for datacoming from single systems and from several sim-ilar systems. The possible extension of virtual agemodels to the marked process case is considered inSection 6. This section is based on some recent pa-pers on the subject. Some concluding remarks aregiven in Section 7, in particular concerning topicsnot covered in the main text.

2. NOTATION AND BASIC DEFINITIONS

We consider a repairable system where time usu-ally runs from t = 0 and where events occur at or-dered times T1, T2, . . . . Here time is not necessarilycalendar time, but can in principle be any suitablemeasurement which is nondecreasing with calendartime, such as operation time, number of cycles, num-ber of kilometers run, length of a crack and so forth.As already mentioned in the Introduction, we shalldisregard time durations of repair and maintenance,so we assume that the system is always restartedimmediately after failure or a maintenance action.Types of events (type of maintenance, type of fail-ure, etc.) are, when applicable, recorded as J1, J2, . . .with Ji ∈ J for some mark space J which will de-pend on the current application. For simplicity wewill here always assume that J is a finite set. Theobservable process (T1, J1), (T2, J2), . . . will be called

4 B. H. LINDQVIST

Fig. 1. Event times (Ti), event types (Ji) and sojourn times (Xi) of a maintained system.

the marked event process or occasionally the failureprocess. The interevent, or interfailure, times will bedenoted X1,X2, . . . . Here Xi = Ti−Ti−1, i = 1,2, . . . ,where for convenience we define T0 ≡ 0. Figure 1illustrates the notation. We also make use of thecounting process representation Nj(t) equal to thenumber of events of type j in (0, t], which countsthe number of events of type j ∈ J , and N(t) =∑

j∈J Nj(t), which counts the number of events ir-respective of their types.

To describe probability models for repairable sys-tems we use some notation from the theory of pointprocesses. A key reference is Andersen, Borgan, Gilland Keiding [4]. Let Ft− denote the history of themarked event process up to, but not including, timet. In models without covariates we assume that Ft−

includes all information on event times and eventtypes before time t. Formally, Ft− is generated bythe set {Nj(s) : 0≤ s < t, j ∈ J }.

Suppose then that a possibly time-dependent co-variate vector Z(t) is observed for the system. In thiscase the covariate history {Z(s) : 0 ≤ s ≤ t} shouldbe added to the history Ft− for each t > 0. This willimply that just before any time t we have the com-plete information on the previous events, as well asthe complete covariate history including the valueof the covariate at time t. In the case of a time-constant covariate vector Z, the information in Z isadded to each history Ft−.

The conditional intensity of the process with re-spect to events of type j ∈ J is now defined as

γj(t)

= lim∆t↓0

Pr(event of type j in [t, t + ∆t)|Ft−)

∆t,(1)

which we call the type-specific intensity for j. Thus,γj(t)∆t is approximately the probability of an eventof type j in the time interval [t, t+∆t) given the his-tory before time t. Further, we let γ(t) =

∑

j∈J γj(t)so that γ(t)∆t is approximately the conditional prob-ability of an event of any type in the time interval[t, t + ∆t), where it has been tacitly assumed that

the probability of more than one event in an inter-val [t, t+∆t) is o(∆t). Note that the γj(·) and hencethe γ(·) may be functions of the covariate vector Z(·)when appropriate. In typical applications, γj(t) maydepend on the covariate history only through thevalue Z(t) at time t. Further, it is common to assumethat γj(t) = γ0

j (t)g(Z(t)), with γ0j (t) depending only

on the pure event history {Nj(s) : 0 ≤ s < t, j ∈ J },and with g(·) being some parametric function ofthe covariate vector such as the exponential one,g(z) = exp(β′

z), where β is a parameter vector.For statistical inference we need an expression for

the likelihood function. Suppose that a single systemwith a marked event process as described above isobserved from time 0 to time τ , resulting in observa-tions (T1, J1), (T2, J2), . . . , (TN(τ), JN(τ)), in additionto the covariate vector Z(s) for 0 ≤ s ≤ τ if appli-cable. The likelihood function is then given by ([4],Section II.7)

L =

{N(τ)∏

i=1

γJi(Ti)

}

exp

{

−

∫ τ

0γ(u)du

}

.(2)

A rough verification of (2) can be given as fol-lows. First, partition the interval (0, τ ] into s equalpieces, each of length h = τ/s. Assume that s isso large that at most one event can happen in aninterval of length h. Then the conditional proba-bility of an event of type j in the interval [(k −1)h,kh), k = 1, . . . , s, given the history before (k −1)h, is roughly γj(kh)h, while the conditional prob-ability of no event in this interval is roughly 1 −γ(kh)h. The probability of a realization of the pro-cess from 0 to τ will therefore include a productof N(τ) terms of the type γj(kh)h, correspondingto the observed events, and which in the limit ash → 0 (after dividing by the normalization hN(τ))gives the product term on the right-hand side of(2). The exponential part of (2) comes from takingthe limit of the product of the terms 1 − γ(kh)h ≈

exp{−∫ kh(k−1)h γ(t)dt} for the intervals that contain

no event, assuming continuity of γ(·).The likelihood function (2) is valid under the as-

sumption that τ is a stopping time, which means

ANALYSIS OF REPAIRABLE SYSTEMS 5

that its value depends stochastically only on thepast history. This property holds for the standardcensoring schemes used in practice and in particularwhen τ is independent of the event process. Thereis, however, an increasing awareness of the need toallow for dependent censoring in many applications(see, e.g., [33]).

In typical applications, data will be available forseveral similar systems, with stopping times τ usu-ally varying from system to system. Under the as-sumption of stochastic independence and identicalprobability mechanisms for the systems, the totallikelihood will be the product of expressions (2) com-puted for all systems. For both parametric and non-parametric models of this kind there is a well de-veloped theory for estimation based on the martin-gale approach to point processes ([4] gives a com-prehensive account). Relevant references for statis-tical inference in reliability models are, among oth-ers, Ascher and Feingold [5], Rausand and Høyland[59], Crowder, Kimber, Smith and Sweeting [21] andMeeker and Escobar [52].

3. MODELS FOR REPAIRABLE SYSTEMS

WITH A SINGLE TYPE OF EVENT

In the present section we assume that the observa-tions are just the failure times T1, T2, . . . . Thus themark space J will be ignored.

A large number of models can be obtained in termsof a given hazard function z(t), which we think of asbeing the hazard function of the time to first failureof a new system. The corresponding density and cu-mulative distribution function are denoted, respec-tively, f(t) and F (t), so z(t) = f(t)/(1−F (t)). Theidea is to use the function z(t) together with a spec-ification of the repair strategy to define the condi-tional intensity function γ(t) of the failure process.Models of this type are considered in Sections 3.1and 3.2. The corresponding models may be extendedto the case with observed covariates, although thiswill not be made explicit. As described in Section 2,the conditional intensities of the form γ(t) as consid-ered below may be multiplied with a factor g(Z(t))that defines the dependence of the covariate valueat time t.

3.1 Perfect and Minimal Repair Models

Suppose first that after each failure, the systemis repaired to a condition as good as new. In this

case the failure process is modeled by a renewal pro-cess with interevent time distribution F , denotedRP(F ). Clearly

γ(t) = z(t− TN(t−)),

where t − TN(t−) is the time since the last failurestrictly before time t.

Suppose instead that after a failure, the systemis repaired only to the state it had immediately be-fore the failure, called a minimal repair. This meansthat the conditional intensity of the failure processimmediately after the failure is the same as it wasimmediately before the failure, and hence is exactlyas it would be if no failure had ever occurred. Thuswe must have

γ(t) = z(t),

so that the process is a NHPP with intensity z(t),denoted NHPP(z(·)). In practice a minimal repairusually corresponds to repairing or replacing only aminor part of the system.

3.2 Imperfect Repair Models and the Virtual

Age of a System

A classical model, suggested by Brown and Proschan[13], assumes that at the time of each failure a per-fect repair occurs with probability p and a minimalrepair occurs with probability 1− p, independentlyof the previous failure history. This model is a sim-ple example of what has been called an imperfectrepair model, and was later generalized in severaldirections.

Kijima [34] suggested two imperfect repair mod-els, both involving what is called the virtual age (oreffective age) of the system. The idea is to distin-guish between the system’s age, which is the timeelapsed since the system was new, usually at timet = 0, and the virtual age of the system, which de-scribes its present condition when compared to anew system. The virtual age is redefined at failuresaccording to the type of repair performed and it runsalong with the true time between repairs. More pre-cisely, a system with virtual age v ≥ 0 is assumed tobehave exactly like a new system which has reachedage v without having failed. The hazard rate of asystem with virtual age v is thus zv(t) = z(v + t) fort > 0, where z(·) is the hazard rate of the time tofirst failure of the system.

It should be clear at this stage that models basedon virtual ages make sense only if the underlyinghazard functions z(·) are nonconstant. In fact, if z(·)

6 B. H. LINDQVIST

is constant, then a reduction of virtual age would notinfluence the rate of failures.

A variety of imperfect repair models can be ob-tained by specifying properties of the virtual ageprocess in addition to the hazard function z(t) ofa new system. For this, suppose v(i) is the virtualage of the system immediately after the ith event,i = 1,2 . . . . The virtual age at time t > 0 is then de-fined by A(t) = v(N(t−)) + t− TN(t−), which is thesum of the virtual age after the last event before tand the time elapsed since the last event. The pro-cess A(t), called the virtual age process by Last andSzekli [40], thus increases linearly between eventsand may jump only at events. It follows that

γ(t) = zv(N(t−))(t− TN(t−)) = z(A(t)),(3)

assuming that A(t) is included in Ft− for all t. Thismeans in turn that v(i) is contained in FTi

for eacht so that v(i) depends on the history up to and in-cluding Ti. The likelihood then becomes

L =

{N(τ)∏

i=1

z(v(i− 1) + Xi)

}

· exp

{

−

N(τ)∑

i=1

∫ Xi

0z(v(i− 1) + u)du

−

∫ τ−TN(τ)

0z(v(N(τ)) + u)du

}

.

This can be recognized as being the same as{N(τ)∏

i=1

fv(i−1)(Xi)

}

{1−Fv(N(τ))(τ − TN(τ))},

where fv(t) = f(v+t)/(1−F (v)) and Fv(t) = (F (v+t) − F (v))/(1 − F (v)) are, respectively, the densityand the cumulative distribution function of time tonext failure for a system with virtual age v andhence with hazard rate zv(·).

It is clear that the perfect repair and minimal re-pair models are the special cases where, respectively,v(i) = 0 and v(i) = Ti, i = 1,2, . . . . In Kijima’s [34]model I, the virtual age v(i) equals

∑ik=1 DkXk,

where D1,D2, . . . is a sequence of random variableson the interval [0,1] such that Dk is independent ofFTk− for each k. Note that FTk− includes D1,D2,. . . ,Dk−1 so that in particular the Dk are indepen-dent. In Kijima’s model II the virtual age v(i) is setto∑i

k=1(∏i

j=k Dj)Xk with the same conditions forthe Dk. This means that the virtual age after theith failure equals Di multiplied by the virtual age of

the system just prior to the ith failure. The modelof Brown and Proschan [13] is obtained when Di is1 with probability 1−p and 0 with probability p forall i.

Dorado, Hollander and Sethuraman [22] studiednonparametric statistical inference in a model slightlymore general than Kijima’s models described above.Nonparametric statistical inference in the Brown–Proschan model was first studied by Whitaker andSamaniego [63] and later by Hollander, Presnell andSethuraman [31].

Recall that for the above models, the Di needto be observed for likelihood inference using (2) tobe valid. This means in effect that the type of re-pair (minimal or perfect) must be reported for eachrepair action. In real applications, however, exactinformation on the type of repair is rarely avail-able. The estimation problem in the case of unob-served Di has been considered by, for example, Lim[45] (suggesting an EM algorithm approach) andLangseth and Lindqvist [38, 39].

Doyen and Gaudoin [23] studied classes of virtualage models based on deterministic reduction of vir-tual age due to repairs, and hence not requiring theobservation of repair characteristics. The basic mod-els of this type can be obtained simply by letting theDi in Kijima’s models above be replaced by para-metric functions. A simple example of [23] is to use1 − ρ for Di, where 0 < ρ < 1 is a so-called age re-duction factor.

There is a large literature on reliability modelingusing the virtual age process. For a review we referto Pham and Wang [57] and Lindqvist [46]. Section6 presents an attempt to define a multivariate vir-tual age process and corresponding repairable sys-tem models with several types of events.

3.3 Generalized Linear Model Types

Berman and Turner [10] considered estimation inparametric models with the conditional intensity be-ing of the generalized linear model type

γ(t) = g

{ p∑

i=0

βizi(t)

}

,(4)

where g is a known monotonic continuous function,the zi(t) are known functions of t and the historyFt−, and the βi are unknown parameters. Note thatthe functions zi(t) may be functions of the covari-ates if available. One aim of the paper was to demon-strate how to use software for generalized linear mod-els to analyze repairable systems data. The model

ANALYSIS OF REPAIRABLE SYSTEMS 7

(4) is closely related to the modulated renewal pro-cess introduced in [18] for which Cox suggested asemiparametric approach for inference using a par-tial likelihood.

The special case of (4) obtained when g(y) = ey

was applied to repairable systems by Lawless andThiagarajah [43]. In particular, they considered themodel

γ(t) = eβ0+β1g1(t)+β2g2(t−TN(t−)),(5)

where g1 and g2 are known functions. This con-ditional intensity is a function of both the calen-dar time and the time since last failure. Note thatβ1 = 0 gives a RP and β2 = 0 gives a NHPP, whileβ1 = β2 = 0 gives a HPP.

3.4 The Trend-Renewal Process

A class of processes called inhomogeneous gammaprocesses was suggested by Berman [9]. Berman mo-tivated the inhomogeneous gamma process by firstconsidering the process T1, T2, . . . obtained by ob-serving every κth event of a NHPP, where κ is apositive integer. He then showed how to generalizeto the case when κ is any positive number.

We present now a generalization of Berman’s idea,called the trend-renewal process, which was exten-sively studied by Lindqvist, Elvebakk and Heggland[48]. We will use this process in particular to de-scribe the three dimensions related to the propertiesof repairable systems.

The idea behind the trend-renewal process is togeneralize the following well-known property of theNHPP. First let the cumulative intensity functionthat corresponds to an intensity λ(·) be defined byΛ(t) =

∫ t0 λ(u)du. Then if T1, T2, . . . is a NHPP(λ(·)),

the time-transformed stochastic process Λ(T1),Λ(T2), . . . is HPP(1).

The trend-renewal process (TRP) is defined sim-ply by allowing the above HPP(1) to be any renewalprocess RP(F ). Thus, in addition to the intensity

function λ(t), for a TRP we need to specify a dis-tribution function F of the interarrival times of thisrenewal process. Formally we can define the processTRP(F,λ(·)) as follows:

Let λ(t) be a nonnegative function defined fort≥ 0, and let Λ(t) =

∫ t0 λ(u)du. The process T1, T2, . . .

is called TRP(F,λ(·)) if the transformed processΛ(T1),Λ(T2), . . . is RP(F ), that is, if the Λ(Ti) −Λ(Ti−1), i = 1,2, . . . , are i.i.d. with distribution func-tion F . The function λ(·) is called the trend func-tion, while F is called the renewal distribution. Tohave uniqueness of the model, it is usually assumedthat F has expected value 1.

Figure 2 illustrates the definition. For a NHPP(λ(·)),the RP(F ) would be HPP(1). Thus TRP(1 − e−x,λ(·)) = NHPP(λ(·)). Also, TRP(F,1) = RP(F ), whichshows that the TRP class includes both the RP andNHPP classes.

As a motivation for the TRP model, suppose thatfailures of a particular system correspond to replace-ment of a major part, for example, the engine of atractor (as in the data given by Barlow and Davis[6]), while the rest of the system is not maintained.Then if the rest of the system is not subjected towear, a renewal process would be a plausible modelfor the observed failure process. In the presence ofwear, on the other hand, an increased replacementfrequency is to be expected. This is achieved in aTRP model by accelerating the internal time of therenewal process according to a time transformationΛ(t) =

∫ t0 λ(u)du which represents the cumulative

wear. The TRP model thus has some similarities toaccelerated failure time models.

It can be shown [48] that the conditional intensityfunction for the TRP(F,λ(·)) is

γ(t) = z(Λ(t)−Λ(TN(t−)))λ(t),(6)

where z(·) is the hazard rate that corresponds to F .This is a product of one factor, λ(t), which dependson the age t of the system and one factor which de-pends on a transformed time from the last previous

Fig. 2. The defining property of the trend-renewal process.

8 B. H. LINDQVIST

failure. However, time since last failure is measuredon a scale that depends on the cumulative intensityof failures. This shows that the TRP class does notcontain, nor is contained in, the classes of processesconsidered in the previous subsection.

Suppose now that a single system has been ob-served in [0, τ ], with failures at T1, T2, . . . , TN(τ). If aTRP(F,λ(·)) is used as a model, then substitutionof (6) into (2) gives the likelihood

L =

{N(τ)∏

i=1

z[Λ(Ti)−Λ(Ti−1)]λ(Ti)

}

(7)

· exp

{

−

∫ τ

0z[Λ(u)−Λ(TN(u−))]λ(u)du

}

.

Equivalently, if f is the density function that corre-sponds to F , we can write this likelihood as

L =

{N(τ)∏

i=1

f [Λ(Ti)−Λ(Ti−1)]λ(Ti)

}

(8)· {1−F [Λ(τ)−Λ(TN(τ))]}.

The latter form of the likelihood follows directlyfrom the definition of the TRP, since the condi-tional density of Ti given T1 = t1, . . . , Ti−1 = ti−1 isf [Λ(ti)−Λ(ti−1)]λ(ti), and the probability of no fail-ures in the time interval (TN(τ), τ ], given T1, . . . , TN(τ),is 1− F [Λ(τ)−Λ(TN(τ))].

A possible extension of the TRP to include co-variates would be to multiply the trend functionλ(t) by a factor g(Z(t)), for example, of the formexp(β′

Z(t)) as suggested in Section 2. The λ(t) wouldthen play the role of a baseline trend function. Thisdefinition generalizes in a natural way the commonlyused NHPP model with covariates; see, for example,[41].

4. UNOBSERVED HETEROGENEITY IN

REPAIRABLE SYSTEMS

Analyses of reliability data often lead to an appar-ent decreasing failure rate which could be counter-intuitive in view of wear and aging effects. Proschan[58] pointed out that such observed decreasing ratescould be caused by unobserved heterogeneity.Proschan presented failure data from 17 air condi-tioner systems on Boeing 720 airplanes. ApplyingMann’s [51] nonparametric trend test to each sys-tem and then combining to a global test statistic,he argued that there is no significant trend in the

failure times for each separate plane. He then con-cluded by a similar test that “it seems safe to acceptthe exponential distribution as describing the fail-ure interval, although to each plane may corresponda different failure rate.” He demonstrated this lastfact statistically by using a result from Barlow, Mar-shall and Proschan [7] which implies that a mixtureof exponential distributions has a decreasing failurerate. More precisely, he applied again the Mann test,which is sensitive to a decreasing failure rate, on thepooled interfailure times from all the planes. In thisway he obtained a p-value of 0.007 for the null hy-pothesis of identical exponential distributions of theinterfailure times.

Heterogeneity in connection with Poisson processeswas in fact studied as early as 1920 by Greenwoodand Yule [27], who used a compound Poisson dis-tribution. Later, Maguire, Pearson and Wynn [50],studying occurrences of industrial accidents, showedhow Laplace transforms enter general expressionsfor resulting distributions of intervals and counts.Cox [17] considered the possibility of heterogeneity,which he called variance components, between ho-mogeneous Poisson processes and listed several rea-sons for the interest in such models for repairablesystems data.

It has similarly long been known in biostatisticsthat neglecting individual heterogeneity may lead tosevere bias in estimates of lifetime distributions. Theidea is that individuals or components have differ-ent “frailties” and that those who are most “frail”will die or fail earlier than the others. This in turnleads to a decreasing population hazard, which hasoften been misinterpreted. Important references onheterogeneity in the biostatistics literature are [62],[32] and [2]. It should be noted that heterogeneityis, in general, unidentifiable if it is considered asan individual quantity. For identifiability it is neces-sary that frailty be common to several individuals,for example, in family studies in biostatistics, or ifseveral events are observed for each individual, suchas for the repairable systems considered in this pa-per and more generally for recurrent events data.The presence of heterogeneity is often apparent fordata from repairable systems if there is a large vari-ation in the number of events per system. However,it is not really possible to distinguish between het-erogeneity and dependence of the intensity on pastevents for a single process. It is a fact, though, thatignorance of an existing heterogeneity may lead tosuboptimal or even wrong decisions.

ANALYSIS OF REPAIRABLE SYSTEMS 9

4.1 Modeling Heterogeneity for Repairable

Systems

The common way to model heterogeneity is to in-clude an unobservable multiplicative constant in theconditional intensity of the process; see, for exam-ple, [62]. For systems with a single type of event thisis done by first replacing the conditional intensitiesγ(t) in (1) by aγ(t), where a is a random variablethat represents the frailty of the system and suchthat a is included in Ft− for each t. Note that γ(t) asdescribed in Section 2 may well be a function of co-variates. Now a can be viewed as being the effect ofan unobserved covariate. Systems with a large valueof a will have a larger failure proneness than sys-tems with a low value of a. Intuitively, the variationin the a between systems implies that the variationin observed number of failures among the systemsis larger than would be expected if the failure pro-cesses were identically distributed. Now, since a isunobservable, one needs to take the expectation ofthe likelihood that results from (2) with respect tothe distribution of a in order to have a likelihoodfunction for the observed data.

In the marked point process formulation of Sec-tion 2 we may more generally assume that there aredifferent frailty variables for each event type j ∈ J .More precisely, we assume that there is a randomvector a = (aj , j ∈ J ) such that the type-specific in-tensities for given a are ajγj(t), respectively, whereγj(t) corresponds to the type-specific conditional in-tensity defined in Section 2. The resulting likelihoodincluding heterogeneity is thus

L = Ea

[(N(τ)∏

i=1

aJiγJi

(Ti)

)

(9)

· exp

{

−∑

j∈J

aj

∫ τ

0γj(u)du

}]

,

where the expected value is taken with respect tothe joint distribution of a. Multivariate frailty dis-tributions are considered by, for example, Hougaard[32] and Aalen [1].

In the case of several independent systems, it isassumed that the a’s that correspond to the sys-tems are i.i.d. from the given joint distribution. Thetotal likelihood is then the product of factors (9),one for each system. Note that for identifiability itmay be necessary to introduce a normalization of a,for example, to assume that E(‖a‖) = 1. This is be-cause otherwise a scale factor may be moved from

aj to γj(·) or vice versa without changing the valueof (9). Alternatively one may let the aj act as freerandom scale parameters in the model if the γj(·)themselves do not include scale parameters.

For the special case of a single type of event oneobtains simplification of the likelihood function in (9),

L = Ea

[

aN(τ)

(N(τ)∏

i=1

γ(Ti)

)

(10)

· exp

{

−a

∫ τ

0γ(u)du

}

]

,

where the expectation is with respect to the dis-tribution of the random variable a and where fornormalization one will usually assume E(a) = 1.

Expression (10) suggests that a gamma distribu-tion for a is mathematically convenient, since a closedform expression of the likelihood is obtained. Moregenerally, for the version (9), a multivariate gammadistribution for a leads to a simplified expression(see, e.g., [1] and [32] regarding multivariate gammadistributions).

Consider now the likelihood (10) and suppose thata is gamma distributed with E(a) = 1,Var(a) = δ.Then a straightforward computation gives

L =

{N(τ)∏

i=1

γ(Ti)

}

·Γ(N(τ) + 1/δ)

δ1/δΓ(1/δ)[1/δ +∫ τ0 γ(u)du]N(τ)+1/δ

(11)

=

{N(τ)∏

i=1

γ(Ti)

}

·[δ(N(τ) − 1) + 1][δ(N(τ) − 2) + 1] · · ·1

[δ∫ τ0 γ(u)du + 1]N(τ)+1/δ

,

where we have used the fact that Γ(r + 1) = rΓ(r).Recall that γ(Ti) may well include covariates. Thislikelihood expression is applicable, for example, to-gether with the virtual age model (3) and the gen-eralized linear model types (4) and (5). It is also thelikelihood function for NHPPs with heterogeneityand possibly covariates, as studied in Lawless [41],and results in the likelihood of the so-called com-pound power law model studied by Engelhardt andBain [26].

We remark that (11) converges to (2) (assuminga single type of event) as δ → 0.

10 B. H. LINDQVIST

4.2 Heterogeneity in the TRP Model, the

HTRP Model

Lindqvist, Elvebakk and Heggland [48] introducedheterogeneity into the TRP model by including anunobservable random multiplicative constant a inthe trend function λ(t), thus considering the condi-tional model TRP(F,aλ(·)) with a renewal distribu-tion F that does not depend on a. This definitionis consistent with the regression version of TRP assuggested at the end of Section 3.4. Now the a re-places the function g(Z(t)) used there. Note that inpractice one may want to include both the frailty aand a covariate factor g(Z(t)). To simplify the dis-cussion, we will, however, not consider covariates inour presentation.

Considering (6), it is seen that the conditional in-tensity function given a is no longer of the simplemultiplicative form aγ(t) which was assumed in theprevious subsection. This is because the Λ(·) in (6) isalso multiplied by a. Instead of the expression (10),the relevant likelihood from one system becomes,using (7),

L = Ea

[{N(τ)∏

i=1

z[a(Λ(Ti)−Λ(Ti−1))]aλ(Ti)

}

· exp

{

−a

∫ τ

0z[a(Λ(u)−Λ(TN(u−)))](12)

· λ(u)du

}

]

or, using (8),

L = Ea

{N(τ)∏

i=1

f [a(Λ(Ti)−Λ(Ti−1))]aλ(Ti)

}

(13)· {1−F [a(Λ(τ)−Λ(TN(τ)))]}.

Here f and z are, respectively, as before, the densityand hazard function of the distribution F .

The expressions (12) and (13) appear to be lesstractable than the expression (10). Lindqvist, Elve-bakk and Heggland [48] obtained, however, a rathersimple expression for the likelihood in the case ofan inhomogeneous gamma process with gamma dis-tributed heterogeneity factor a, under the furtherassumption that the stopping times τ coincide withfailure times. In this case the last factor of (13) dis-appears, and letting F be the gamma distributionwith unit expectation and variance γ, while a is

gamma distributed with unit expectation and vari-ance δ, one obtains

L =

{N(τ)∏

i=1

(Λ(Ti)−Λ(Ti−1))1/γ−1λ(Ti)

}

· (Γ(N(τ)/γ + 1/δ))

·{

γN(τ)/γ [Γ(1/γ)]N(τ)δ1/δ

· Γ(1/δ)[1/δ + (1/γ)Λ(TN(τ))]N(τ)/γ+1/δ

}−1.

Note that for γ = 1 this is of the same form as in(11).

We use the notation HTRP(F,λ(·),H) for the modelwith likelihood (12) or, equivalently, (13). The “H”in HTRP here stands for heterogeneity, and the Hwhich is added to (F,λ(·)) in the notation is the dis-tribution of the variable a, which can be any positivedistribution with expected value 1.

4.3 The Three Dimensions of a Repairable

System Description: The Model Cube and

the Log-Likelihood Cube

A useful feature of the HTRP model is that severalmodels for repairable systems can be represented as

Fig. 3. The model cube illustrating the HTRP(F,λ(·),H)and the submodels obtained by restricting one or more ofF,λ(·),H to their basic versions, respectively, F being stan-dard exponential (using the notation - in the figure), λ(t) ≡ λ

being constant in time and H being the distribution determin-istic at 1 (- in the figure).

ANALYSIS OF REPAIRABLE SYSTEMS 11

submodels. With the notation HPP, NHPP, RP andTRP used as before, and with an H in front mean-ing the model which includes heterogeneity, Figure 3shows how the HTRP and the seven sub-models canbe represented in a cube [25]. Each vertex of thecube represents a model, and the lines that connectthem correspond to changing one of the three “coor-dinates” (F,λ(·),H) in the HTRP notation. Goingto the right corresponds to introducing a time trend,going upward corresponds to entering a non-Poisson(renewal) case and going backward (inward) corre-sponds to introducing heterogeneity.

In analyzing data by parametric HTRP models wemay use the cube to facilitate the presentation ofmaximum log-likelihood values and parameter esti-mates for the different models in a convenient, visualmanner which may guide model choice (see [48]).Figures 4 and 5 show maximum likelihood valuescomputed from the data of Proschan [58] and Aalenand Husebye [3], respectively. The latter data set istaken from a medical study and is included here todemonstrate results for data which are clearly non-Poisson distributed.

Fig. 4. The log-likelihood cube for the data of Proschan [58]concerning failures of air conditioner systems on airplanes,fitted with a parametric HTRP(F,λ(·),H) model and its sub-models. Here F is a Weibull distribution with expected value1 and shape parameter s, λ(t) = cbtb−1 is a power functionof t and H is a gamma distribution with expected value 1and variance v. The maximum value of the log-likelihood isdenoted l.

Fig. 5. The log-likelihood cube for the data of Aalen andHusebye [3] concerning migratory motor complex periods, fit-ted with a parametric HTRP(F,λ(·),H) model and its sub-models. Here F is a Weibull distribution with expected value1 and shape parameter s, λ(t) = cbtb−1 is a power functionof t and H is a gamma distribution with expected value 1and variance v. The maximum value of the log-likelihood isdenoted l.

For the Proschan data we conclude that the re-newal distribution can be taken to be exponential,leaving us with the bottom face of the cube. Fur-ther, when comparing the front face to the back facethere is clear reason to conclude that there is het-erogeneity between the systems, with Var(a) beingestimated to approximately 0.11. The conclusionsso far are thus in accordance with the conclusions ofProschan [58]. However, a comparison of the left andright faces of the cube reveals a slight time trend. Infact, twice the log-likelihood difference from HHPPto HNHPP amounts to 5.28, giving a p-value of0.022 assuming a chi-squared distribution with onedegree of freedom of the corresponding likelihoodratio test statistic. The power parameter b of thetrend function is, furthermore, estimated as 1.16.

The most obvious conclusion for the Aalen andHusebye [3] data is that the renewal distributionis not exponential, implying that the upper faceof the cube applies. Further, the differences in log-likelihood obtained by introducing heterogeneity areseen to be small enough to conclude there is no sig-nificant heterogeneity. However, as for the Proschan

12 B. H. LINDQVIST

Fig. 6. Plot of cumulative number of failures, N(t), for air conditioner failures of plane 7913 in the Proschan [58] data.

data, there seems to be a slight time trend. Here,

twice the log-likelihood difference from RP to TRP

amounts to 4.18, giving a p-value of 0.041, while the

power parameter b is estimated as 1.14 for the TRP

model. Note the large difference in log-likelihood

value between, for example, the TRP and NHPP

models. As shown by the parameter estimates (Fig-

ure 5), the NHPP estimates seem to compensate for

the large estimated shape parameter for the renewal

distribution of the TRP by increasing the power pa-

rameter b of the trend function (from 1.14 to 1.45).

It is also seen that for the Poisson models (bottom

face) there is no gain in log-likelihood by introduc-

ing heterogeneity. Thus the maximum likelihood es-

timates of the heterogeneity variance v are given by

the border value 0. This is so since the profile likeli-

hood of v can be shown to be a decreasing function

of v > 0 near 0 (see [48] for a further discussion of

this effect).

5. TREND TESTING

In many applications involving repairable systems,

the main aim is to detect trends in the pattern of

failures that occur over time. These may often be re-

vealed as monotonic trends in the interfailure times,

corresponding to either improving or deteriorating

systems. Various types of nonmonotonic trends may

also be present, for example, a cyclic trend or a bath-

tub shaped trend.

5.1 Graphical methods

A simple but informative way to check for a possi-ble trend in the pattern of failures is to study plotslike Figure 6, which is a plot of cumulative failurenumber versus failure time for a single system. Theunderlying data are failures of the air conditionersystem of airplane 7913 of the Proschan [58] data.A convex plot would be indicative of a deteriorat-ing system, while a concave plot would indicate animproving system. In Figure 6 there seems to be nosignificant deviation from a straight line, however,thus indicating no trend in interfailure times.

5.1.1 Nelson–Aalen plot. The plot of Figure 6 is aspecial case of the Nelson–Aalen plot to be describednext. Assume that m systems are observed, withthe individual failure processes being independentand identically distributed. Suppose further that theith process is observed on the time interval (0, τi]and let y(t) denote the number of processes underobservation at time t. Note that y(t) is a functionof the τi and not of the failure times. Let Tk denotethe kth arrival time in the superposed process, thatis, Ti is a failure time in one of the processes and0 < T1 ≤ T2 ≤ · · · ≤ TN ≤ τ , where τ = max{τi : i =1, . . . ,m}. Define the cumulative mean function of asingle process to be M(t) = E(N(t)). The Nelson–Aalen estimator of M(t) is given by

M(t) =∑

Tk≤t

1

y(Tk),

ANALYSIS OF REPAIRABLE SYSTEMS 13

where the sum is taken over all failure times Tk be-fore or at time t. Figure 7 shows the plot of M(t)for the data on times of valve-seat replacements ina fleet of m = 41 diesel engines, taken from [53].The plot indicates that the replacement frequencyis fairly constant up to 550 days and then increasesas revealed from the convex shape of the curve atthe right end.

The plot as defined here is studied, for example, in[53] and [42]. These papers also derive robust non-

parametric estimates of the variance of M(t), validunder any distributional properties of the individualprocesses N(t).

5.1.2 TTT plot. Consider the special case of theabove where the m processes are independent NHPPswith a common intensity function λ(t). The super-posed process is now a NHPP with intensity func-tion φ(t) = λ(t)y(t), and hence (see Section 3.4) the

process∫ T10 φ(u)du,

∫ T20 φ(u)du, . . . is HPP(1) on (0, τ).

Define the total time on test (TTT) at time t by

r(t) =

∫ t

0y(u)du.

Barlow and Davis [6] introduced the TTT plot forrepairable systems data as a plot of the points

(

i

N,r(Ti)

r(τ)

)

, i = 1, . . . ,N.

The idea is that if λ(t) is a constant, so that theprocesses are HPP, then the r(Ti)/r(τ), i = 1, . . . ,N ,

form a HPP(1) on [0,1]. In this case the TTT plotis by its definition expected to be located near themain diagonal of the unit square. Under the alterna-tives of decreasing, increasing and bathtub-shapedintensity λ(t), on the other hand, the TTT plotsappear to be, respectively, convex, concave and S-shaped. Figure 8 shows the TTT plot of the valve-seat replacement data of Nelson [53]. The plot ap-pears to be fairly straight, but with a slightly con-cave shape near the end corresponding to the in-creasing intensity here as revealed by the Nelson–Aalen plot in Figure 7.

5.2 Statistical Trend Tests

Statistical trend tests for repairable systems datawere extensively discussed by Ascher and Feingold[5], Chapter 5B. A trend test is a statistical testfor the null hypothesis that the failure process isstationary, in some sense to be made precise, versusalternatives that depend on the kind of trend onewould like to detect. Here we give main attentionto the null hypothesis that the process is a HPP ormore generally a RP. However, as will be discussedbelow, some care should be taken when determiningthe relevant null hypothesis.

The null hypothesis of HPP is the most commonand often the most useful in reliability applications.The corresponding null property, under the name“randomness,” was studied in several papers in the1950s, and various tests for randomness in time were

Fig. 7. Nelson–Aalen plot of the estimated cumulative mean function M(t) for the valve-seat replacement data as given byNelson [53].

14 B. H. LINDQVIST

devised. Here randomness pertained to the prop-erty that counts in given time intervals are Poisson-distributed. Maguire, Pearson and Wynn [50], how-ever, discussed the advantages of using intereventtimes rather than counts to test for changes withtime of the occurrence rate of events. Cox [17] statedeight different kinds of possible alternatives to ran-domness, one of them being trend in the sense thatthe conditional intensity is a smooth function oftime.

5.2.1 Tests of the null hypothesis of HPP. Sin-

gle process. Suppose first that the null hypothesisis “the process is a HPP,” with the alternative be-ing a NHPP with monotone intensity. Two classicaltrend tests for this case are the Laplace test and theMilitary Handbook test (see, e.g., [5], page 79). Tosee how they are obtained, consider a single systemobserved on [0, τ ]. If the failure process is a HPP,then given N(τ) = n, the failure times T1, T2, . . . , Tn

are distributed as the ordering of n i.i.d. uniformrandom variables on [0, τ ]. Equivalently, the Ti/τ(i = 1, . . . , n) are distributed as ordered i.i.d. uni-forms on [0,1] conditionally given N(τ) = n. Fromthis we can in principle obtain trend tests from anytest for detecting deviations from a uniform sample.The Laplace test statistic is simply a normalizationof∑n

i=1 Ti, while the Military Handbook test statis-tic is similarly a normalization of

∑ni=1 logTi. The

Laplace test and the Military Handbook test are op-timal tests against the alternatives of NHPPs with,respectively, log linear intensity and power intensityfunctions ([5], page 79).

Several processes. As in Section 5.1.2, assume thatm independent NHPPs with a common intensityfunction λ(t) are observed, where the ith processis observed on the time interval (0, τi]. Recall that,under the null hypothesis that λ(t) is a constant,the r(Ti)/r(τ), i = 1, . . . ,N , form a HPP(1) on [0,1].Kvaløy and Lindqvist [35] suggested from this thatformal trend tests could be defined by substitutingthe r(Ti)/r(τ) into the Laplace and Military Hand-book test statistics. While these TTT-based testsare powerful against monotone alternatives, the au-thors suggested using a test statistic based on theAnderson–Darling statistic as a general test withpower against several kinds of trend.

For many applications, the null hypothesis needsto be weakened to state that each process is a HPP,but that intensities may differ from system to sys-tem. For example, in the data of Proschan [58], onemay be interested in a simultaneous trend test forthe systems, allowing there to be heterogeneities be-tween them. Kvaløy and Lindqvist [35] suggestedtests for this case called combined tests. A precisesetting for these tests was recently defined by Kvist,Andersen and Kessing [37], who considered a modelwhere the conditional intensity function for a partic-ular system is given by ag(Z)λ(t), where a is an un-observable frailty variable as considered in Section4.1, Z is a fixed-time covariate vector observed foreach system, g is a parametric regression functionand λ(t) is a baseline intensity function. Supposethat such a process is observed on the time interval[0, τ ] with events at times T1, T2, . . . , TN(τ). Then,

Fig. 8. TTT plot of valve-seat data as given in [53].

ANALYSIS OF REPAIRABLE SYSTEMS 15

conditional on (a,Z, τ,N(τ)), the T1/τ,T2/τ, . . . ,TN(τ)/τ are distributed as N(τ) ordered standarduniform variables on [0,1] if λ(t) is constant. Weare hence back to the setting of the beginning of thissubsection. In practice one observes m independentprocesses of this kind, with a common λ(t), with thea being i.i.d. unobservable random variables and theZ being observed covariate vectors for each system.The above-mentioned combined tests by Kvaløy andLindqvist [35] can thus be used to test the null hy-pothesis that λ(t) does not depend on t. Kvist, An-dersen and Kessing [37] applied the Laplace typetest of this kind on data from the Danish registeron psychiatric hospital admissions.

5.2.2 Tests of the null hypothesis of RP. The La-place test and the Military Handbook test are testsfor the null hypothesis that the data come fromHPPs. Thus rejection of the null hypothesis meansmerely that the process is not a HPP. It could still,however, be a RP and thus still have “no trend.”Lawless and Thiagarajah [43] and Elvebakk [25] con-cluded from simulations that the Laplace and Mil-itary Handbook tests in fact may be seriously mis-leading when used to detect trend departures fromgeneral renewal processes. Similarly, Lewis and Robin-son [44] noted that these tests are not able to dis-criminate properly between trends in the data andthe appearance of sequences of very long intervals.

To test the null hypothesis of RP, Lewis andRobinson [44] suggested modifying the Laplace testby dividing the test statistic by an estimate of thecoefficient of variation of the interfailure times un-der the null hypothesis of a RP. This test, called theLewis–Robinson test, is thus a simple modificationof the Laplace test. Another classical trend test forthe null hypothesis of RP is the rank test developedby Mann [51] and used by Proschan [58] (see Section4).

Kvaløy and Lindqvist [36] presented a general classof tests for renewal process versus both monotonicand nonmonotonic trend for which the Lewis–Robinsonand a useful Anderson–Darling type test are specialcases.

Elvebakk [25] demonstrated how tests for the nullhypothesis of RP can be obtained from tests for thePoisson case by adjusting their critical values by re-sampling failure data under the RP hypothesis. Thegeneral conclusion of Elvebakk [25] was to recom-mend the use of such resampled trend tests when-ever it is not clear that the failure processes are of

Poisson type. In particular he showed in a simula-tion study that the resampled tests are usually fa-vorable to the Lewis–Robinson test, and that theydo not lose much power under NHPP alternativeswhen compared to the standard tests.

5.2.3 Tests of the null hypothesis of stationary in-

terfailure times. Lewis and Robinson [44] presenteda test for distinguishing between a general station-ary sequence of interfailure times Xi and a mono-tonic trend in interfailure times. Elvebakk [25] ex-tended the resampling trend testing approach de-scribed in the previous subsection, to cover the casewhen “no trend” corresponds to stationary interfail-ure times. The idea is to resample data under thisnew null hypothesis assumption. Elvebakk did thisboth by a parametric approach assuming an under-lying autoregressive model and by employing a blockbootstrap technique adapted from Hall [28]. Simu-lations indicated rather satisfactory performance ofthe method.

5.2.4 Trend tests obtained as likelihood ratio tests.

In parametric models which include separate param-eters for trend, trend tests may be performed as like-lihood ratio tests that involve these parameters. Anexample is to test the null hypothesis β1 = 0 in (5)which was suggested in [43]. Trend tests can also beobtained in models of the form HTRP(F,λ(·),H) bytesting the null hypothesis that λ(·) ≡ λ using likeli-hood ratio tests. Note that this leads to tests of thenull hypothesis that the processes are all renewalprocesses with a possibility of heterogeneity.

A nonparametric likelihood ratio test for the nullhypothesis of a HPP versus the alternative of a NHPPwith monotone intensity λ(·) was derived by Boswell[11]. A generalization to the null hypothesis of RPcan be obtained using the nonparametric monotoneestimator of λ(·) in the TRP model derived by Heg-gland and Lindqvist [29].

6. REPAIRABLE SYSTEMS WITH SEVERAL

TYPES OF EVENTS

In this section we consider the general markedevent process described in Section 2. The purposeis to show how new classes of maintenance and re-pair models can be obtained by generalizing the ap-proach of the imperfect repair models for single typeevents considered in Section 3.2. To simplify the pre-sentation we shall not allow covariates or hetero-geneity in the models considered here.

16 B. H. LINDQVIST

As in Section 3.2, we consider first a nonrepairableunit. Assume that this unit may fail due to one ofseveral causes or may be stopped for PM before itfails, in which case failure is prevented.

We can formally think of this as having a systemwith, say, n components, denoted {C1,C2, . . . ,Cn},where a unique failing component can be identifiedat failures of the system and where PM, if appli-cable, is represented by one of these componentsso as to simplify notation. Let Wj be the poten-tial failure time due to failure of component Cj ,j = 1,2, . . . , n. What is observed is the failure timeT = min(W1, . . . ,Wn) and the identity of the failingcomponent, say J = j if the component Cj fails. Thisdetermines a competing risks situation with n com-peting risks and with the observed outcome (T,J)([20], Chapter 3). The joint distribution of (T,J)is thus identifiable from data, as are the so-calledtype-specific hazards defined by

hj(t) = lim∆t↓0

Pr(t < T ≤ t + ∆t, J = j|T > t)

∆t.(14)

However, neither the joint nor the marginal distribu-tions of the individual potential failure times W1, . . . ,Wn are identifiable in general from observation of(T,J) only. This follows from the so-called Cox–Tsiatis impasse; see [20], Chapter 7. On the otherhand, these marginal and joint distributions are in-deed of interest in reliability applications, for exam-ple, in connection with maintenance optimization.An example is given in the next paragraph.

Consider the setup of Cooke [15, 16] that involvesa competing risks situation with a potential failureof a unit at some time W1 and a potential actionof preventive maintenance to be performed at timeW2. Thus n = 2, while C1 corresponds to failureof the unit (J = 1) and C2 (J = 2) corresponds tothe action of PM. Knowing the marginal distribu-tion of W1 would be particularly important sinceit is the basic failure time distribution of the unitwhen there is no PM. However, as noted above, themarginal distributions of W1 and W2 are not iden-tifiable unless specific assumptions are made on thedependence between W1 and W2. The most com-mon assumption of this kind is that W1 and W2

are independent, in which case identifiability follows([61]; [20], Chapter 7). However, this assumption isunreasonable in the present application, since themaintenance crew is likely to have some informationregarding the unit’s state during operation. This in-sight is used to perform maintenance so as to avoid

a failure. Thus we are in practice faced with a situa-tion of dependent competing risks between W1 andW2, and hence identifiability of marginal distribu-tions requires additional assumptions.

Lindqvist, Støve and Langseth [49] suggested amodel called the repair alert model to describe thejoint behavior of the failure time W1 and time W2

of PM. This model is a special case of random signscensoring [15, 16] under which the marginal distri-bution of W1 is always identifiable. Recall that W2

is said to be a random signs censoring of W1 if theevent {W2 < W1} is stochastically independent ofW1, that is, if the event of having a PM before fail-ure is not influenced by the time W1 at which thesystem fails or would have failed without PM. Theidea is that the system emits some kind of signal be-fore failure and that this signal is discovered with aprobability which does not depend on the age of thesystem. The repair alert model extends this idea byintroducing a so-called repair alert function whichdescribes the “alertness” of the maintenance crewas a function of time.

Another possibility to obtain identifiability of thedistributions of W1 and W2 would be to use the re-sult of Zheng and Klein [64], which shows identifia-bility of marginal distributions when the dependenceis given by a known copula.

Return now to the general case. Suppose that thesystem is repaired after failure and then put intooperation, then may fail again and so on. This leadsto a marked event process as described in Section 2with marks in J = {1,2, . . . , n}, so that the mark ateach event time is the number of the failing compo-nent (or more generally the type of event).

The properties of this process depend on the re-pair strategy. Several classes of interesting modelscan be described in terms of a generalization of thevirtual age concept introduced in Section 3.2, as dis-cussed in the next subsection.

6.1 Virtual Age Models with Several Types of

Events

Recall from Section 3.2 that the class of virtualage models generalizes the perfect repair and mini-mal repair models, and that the approach more gen-erally leads to a large class of models. The main in-puts are a hazard function z(·), which is thoughtof as the hazard function of a new unit, and a vir-tual age process which is a stochastic process whichdepends on the actual repair actions performed.

ANALYSIS OF REPAIRABLE SYSTEMS 17

Several generalizations of the standard imperfectrepair models are found in the literature. Shakedand Shanthikumar [60] suggested a multicomponentimperfect repair model with components that havedependent life-lengths. Langseth and Lindqvist [38]suggested a model which involves imperfect mainte-nance and repair in the case of several componentsand several failure causes. In a recent paper, Doyenand Gaudoin [24] developed the ideas further by pre-senting a general point process framework for mod-eling imperfect repair by a competing risks situationbetween failure and PM. Bedford and Lindqvist [8]considered a series system of n repairable compo-nents where only the failing component is repairedat failures.

Inspired by the mentioned approaches, we suggestin this section a generalization of the imperfect re-pair models to the case where there is more thanone type of event and where the virtual age processis multidimensional.

We let the first part of a virtual age model forn components be given by a vector process A(t) =(A1(t), . . . ,An(t)) that contains the virtual ages ofthe n components at time t. The crucial assumptionis that A(t) = (A1(t), . . . ,An(t)) ∈ Ft−, which meansthat the component ages are functions of the historyup to time t.

As for the case with n = 1 in Section 3.2, it isassumed that the Aj(t) increase linearly with timebetween events, and may jump only at event times.We define vj(i) to be the virtual age of componentj immediately after the ith event. The virtual ageprocess for component j is therefore defined by

Aj(t) = vj(N(t−)) + t− TN(t−).

The second part of a virtual age model in the casen = 1 consists of the hazard function z(·). For gen-eral n we replace this by functions νj(v1, . . . , vn) forv1, v2, . . . , vn ≥ 0, such that the conditional intensityof type j events, given the history Ft−, is

γj(t) = νj(A1(t), . . . ,An(t)).

Thus νj(v1, . . . , vn) is the intensity of an event oftype j when the component ages are v1, . . . , vn, re-spectively. The conditional intensity thus dependson the history only through the virtual ages of thecomponents.

The family {νj(v1, . . . , vn) :v1, v2, . . . , vn ≥ 0} de-scribes the failure mechanisms of the componentsand the dependence between them in terms of theages of all the components. The basic statistical

inference problem therefore consists of estimatingthese functions from field data. The case n = 1 hasalready been discussed in Section 3.2, but we shallsee that identifiability problems can occur when n >1.

6.2 Repair Models and their Virtual Age

Processes

Most of the virtual age processes considered forthe case n = 1 can be generalized to the presentcase of several event types. There are, however, of-ten several ways to do this. Some examples are givenbelow. Additional examples include generalizationsof Kijima’s [34] models, which may be plausible inapplications.

6.2.1 Perfect repair of complete system. Supposethat all the components are repaired to as good asnew at each failure of the system. In this case wehave vj(i) = 0 for all j and i, and hence Aj(t) =t−TN(t−) for all j. It follows that we can only iden-tify the “diagonal” values νj(t, . . . , t) of the functionsνj . As noted in Section 6.3, these are given by thetype-specific hazards defined in (14) for the nonre-pairable competing risks case. This is not surprisingin view of the fact that the present case of perfectrepair essentially corresponds to observation of i.i.d.realizations of the nonrepairable competing risks sit-uation.

6.2.2 Minimal repair of complete system. In thegiven setting a minimal repair will mean that fol-lowing an event, the process is restarted in the samestate as was experienced immediately before the event.In mathematical terms, this implies that vj(i) = Ti

for all i, j and hence that Aj(t) = t for all j. Notethat the complete set of functions νj is again notidentifiable. Moreover, for a single component it iswell known that minimal repair results in a failuretime process which is a NHPP. In the present case ofseveral components which are minimally repaired, itfollows similarly that the failure processes of the in-dividual components are independent NHPPs withthe intensity for component j given by νj(t, . . . , t),which as already noted equals the type-specific haz-ard (14).

6.2.3 A partial repair model. Bedford and Lindqvist[8] suggested a partial repair model for the n com-ponent case. The virtual age process is defined byletting Aj(t) = time since last event of type j. Equiv-alently, the process could be defined by

vj(i) =

{

0, if Ji = j,vj(i− 1) + Xi, if Ji 6= j.

18 B. H. LINDQVIST

Thus, the age of the failing component is reset to0 at failures, whereas the ages of the other compo-nents are unchanged. The authors considered a sin-gle realization of the process, with the main resultbeing that under reasonable conditions pertainingto ergodicity, the functions νj(v1, . . . , vn) are identi-fiable. The intuitive idea of their proof is that theages v1, . . . , vn will mix in such a manner that thecomplete set of νj(v1, . . . , vn) can be identified.

6.2.4 Age reduction models. Doyen and Gaudoin[24] considered a single component or system andtwo types of events: C1 = failure, C2 = PM. In theirbasic model the virtual ages of the two types ofevents are equal: A1(t) = A2(t) = A(t). They indi-cated, however, that this restriction is not necessary.Various choices of virtual age processes were con-sidered. In particular they considered age reductionmodels that generalize these mentioned at the endof Section 3.2. More precisely, assume that there aregiven age reduction factors 0 < ρ1, ρ2 < 1 for the twotypes of events. The virtual age immediately afterthe ith repair is then

v(i) = (1− ρJi)(v(i− 1) + Xi),

which means that the virtual age immediately be-fore the ith failure, v(i− 1) + Xi, is reduced due torepair by the factor 1−ρJi

. Alternatively, if only theadditional age Xi is reduced by the repair, it couldbe assumed that v(i) = v(i− 1) + (1− ρJi

)Xi.

6.3 Modeling the Intensity Functions νj

In principle the functions νj(v1, . . . , vn) could beany functions of the component ages. Bedford andLindqvist [8] motivated these functions by writing,for j = 1, . . . , n,

νj(v1, . . . , vn) = λj(vj) + λj∗(v1, . . . , vn)(15)

with the convention that λj∗(v1, . . . , vn) = 0 whenall the component ages except the jth are 0, so asto have uniqueness. Then λj(vj) is thought of as theintensity of component j when working alone or to-gether with only new components, while λj∗(v1, . . . ,vn) is the additional failure intensity imposed oncomponent j caused by the other components whenthey are not all new. Note that any functions ofv1, . . . , vn can be represented this way, by allowingthe λj∗ to be negative as well as positive.

Langseth and Lindqvist [38] and Doyen andGaudoin [24] extended the competing risks situa-tion between failure and PM, as described at the

beginning of the present section, and suggested howto define suitable functions νj . The main ideas ofthese approaches can be described for general n asfollows. Starting from a state where the componentages are, respectively, v1, . . . , vn, let the time to nextevent be governed by the competing risks situationbetween the random variables W ∗

1 , . . . ,W ∗n with dis-

tribution equal to the conditional distribution ofW1 − v1, . . . ,Wn − vn given W1 > v1, . . . ,Wn > vn,where the Wi are defined in the nonrepairable casedescribed at the beginning of the section. It is thenrather straightforward to show that this implies

νj(v1, . . . , vn) =−∂jR(v1, . . . , vn)

R(v1, . . . , vn),(16)

where R(v1, . . . , vn) = P (W1 > v1, . . . ,Wn > vn) isthe joint survival function of the Wi, and ∂j meansthe partial derivative with respect to the jth entry inR. Note that this generalizes the usual hazard ratein the case n = 1 considered in Section 3.2. Further,we have νj(t, t, . . . , t) = hj(t), where the latter is thetype-specific hazard rate given in (14).

A final remark on the suggested construction ofthe functions νj is due. It was demonstrated by Bed-ford and Lindqvist [8] that, even in the case withn = 2, it is not always possible to derive a generalset of functions νj(v1, . . . , vn) from a single joint sur-vival distribution as in (16). A simple counterexam-ple was given in [8]. Thus for generality one shouldstick to completely general representations like (15).

7. CONCLUDING REMARKS

In the present paper we have reviewed some mainapproaches for the analysis of data from repairablesystems. To a large extent the emphasis has been ondescribing the underlying principles and structuresof common models. Essential features of such mod-els correspond to the three dimensions of the modelcube in Figure 3: renewal property, time trend andheterogeneity. The presentation places less empha-sis on statistical inference than on modeling. How-ever, it has been an intention to show how likelihoodfunctions are obtained for the different models. It isalso indicated how covariates can be included in themodels and the corresponding likelihood functions.While the derived likelihood functions can be used ina rather straightforward manner in parametric sta-tistical inference, there turn out to be several chal-lenging problems connected to nonparametric esti-mation in some of the models.

ANALYSIS OF REPAIRABLE SYSTEMS 19

Two main types of models with rather simple andtransparent basic structures have been considered.These are the virtual age type models and the TRPtype models. The former type combines two basicingredients: a hazard rate z(·) of a new system to-gether with a particular repair strategy which gov-erns the virtual age process A(t). The renewal di-mension is taken care of by the virtual age process,while trend is determined by the distribution of anew system. For the TRP(F,λ(·)), the renewal di-mension corresponds to the renewal distribution F ,while the trend is explicitly given by the trend func-tion λ(·). For both types of processes, heterogeneitycan be included by multiplicative factors working onthe intensities. A noticeable difference between thetwo types of models as regards statistical inference isthat the virtual age type model usually requires thatthe virtual age process be observable. Such observa-tions may, however, often be lacking in real data.

Many processes show some degree of clustering offailures. This may be due to various causes; see, forexample, [17]. Several models have been suggestedin the literature, a classical one being the Neymanand Scott [54] model. As pointed out by a referee,even the TRP model can pick up the clustering effectby allowing the renewal distribution to be a mixturewith a substantial amount of probability near zero.

Pena [55] has reviewed a class of models suggestedin [56]. These are virtual age models which includethe possibility of heterogeneity between systems, time-dependent covariates, and for which in addition theconditional intensities may depend on the number ofprevious events. This last feature adds an interest-ing flexibility to the model. In particular it enablesmodeling of certain load sharing processes and soft-ware failure processes.

Certain systems, for example, alarm systems, aretested only at fixed times which are usually peri-odic. If the system is found in a failed state, thenit is repaired or replaced. Thus repair is not doneat the same time as the failure, and the situation isnot covered by the methods considered in the pa-per. A simple model of this situation was suggestedby Hokstad and Frøvig [30] and further studied andextended by Lindqvist and Amundrustad [47]. Con-sider a system which starts operation at time t = 0and is tested at time epochs τ,2τ,3τ, . . . . When timeis running between testing epochs, the state of thesystem is modeled by an absorbing Markov chain.Having thus defined the probabilistic behavior of thesystem state between testing, one needs to add to

the model a specification of the repair policy. In [47]this is modeled in the form of a transition matrix onthe state space of the Markov chain, which definesthe possible changes of state and their probabilitiesfollowing the repair actions.

In a given study there is usually a choice betweenseveral types of models. It is thus important to havetools for model checking and goodness-of-fit pro-cedures. For model checking in parametric estima-tion of the HTRP model, we refer to [48], whichused a type of Cox–Snell residuals together withplots using the TTT technique. The general un-derlying idea, which in principle can be used withall estimation methods considered in this paper, isthat the process of integrated conditional intensities,∫ T10 γ(t)dt,

∫ T20 γ(t)dt, . . . , is HPP(1) [12]. In turn

this gives rise to computable residual processes whenestimates are inserted for parameters and distribu-tions. The use of these processes in model checkingis demonstrated for three different data sets in [48].Typically, one would check (i) the distribution ofthe residuals with respect to departures from theunit exponential distribution, (ii) the possible pres-ence of time trends in residuals within each systemand (iii) the possible presence of autocorrelation intimes between events in the residual processes.

ACKNOWLEDGMENTS

The author is grateful to Dr. Sallie Keller-McNultyand Dr. Alyson Wilson for the invitation to con-tribute to this special issue of Statistical Science.The paper was prepared while the author was work-ing in an international research group studying eventhistory analysis at the Centre for Advanced Studyat the Norwegian Academy of Science and Lettersin Oslo during the academic year 2005/2006. Theauthor is thankful for the pleasant hospitality ex-tended to him. Discussions with other members ofthe group, in particular Odd Aalen, Per Kragh An-dersen and Ørnulf Borgan, have been very helpfuland are greatly appreciated. Thanks are also duethe author’s former PhD students Georg Elvebakk,Jan Terje Kvaløy and Helge Langseth for long timecollaborations and important contributions to thetheory presented here. The author finally thankstwo anonymous referees for careful reading of themanuscript and for their important and constructivecomments, which led to a much improved paper.

20 B. H. LINDQVIST

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