This article was downloaded by: [Nova Southeastern University]On: 07 October 2014, At: 12:08Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

TechnometricsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/utch20

Unconditional Tests of Goodness of Fit for the Intensityof Time-Truncated Nonhomogeneous Poisson ProcessesM Bhattacharjeea, J.V Deshpandeb & U.V Naik-Nimbalkarb

a Rolf Nevanlinna Institute, University of Helsinki, Finlandb Department of Statistics, University of Pune, Pune 411 007, IndiaPublished online: 01 Jan 2012.

To cite this article: M Bhattacharjee, J.V Deshpande & U.V Naik-Nimbalkar (2004) Unconditional Tests of Goodnessof Fit for the Intensity of Time-Truncated Nonhomogeneous Poisson Processes, Technometrics, 46:3, 330-338, DOI:10.1198/004017004000000356

To link to this article: http://dx.doi.org/10.1198/004017004000000356

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Unconditional Tests of Goodness of Fit for theIntensity of Time-Truncated NonhomogeneousPoisson Processes

M. BHATTACHARJEE

Rolf Nevanlinna InstituteUniversity of Helsinki

Finland

J. V. DESHPANDE and U. V. NAIK-NIMBALKAR

Department of StatisticsUniversity of Pune

Pune 411 007, India(uvnaik@stats.unipune.ernet.in)

Procedures for testing trends in the intensity functions of nonhomogeneous Poisson processes are basedmostly on conditioning on the number of failures observed in (0, t] with fixed t. We study an unconditionaltest based on the time-truncated data that enables meaningful asymptotics as t → ∞. We show that theasymptotic test is conservative and that its power quickly comes close to the power of the uniformly mostpowerful unbiased test for the power-law alternatives. Moreover, for the goodness of fit of a specifiedintensity, the exact test has more power than the test based on the conditional approach. We illustrate theprocedure using a real dataset.

KEY WORDS: Asymptotic relative efficiency; Consistency; Intensity function; Minimal-repair; Non-homogeneous Poisson process; Trend testing.

1. INTRODUCTION

The use of nonhomogeneous Poisson processes to model theoccurrence of successive failures of a repairable system un-dergoing minimal repair is well known (see, e.g., Ascher andFeingold 1984). Let N(u) be the number of failures (as wellas the instantaneous repairs) undergone by the repairable sys-tem under consideration up to time u. We assume that N(u) is anonhomogeneous Poisson process (NHPP) with cumulative in-tensity function �(u) (also known as the mean value function)giving the expected number of failures up to any time u. It isalso known that an NHPP is characterized through �(u) or thecorresponding intensity function λ(u), which is the time deriv-ative of �(u). Many such NHPP models are known to be in useas models for repairable systems.

For nonrepairable systems the proportional hazards model isimmensely popular due to its usefulness and versatility. Theconstant of proportionality can be apportioned into effects ofvarious covariates. But for many biological and biochemicalexperiments, it has been observed that the ratio of the hazardsshows increasing or decreasing trends rather than being con-stant. (For a discussion; see Deshpande and Sengupta 1995;Sengupta and Deshpande 1994.) For repairable systems whenan NHPP is assumed to be the model, the intensity functionsolely describes the probabilistic features of the process. Hence,comparing such a repairable system with a standard one re-quires comparing their corresponding intensity functions. If theintensity functions of the two processes are proportional, thenit would mean that the observed system is aging (either posi-tively or negatively) in the same manner as the standard one.In contrast, if the ratio of the two intensity functions shows anonconstant monotonic trend, say increasing, then this meansthat failures (and consequently repairs) are becoming more fre-quent for the process under consideration as compared with thestandard system. Hence in the sense of frequency of failures,the new system is deteriorating relative to the standard system.Therefore, to distinguish between the foregoing two situations,

it is of interest to test the following hypothesis. Let λ0(u) be agiven intensity function and let �0(u) be the corresponding cu-mulative intensity function. Then we want to test the hypotheses

H0 : λ(u) ∝ λ0(u), 0 ≤ u ≤ t

(1)versusH1 : λ(u)/λ0(u) increases on (0, t].

Although we want to test these hypotheses for all t, in the non-parametric setup it is not possible to extrapolate the inferencebeyond the interval of observation.

Note that the time transformation u → �0(u) = ∫ u0 λ0(x)dx

transforms the foregoing problem to the problem of testingwhether the transformed process has constant intensity (i.e.,it is an HPP) or whether the intensity is a nonconstant increas-ing function of time (i.e., it is an NHPP). If T1,T2, . . . ,Tnare the first n failure times from the process, then underH0, �0(T1), . . . ,�0(Tn) are the first n failure times from anHPP with unknown constant intensity.

Testing constant versus monotonic trend for NHPPs has al-ready been considered in literature (see Ascher and Feingold1984). Reviews have been given by Bain, Engelhardt, andWright (1985) and Cohen and Sackrowitz (1993). These testsare described as conditional tests (conditional on the number offailures observed up to a fixed time point). Bain et al. (1985)considered six different tests for testing the constant versus in-creasing intensity problem. Cohen and Sackrowitz (1993) ob-served that out of these six tests, three tests perform very poorlycompared with the others. Among the better three, the test basedon maximum likelihood is computationally very complicated.The remaining two statistics, called L and Z, are defined next.Let t be a fixed time point, and let there be n failures by time tstarting from time 0 [i.e., N(t) = n]. Let T1,T2, . . . ,Tn be thefailure times of the process; then 0 < T1 < T2 < · · · < Tn < t.

© 2004 American Statistical Association andthe American Society for Quality

TECHNOMETRICS, AUGUST 2004, VOL. 46, NO. 3DOI 10.1198/004017004000000356

330

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

12:

08 0

7 O

ctob

er 2

014

TESTING INTENSITY IN NONHOMOGENEOUS POISSON PROCESSES 331

• Laplace test statistic:

L =n∑

i=1

Ti

t.

The test based on this statistic was studied by Cox (1955) andmany others. Under the null hypothesis of constant intensityand conditional on N(t) = n, L is distributed as the sum of nindependent uniform(0,1) random variables.

• Power law test statistic:

Z = 2n∑

i=1

lnt

Ti.

The test based on this statistic was studied by Crow (1974)and others. Under the null hypothesis of constant intensity andconditional on N(t) = n,Z has a chi-squared distribution with2n degrees of freedom.

The test based on L is known to be the uniformly most power-ful unbiased (UMPU) test for the log-linear alternative with cu-mulative intensity �(u) = exp(α + βu),u > 0, β > 0. Whereasthe test based on Z is UMPU for the alternative of the power lawprocess with cumulative intensity function �(u) = λuβ , u > 0,β > 1, λ > 0.

The tests considered by Bain et al. (1985) use the conditionaldistribution of the statistics under H0 for obtaining the criticalpoints. In the setup where the period of observation is fixed tobe [0, t] and the conditional distribution demands conditioningon N(t) = n, the concept of asymptotics is not very clear. Theasymptotic theory is required for approximating the exact crit-ical points. To have a reasonable approximation, one needs alarge number of failures, which is possible if one can observea large number of identical systems or observe a given systemfor a long period. A long observation period is needed becausefor a Poisson process with continuous intensity, the event thata large number of failures occur in an interval of finite lengthcan have only a very small probability. Thus we consider theasymptotic distribution as t → ∞, so that we can design the ex-periment with an appropriate time frame t giving a large num-ber of failures with a reasonably large probability and have theasymptotic distribution as a good approximation.

Peña (1998) and Augustin and Peña (1999) studied a gener-alized version of L as a goodness-of-fit test. Their statistic is

L∗(t) = √12

(√N(t)�0(t)

)−1

[ N(t)∑

i=1

�0(ti) − �0(t)N(t)

2

]

,

where t1, t2, . . . , tN(t) are the failure times of the process ob-served on [0, t].

In this article we consider the statistic

Z∗(t) = −(√N(t)

)−1[∫ t

0

(

ln�0(u)

�0(t)+ 1

)

dN(u)

]

= −(√N(t)

)−1

[ N(t)∑

i=1

ln�0(ti)

�0(t)+ N(t)

]

.

We note that Z∗(t) and L∗(t) are suitably normed versions ofZ(t) and L(t).

By specifying different λ0(t), we can have a goodness-of-fittest for the corresponding proportional intensity family com-prising all of those intensities that are proportional to the spec-ified λ0(t). The test is carried out in the nonparametric familyλ(t) = λ0(t)c(t), where λ0(t) is known and c(t) is assumed to beconstant under the null and increasing under the alternative hy-potheses. We reject the null hypothesis for negative values of Z∗below a critical point. Note that by making suitable choices ofthe parametric functions, one can have a wide range of shapesfor λ(t). For example, one can let λ0(t) = β0tβ0−1, β0 > 0, andc(t) = exp(α0 + α1t), α1 > 0, giving rise to a large class of in-tensity functions.

In Section 2, we study the asymptotic properties (as timet → ∞) of Z∗(t). We obtain the asymptotic distribution andshow that the test is consistent against several large classes ofalternatives. We also extend the concept of Pitman asymptoticrelative efficiency (ARE) to the present context and calculate itfor comparison of L∗ and Z∗.

Following this, in Section 3, we consider the finite time dis-tribution of the statistic Z∗(t) under H0. In the process of thisstudy, we prove two interesting results, (1) the stochastic order-ing property of the truncated Poisson variables and (2) the left-tail ordering property of the standardized gamma/chi-squaredvariables. These help us prove the conservative nature of theproposed test if the critical points are based on the asymptoticdistribution. This implies that for alternatives near the null hy-pothesis, the power of the asymptotic Z∗ test is less than thenominal level of significance, thus leading to a lack of unbi-asedness of the test. However, a Monte Carlo study, reported inSection 4, shows that its power against certain power law andlog-linear alternatives quickly crosses the nominal level as t in-creases. For the power law alternatives, the power comes closeto the unconditional power of the UMPU Z test in a fairly shortperiod. In the same section, we consider the power propertiesof the test based on Z∗(t) using the exact critical point for test-ing the goodness of fit of a specified intensity function, that is,when we have the simple null hypothesis

H0 :λ(u) = λ0(u), 0 ≤ u ≤ t, (2)

where λ0(t) is a given known intensity of interest. It is seen thatagainst a power law alternative, the Z∗(t) test based on the exactcritical point has more power than the Z(t) test. We note that inthe simple H0 versus simple H1 problem, we no longer have theexponential family setup. Section 5 contains an example andSection 6 concludes with some remarks.

2. ASYMPTOTIC PROPERTIES

To obtain the asymptotic null distribution of the proposedstatistics, we need to represent them in terms of the innovationmartingales. Let

M(u) = N(u) − �(u).

Then M(u) is the innovation martingale for the countingprocess N(u) (for further details, see Karr 1991, chap. 2).

Theorem 1. Under H0, Z∗(t) has asymptotically (as t → ∞),standard normal distribution.

TECHNOMETRICS, AUGUST 2004, VOL. 46, NO. 3

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

12:

08 0

7 O

ctob

er 2

014

332 BHATTACHARJEE, DESHPANDE, AND NAIK-NIMBALKAR

Proof. We first obtain the limiting distribution of

Z∗∗(t) = −(√�(t)

)−1[∫ t

0

(

ln�0(u)

�0(t)+ 1

)

dN(u)

]

= −(√�(t)

)−1

×[∫ t

0

(

ln�0(u)

�0(t)+ 1

)(dM(u) + d�(u)

)]

= −√�(t)

×[

(�(t))−1∫ t

0

(

ln�0(u)

�0(t)+ 1

)

dM(u)

+ (�(t))−1

×∫ t

0

(

ln�0(u)

�0(t)+ 1

)

d�(u)

]

. (3)

The asymptotic normality of Z∗(t) follows using

N(t)

�(t)

p−→1, as t → ∞.

Details are given in the Appendix.Note that these results are valid for any λ0(t) and do not de-

pend on the unknown constant of proportionality.

2.1 Consistency of the Test

To prove consistency of the test, we need to show that un-der the alternative, the proposed test statistic diverges to −∞.Under the alternative, let λ(u) = c(u)λ0(u),0 ≤ u ≤ t, for someunknown nonconstant nondecreasing positive function c(u). Wenote that it is sufficient to show that Z∗∗(t) diverges to −∞.

The first integral in (3) has mean 0 for all t, and its varianceconverges to 0. Thus, because

√�(t) → ∞, a sufficient condi-

tion for the divergence of Z∗∗(t) to −∞ is

limt→∞(�(t))−1

∫ t

0

(

ln�0(u)

�0(t)+ 1

)

d�(u) > 0

≡ limt→∞(�(t))−1

(∫ t

0ln �0(u)d�(u)

− �(t) ln�0(t) + �(t)

)

> 0

≡ limt→∞(�(t))−1

(∫ t

0�(u)d

(ln�0(u)

))

< 1

≡ limt→∞(�(t))−1

(∫ t

0

�(u)λ0(u)

�0(u)du

)

< 1

≡ limt→∞

�(t)λ0(t)

λ(t)�0(t)< 1

≡ limt→∞

�(t)

c(t)�0(t)< 1

≡ limt→∞

c′(t)�0(t)

c(t)λ0(t)> 0. (4)

It can be seen that a large class of alternatives satisfies theforegoing conditions. For example, if c(t) is an increasing func-

tion in t, then c(�−10 (t)) is also increasing, and if the latter

is convex then the limit condition is satisfied trivially. For ex-ample, the limit condition holds for �0(t) = t, c(t) = βtβ−1,β > 1. The foregoing results can be summarized as follows.

Theorem 2. The tests based on Z∗(t) are consistent for thealternatives satisfying condition (4).

Next we show that the test is consistent against the mixedmodel of PLP–LLP (see Lee 1980) as well. Let

c(t) = λβtβ−1 exp(α1t), β > 1, α1 > 0;λ0(t) = 1 [i.e., �0(t) = t];

and

�(t) =∫ t

0c(u)du.

Then

c′(t)�0(t)

c(t)λ0(t)= (β − 1) + α1t.

Therefore,

limt→∞

c′(t)�0(t)

c(t)λ0(t)> 0,

and hence the proof.Here we remark that the asymptotic variance of the statistic

is finite and that the foregoing conditions essentially say thatthe asymptotic mean diverges to minus infinity, leading to con-sistency. Consistency of L∗ can be established under (4) in asimilar manner.

2.2 Asymptotic Relative Efficiency

The concept of Pitman (ARE) is well defined for the iid sam-ple case. In our situation we have non-iid observations and alsohave time-truncated data yielding a random number of obser-vations. We therefore adapt the general idea of Pitman ARE toour case in the following manner.

We define ARE as the “limiting ratio of the truncation times”required to achieve the same limiting power against the samesequence of alternatives (converging to the null hypothesis)when the limiting significance levels of the two tests are equal.If conditions similar to the conditions of Noether’s theorem(Randles and Wolfe 1979, thm. 5.2.1) are satisfied, replacingthe sample sizes by truncation times, then in our setup the fore-going ARE also can be obtained. Let {θti} be a sequence withlimi→∞ θti = θ0, the null hypothesis value of θ.

Theorem 3. Let {Sti} and {Tt′i} be two sequences of tests withassociated sequences of numbers {µSti

(θ)}, {µTt′i(θ)}, {σ 2

Sti(θ)},

and {σ 2Tt′i

(θ)} satisfying assumptions (a)–(f):

(a)Sti − µSti

(θti)

σSti(θti)

andTt′i − µTt′i

(θti)

σTt′i(θti)

have the same interval of support when θti is the true value of θ ,and they have the same continuous limiting distribution withcdf H(·).

TECHNOMETRICS, AUGUST 2004, VOL. 46, NO. 3

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

12:

08 0

7 O

ctob

er 2

014

TESTING INTENSITY IN NONHOMOGENEOUS POISSON PROCESSES 333

(b) Same as (a) but with θti replaced by θ0 (the null hypoth-esis value of θ ) throughout.

(c) limi→∞

σSti(θti)

σSti(θ0)

= limi→∞

σTt′i(θti)

σTt′i(θ0)

= 1.

(d)d

dθ

[µSti

(θ)] = µ′

Sti(θ)

and

d

dθ

[µTt′i

(θ)] = µ′

Tt′i(θ)

are assumed to exist and to be continuous in some closed inter-val about θ = θ0 with µ′

Sti(θ0) and µ′

Tt′i(θ0) both being nonzero.

(e) limi→∞

µ′Sti

(θti)

µ′Sti

(θ0)= lim

i→∞

µ′Tt′i

(θti)

µ′Tt′i

(θ0)= 1.

(f ) limi→∞

µ′Sti

(θ0)√

ti σ 2Sti

(θ0)= KS

and

limi→∞

µ′Tt′i

(θ0)

√t′i σ 2

Tt′i(θ0)

= KT ,

where KS and KT are positive constants.

Then ARE(S,T) = K2S/K2

T .

Proof. The proof and motivation of this theorem are almostsame as those of Noether’s theorem (Randles and Wolfe 1979,thm. 5.2.7), and hence the proof is omitted.

We illustrate this concept by computing the ARE of L∗(t) andZ∗(t) against the power-law alternative. Let

L∗∗(t) = √12

(√�(t)�0(t)

)−1[∫ t

0

(

�0(u) − �0(t)

2

)

dN(u)

]

.

We note that asymptotically, L∗(t) and L∗∗(t) have the samedistributions, and hence the same asymptotic properties. Thisalso holds for Z∗(t) and Z∗∗(t).

Let �0(u) = u [i.e., λ0(u) = 1] and λ(u) = βuβ−1. Thenc(u) = λ(u). Note that if β > 1, then both λ(u) and c(u) areconvex. Therefore, from the discussion of the previous section,L∗(t) and Z∗(t) are both consistent against the power law alter-native. We have �(u) = uβ. Now

L∗∗(t) = √12

(t√

�(t))−1

[∫ t

0

(

u − t

2

)

dN(u)

]

;

E(L∗∗(t)

) = √3tβ/2 β − 1

β + 1= µL(t), say;

d

dβ

[µL(t)

] = √3tβ/2

[(β − 1) ln(t)

2(β + 1)+ 2

(β + 1)2

]

;

var(L∗∗(t)

) = 3β2 − β + 2

β2 + 3β + 2= σ 2

L(t), say;

Z∗∗(t) = −(√�(t)

)−1[∫ t

0

{

lnu

t+ 1

}

dN(u)

]

;

E(Z∗∗(t)

) = tβ/2 1 − β

β= µZ(t), say;

d

dβ

[µZ(t)

] = −tβ/2[(β − 1) ln(t)

2β+ 1

β2

]

;

and

var(Z∗∗(t)

) = β2 − 2β + 2

β2 = σ 2Z(t), say.

Conditions (a)–(d) of Theorem 3 can be easily verified. [Theproof of asymptotic normality of (L∗∗(t) − µL(t))/σL(t) and(Z∗∗(t) − µZ(t))/σZ(t) is similar to that of Theorem 1.]

Let the sequence {θ(t)} be given by {β(t)} such that β(t) → 1and (β(t) − 1) ln(t) → 0 as t → ∞. For example, one can solvethe following for each t to get the corresponding value of β(t):where

t =[

1

β(t) − 1

]2/β(t)

.

The foregoing yields two values of β(t) for each t. Oneof these values approaches 1 from below and the other ap-proaches 1 from above as t → ∞. An alternative of increasingintensity will imply that one must choose the sequence of β(t)converging to 1 from above.

The foregoing conditions on β(t) imply that the and condi-tions (e) and (f) of Theorem 3 are satisfied, and hence the AREis given by

K2L

K2Z

= 3

4,

which shows that, asymptotically, Z∗∗(t) is more efficientin this sense than L∗∗(t). A similar conclusion holds forZ∗(t) and L∗(t).

Recall that the conditional analog of Z∗(t) (i.e., Z) is UMPUagainst the power law alternative.

3. FINITE TIME DISTRIBUTION OF THE PROPOSEDTEST STATISTIC Z ∗(t) UNDER THE

NULL HYPOTHESIS

Using the fact that for an NHPP with cumulative in-tensity function �0(t), the conditional distribution of2∑N(t)

i=1 ln(�0(t)/�0(ti)) given N(t) = n is chi-squared with2n degrees of freedom, it is easy to obtain the exact (finite time)distribution of Z∗(t). Let c0 be some constant. Then under H0,

P[Z∗(t) ≤ c0] =∞∑

n=1

P[Z∗(t) ≤ c0|N(t) = n]P[N(t) = n,n > 0]

=∞∑

n=1

P[χ22n ≤ 2(

√nc0 + n)]

× (λ�0(t))n exp(−λ�0(t))

n!(1 − exp(−λ�0(t))), (5)

where λ is the constant of proportionality.From this relation, we observe that the finite time distribution

depends on both λ and t, but only through the product λ�0(t),which is also the mean function under H0. The practitioner may

TECHNOMETRICS, AUGUST 2004, VOL. 46, NO. 3

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

12:

08 0

7 O

ctob

er 2

014

334 BHATTACHARJEE, DESHPANDE, AND NAIK-NIMBALKAR

Table 1. Exact Critical Points of Z ∗(t) for Size Close to .05

λΛ0(t) Critical point Exact size

1 −1.009 .04991.5 −1.094 .05042 −1.15 .05002.5 −1.19 .05023 −1.224 .05003.5 −1.253 .04994 −1.278 .05004.5 −1.3 .05015 −1.3205 .0500

10 −1.4345 .050015 −1.48 .050020 −1.505 .050050 −1.558 .0502

well have some idea about the mean number of failures to be ob-served up to time t even if λ, the constant of proportionality, isunknown. The exact 5% critical point for that value of λ�0(t)can then be obtained from Table 1. If this is not known, thenonly the asymptotic critical point as described in the next sec-tion will be available. In particular, the table may be used fortesting the hypothesis described in (2), that is, when λ = 1 and�0(t) is known. For values of λ�0(t) other than those tabulated,the critical points may be obtained by linear (or otherwise) in-terpolation in the corresponding interval. Also from Table 1,note that the critical points monotonically decrease (a fact sug-gested by Thm. 4) to the corresponding normal critical point.

Using the same result, we have tabulated in Table 2 the actualsize of the Z∗(t) test statistic under the null hypothesis using thecritical points of its limiting distribution, that is, quantiles of thestandard normal distribution. It is apparent from Table 2 thatsuch an approximation is resulting in an actual level lower thanthe corresponding nominal level α; that is, we will always havea conservative test. Also, this actual size increases monotoni-cally in relation to the nominal size as �(t) becomes large. Thisfact is established by Theorem 4 for α ≤ .05. To prove this, wefirst state two important lemmas. The proofs of all of the resultsare given in the Appendix.

Lemma 1. Truncated Poisson random variables are stochas-tically ordered.

Lemma 2. Standardized χ22n are lower-tail ordered; that is,

∀ c0 < −1.5,P[χ22n ≤ 2(

√nc0 + n)], increases with n.

Table 2. Actual Size of the Z* Test at the Standard Normal CriticalPoints for Different Values of λΛ0(t)

Values ofλΛ0(t)

Significance Levels (critical points).1 (−1.2816) .05 (−1.6449) .025 (−1.9600) .01 (−2.3264)

1 .0106 .0003 3.8E−6 6.2E−92 .0243 .0015 6.0E−5 3.3E−74 .0492 .0071 .0006 1.0E−56 .0648 .0138 .0020 .00018 .0735 .0194 .0037 .0003

10 .0785 .0236 .0055 .000520 .0876 .0333 .0112 .002150 .0933 .0404 .0166 .0047

100 .0956 .0435 .0192 .0062200 .0970 .0456 .0209 .0073500 .0982 .0473 .0225 .0083

Theorem 4. P[Z∗(t) ≤ c0] is increasing in �(t) for givenconstant c0, c0 < −1.5.

The result in Lemma 2 may seem to have been indicated inJohnson and Kotz (1970, fig. 2, p. 170), but these authors didnot claim the monotonic nature of the convergence.

4. POWER COMPARISON

Because of the complex nature of the exact distributions ofthe statistics under the null and alternative hypotheses and thatof the power functions of the proposed tests, apart from a fewparticular cases, we used a Monte Carlo simulation technique toestimate the powers against some alternatives. Powers based onsimulations have been designated “estimated powers,” whereaspowers based on the exact distributions, namely for the statis-tics Z∗ and Z against the power law alternative, have been des-ignated “exact powers.” The powers presented correspond toa 5% size test.

We generated samples from the required process, using thefollowing method. Let �(t) be the mean value function of theprocess from which we need to generate a time-truncated sam-ple, truncated at t. We generate y1, y2, . . . , yn, a time truncatedsample from an HPP with intensity 1, time-truncated at �(t).Thus �−1( y1),�

−1( y2), . . . ,�−1( yn) provide a random sam-

ple from the process with mean value function �(t) time trun-cated at �−1(�(t)) = t.

We considered powers for L∗ and Z∗ for two sets of criticalvalues. One set of critical points is based on the standard nor-mal distribution. For L∗(t), the other critical point is based onits empirical distribution function under H0 and is referred to asan “estimated critical point.” For Z∗(t), the exact critical pointis calculated (as in Table 1) and is referred to as an “exact criti-cal point.” Those from the limiting distribution are identified as“asymptotic critical points.”

Table 3 contains the estimated powers against the log-linearalternative with λ(t) = exp(t). The powers are estimated by theproportion of rejections out of 500,000 values of the statisticsgenerated under the alternative.

From Table 3, we see that initially L∗(t) with the estimatedcritical points has less power than Z∗(t) but by truncationtime 3, it overtakes the power of Z∗(t), and even its power basedon the normal critical points overtakes those of Z∗(t). A remark-able fact that needs to be noted here is that the power of theseunconditional tests for the stated alternative approaches the lim-iting value of 1 very fast. We only need to observe the systemup to four time units to reach almost power 1; that is, the power

Table 3. Powers of the Statistics for H1: Log-Linear Intensity,exp(t) at a 5% Level of Significance

L∗ (estimated power) Z ∗ (estimated power)Truncation Estimated Asymptotic Exact Asymptotictime t critical point critical point critical point critical point

1 .1095 .0908 .1536 .00431.5 .2042 .1884 .2873 .04022 .4069 .3919 .5066 .20262.5 .7286 .7199 .7784 .55823 .9627 .9608 .9598 .89633.5 .9996 .9995 .9991 .99624 1 1 1 1

TECHNOMETRICS, AUGUST 2004, VOL. 46, NO. 3

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

12:

08 0

7 O

ctob

er 2

014

TESTING INTENSITY IN NONHOMOGENEOUS POISSON PROCESSES 335

of the test is very close to 1 if we are allowed a time that isfour times the expected time to one failure of the system. So interms of resources, it may be said to be equivalent to observingfour systems up to the first failure of each of them. That the pro-cedure’s asymptotic power is approached with small resourcesis a very desirable property. Moreover, as seen in Table 2, theactual size for the asymptotic Z∗ test for λ�0(t) = 4 is .0071.

We now consider the power of the tests based on Z andZ∗ against the power law alternatives. Because through atime transformation we can reduce the problem of testingH0 :�(u) = λ�0(u) to testing H0 :�(u) = λu, we consider thelatter H0. We consider the power in two contexts. In the firstinstance, the statistics are being used for testing

H0 :�(u) = λu, 0 < λ < ∞, 0 < u < t

against

H1 :�(u) = uβ, 0 < u < t, β > 0 (β known).

This is the problem of testing the goodness of fit of the Pois-son process model with constant (unknown) intensity. But H1 isa simple hypothesis. Because the exact distribution of Z∗ un-der H0 depends on the unknown λ, we consider the test us-ing the asymptotic .05 critical point, that is, from the standardnormal distribution. In reality, the level of this Z∗ is smallerthan .05. Tables 4 and 5 (column 3 in both) consider the powerof the asymptotic Z∗ test for two simple alternatives, β = 2 andβ = 4.

The conditional power of Z for each fixed n can be calculatedusing the fact that under the alternative of power law process,βZ, conditionally on N(t) = n, follows a chi-squared distribu-tion with 2n degrees of freedom. Then the unconditional powerof Z can be obtained by taking the linear combination of theseconditional powers with P[N(t) = n] as weights under the alter-native analogous to expression (5). The last column of Table 4and of Table 5 contain the exact unconditional powers of theUMPU test Z. It is seen that by t = 5 and t = 2, the power ofthe asymptotic Z∗ test is very close to that of the UMPU test.

Next we consider the power of these tests for testing

H′0 :�(u) = u, 0 < u < t

against the same alternative hypotheses H1. Here the null aswell as both of the alternatives are simple hypotheses. This isthe problem of testing the goodness of fit of a Poisson process

Table 4. Powers of the Statistics for H1: Power Law Intensity,λ = 1, β = 2 at a 5% Level of Significance

Z ∗ (exact power)Truncation Exact Asymptotic Z (exact andtime t critical point critical point unconditional)

1 .1542 .0026 .13432 .4093 .0922 .29593 .7016 .4080 .57624 .8931 .7430 .82355 .9742 .9276 .9518

10 1 1 1

Table 5. Powers of the Statistics H1: for Power Law Intensity,λ = 1, β = 4 at a 5% Level of Significance

Z ∗ (exact power)Truncation Exact Asymptotic Z (exact andtime t critical point critical point unconditional)

1 .3081 .0137 .31491.5 .8867 .5190 .80982 .9998 .9967 .99922.5 1 1 1

model with unit (λ = 1) intensity. Now the exact distribution ofZ and of Z∗ are known under H0 as well as under H1. The exactcritical points of Z∗ near α = .05 are given in Table 1. Usingthese critical points, we obtain the exact power for these twoalternatives as given in column 2 of Tables 4 and 5, for selectedvalues of t. We notice that the power of the Z∗ test quicklyexceeds that of Z (the UMPU test in the problem of testing thecomposite H0) as t increases. In the simple H0 versus simpleH1 problem, we no longer have the exponential family setup,and the tests based on Z may not be UMPU.

The graphs in Figure 1 show the power of the foregoing threetests as a function of the shape parameter β for four differenttruncation times, t = 5, t = 10, t = 20, and t = 50, and the scaleparameter λ = 1. We see that the power of the unconditionaltest based on Z∗ with the exact critical point is higher than thatof the others throughout. The graphs also show that in actualpractice, one will not need a very large truncation time for thelimiting distribution to become a good approximation.

5. EXAMPLE

To illustrate the use of the test in an instructive manner, weapply it to the first 4,000 hours of the data given by Majumdar(1993) on the failure times of a vertical boring machine. Therequired observations are as follows: 376, 808, 1,596, 1,700,1,701, 1,781, 1,976, 2,076, 2,136, 2,172, 2,296, 2,380, 2,655,2,672, 2,806, 2,816, 2,848, 2,937, 3,158, 3,575, 3,632, 3,686,3,695, 3,705, 3,802, 3,811, and 4,020. Let the null the hypoth-esis be H0 :�0(u) = u/1,000. With truncation time t = 2,000,N(2,000) = 7, the value of the statistics are Z = 6.5116 andZ∗ = −1.4152. The critical points at the .05 level using the(conditional) chi-squared distribution with 14 degrees of free-dom and Table 1 are 6.5706 and −1.15. For truncation timet = 4,000, N(t) = 26, Z = 28.0509, Z∗ = −2.3484, and the cor-responding exact critical points are 36.4371 and −1.278. Thenull hypothesis is rejected in all of the cases. The asymptoticcritical point is −1.64 and the null hypothesis is rejected att = 4,000.

The plot of the N(t) process (Fig. 2) indicates a piecewiselinear cumulative intensity function. We choose such a functionfor testing as the null hypothesis. Let H0 :�0(u) = u/1,500, 0 <

u < 1,500 and �0(u) = 17u/2,000− 11.75, 1,500 < u < 3,000.

We obtain Z∗(3,000) = 0.2734. The null hypothesis is acceptedusing exact and asymptotic critical points. [The exact criticalpoint corresponds to �0(3,000) = 13.75.]

TECHNOMETRICS, AUGUST 2004, VOL. 46, NO. 3

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

12:

08 0

7 O

ctob

er 2

014

336 BHATTACHARJEE, DESHPANDE, AND NAIK-NIMBALKAR

(a) (b)

(c) (d)

Figure 1. Comparison of Power for Power Law Process Alternatives: (a) Truncation Time 5; (b) Truncation Time 10; (c) Truncation Time 20;Truncation Time 50. ( Z ∗ exact critical point; power of unconditional Z ; Z ∗ asymptotic critical point.) Scale parameter = 1.

6. CONCLUDING REMARKS

We note that the power depends on �(t), the value of themean function at the truncation time. It is reasonable to requireat least a moderately large expected number of failures for adecision to be taken with high power. If one has some knowl-edge of �(t) under the alternative hypothesis, then the trunca-tion time can be so designed.

The conditional test based on Z uses a critical point that de-pends only on the number of failures and not on the (truncation)time t0 in which these failures occur, whereas the test based onZ∗ with the critical point obtained from Table 1 depends on t0.From the graphs (Fig. 1), it is seen that with a simple null hy-pothesis, the latter test has more power than the unconditionalpower of the former test against the power law alternatives.

The asymptotic approximation is adequate. Figure 1 showsthat if the truncation time is 20 or more, then we may use thestandard normal critical point. With truncation time 20, for al-

Figure 2. Failure Process of the Boring Machine.

ternatives near the null hypothesis, the asymptotic test has about5% less power but about 33% less probability of type 1 error.

We have demonstrated that the unconditional tests are quitecomparable to conditional tests and may even be preferred incertain cases.

ACKNOWLEDGMENTS

This work was supported by the NBHM of India and theCSIR of India. The authors thank the referees for useful com-ments that helped improve the article.

APPENDIX: PROOFS

Proof of Theorem 1

Z∗(t) = −(√N(t)

)−1[∫ t

0

(

ln�0(u)

�0(t)+ 1

)

dN(u)

]

.

First, we show that

Z∗∗(t) = −(√�(t)

)−1[∫ t

0

(

ln(�0(u)

�0(t)+ 1

)

dN(u)

]

has asymptotically a standard normal distribution.Under H0, E(Z∗∗(t)) = 0 ∀ t. We note that because λ(t) =

cλ0(t),

limt→∞(�(t))−1

[∫ t

0

(

ln�0(u)

�0(t)+ 1

)2

d�(u)

]

= limt→∞(�0(t))

−1[∫ t

0

(

ln�0(u)

�0(t)+ 1

)2

d�0(u)

]

= 1, (A.1)

TECHNOMETRICS, AUGUST 2004, VOL. 46, NO. 3

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

12:

08 0

7 O

ctob

er 2

014

TESTING INTENSITY IN NONHOMOGENEOUS POISSON PROCESSES 337

using limu→0 u ln(u) = 0.

Now, after suitably changing the norming quantity from t1/2

to �1/20 (t), we use theorem B.21 of Karr (1991). It can be eas-

ily seen that a similar proof holds with this norming as well. Inour case the necessary limit condition is shown to satisfy (A.1).Therefore, Z∗∗(t) has asymptotically a standard normal distrib-ution.

Let Dt be the distance between Z∗(t) and Z∗∗(t), defined by

Dt =∣∣∣∣−

(√N(t)

)−1[∫ t

0

(

ln�0(u)

�0(t)+ 1

)

dN(u)

]

+ (√�(t)

)−1[∫ t

0

(

ln�0(u)

�0(t)+ 1

)

dN(u)

]∣∣∣∣.

We show that Dtp−→0, as t → ∞. We have from Thompson

(1981, p. 5, thm. 4.1),

N(t)

�(t)

p−→1, as t → ∞.

Therefore,

Dt =∣∣∣∣

(√�(t)√N(t)

− 1

)(√

�(t))−1

×[∫ t

0

(

ln�0(u)

�0(t)+ 1

)

dN(u)

]∣∣∣∣

p−→0, as t → ∞.

Thus Z∗(t) also has asymptotically a standard normal distribu-tion, and hence the proof.

Proof of Lemma 1

Consider the survival function of a truncated Poisson vari-able (truncated at 0), say X, with parameter �(t) = θ . Usingthe result that

n∑

i=0

λie−λ

i! =(∫ ∞

λ

xne−x dx

)/n!

we have

Sθ (n) = P[X > n|X > 0]= P

[X > n,X > 0]P[X > 0]

=(∫ θ

0xne−x dx

)/(n!(1 − e−θ )

).

Thend

dθ(Sθ (n))

=(

θne−θ (1 − e−θ ) −(∫ θ

0xne−x dx

)

e−θ

)/(n!(1 − e−θ )2)

=(

e−θ

[

θn(1 − e−θ ) −∫ θ

0xne−x dx

])/(n!(1 − e−θ )2)

> 0, as θn(1 − e−θ ) >

∫ θ

0xne−x dx.

Therefore Sθ (n) ↑ with θ ; that is, Fθ (n) = (1−Sθ (n)) ↓ with θ .Hence the proof.

Proof of Lemma 2

We show that

1

n!∫ Cn+1

0e−yyn dy − 1

(n − 1)!∫ Cn

0e−yyn−1 dy ≥ 0,

where Cn = n − C0√

n.

Claim. The foregoing holds for C0 ≥ 1.5 and all n for whichCn is positive.

We note that

1

n!∫ Cn+1

0e−yyn dy = 1

n!(∫ Cn+1

Cn

e−yyn dy − e−yyn|Cn0

)

+ 1

(n − 1)!∫ Cn

0e−yyn−1 dy

= 1

n!(∫ Cn+1

Cn

e−yyn dy − e−CnCnn

)

+ 1

(n − 1)!∫ Cn

0e−yyn−1 dy.

Thus it is sufficient to show that∫ Cn+1

Cn

e−yyn dy − e−CnCnn ≥ 0. (A.2)

We use the inequality∫ b

af ( y)dy ≥ f (a)(b − a) + m(b − a)2/2,

where m ≤ f ′( y) for y ∈ [a,b], (A.3)

given by Mitrinovic (1970, p. 298). Take f ( y) = e−yyn, a = Cn,and b = Cn+1. Then m can be taken as f ′(Cn). Because,

f ′( y) = nyn−1e−y − yne−y

= yn−1e−y(n − y)

and

f ′′( y) = yn−2e−y(n(n − 1) − 2ny + y2)

= yn−2e−y((n − y)2 − n)

= yn−2e−y(n − y + √n)(n − y − √

n)

≥ 0 iff (n − y + √n)(n − y − √

n) ≥ 0.

Thus f ′′( y) ≥ 0 for y ≤ n − √n.

For y ∈ [Cn,Cn+1], y ≤ Cn+1 = n + 1 − C0√

n + 1. But forC0 ≥ 1.5 and n ≥ 3, we have

1 ≤√

n + 1

2+ (

√n + 1 − √

n),

giving

n + 1 − C0√

n + 1 ≤ n − √n.

Therefore, f ′′( y) ≥ 0 on [Cn,Cn+1],�⇒ f ′( y) ↑ in y on [Cn,Cn+1]�⇒ f ′(Cn) ≤ f ′( y) ∀ y ∈ [Cn,Cn+1].

Thus m = f ′(Cn) = e−CnCn−1n (n − Cn).

TECHNOMETRICS, AUGUST 2004, VOL. 46, NO. 3

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

12:

08 0

7 O

ctob

er 2

014

338 BHATTACHARJEE, DESHPANDE, AND NAIK-NIMBALKAR

From inequality (A.3),∫ Cn+1

Cn

e−yyn dy ≥ e−CnCnn(Cn+1 − Cn)

+ e−CnCn−1n

(n − Cn)(Cn+1 − Cn)2

2.

Subtracting e−CnCnn from both sides, we get

∫ Cn+1

Cn

e−yyn dy − e−CnCnn

≥ e−Cn Cn−1n

{

Cn(Cn+1 − Cn)

+ (n − Cn)(Cn+1 − Cn)2

2− Cn

}

.

Thus (A.2) holds if and only if the expression in the bracket≥ 0. We need to substitute Cn = n − C0

√n and Cn+1 = n + 1 −

C0√

n + 1 in the bracketed term.Now

Cn+1 − Cn = 1 − C0(√

n + 1 − √n)

= 1 − C0/(√

n + 1 + √n)

= 1 − δn, say.

We require that{

(n − C0√

n)(1 − δn) + C0√

n(1 − δn)2

2− (n − C0

√n)

}

≥ 0.

Therefore,

−(n − C0√

n)δn + C0√

n(1 − δn)2

2≥ 0

⇐⇒ C0√

nδn + C0√

n(1 − δn)2

2≥ nδn

⇐⇒ C0√

nδn + C0√

n

2− C0

√nδn + C0

√nδ2

n

2≥ nδn

⇐⇒ C0√

n

2+ C0

√nδ2

n

2≥ nδn,

which holds if

C0√

n

2≥ nC0√

n + 1 + √n

⇐⇒√

n

2≥ n√

n + 1 + √n

holds ∀n.

Hence the proof.

Proof of Theorem 4

First, we know that if F(x) ≤ G(x) ∀ x, and U(x) ↑ with x,then

∫U(x)dF(x) >

∫U(x)dG(x).

Then, noting that

P[Z∗(t) < c0] =∞∑

n=1

P[χ22n < 2(

√nc0 +n)] �(t)n exp(−�(t))

n!(1 − exp(−�(t)))

we have the desired result using Lemma 1, Lemma 2, and re-sult (A.4); hence the proof.

[Received January 1997. Revised December 2003.]

REFERENCES

Agustin, Z., and Peña, E. (1999), “Order Statistic Properties, Random Genera-tion, and Goodness-of-Fit Testing for a Minimal Repair Model,” Journal ofthe American Statistical Association, 94, 266–272.

Ascher, H., and Feingold, H. (1984), Repairable Systems Reliability, New York:Marcel Dekker.

Bain, L. J., Engelhardt, M., and Wright, F. T. (1985), “Test for an IncreasingTrend in the Intensity of a Poisson Process: A Power Study,” Journal of theAmerican Statistical Association, 80, 419–422.

Cohen, A., and Sackrowitz, H. B. (1993), “Evaluating Test for Increasing In-tensity of a Poisson Process,” Technometrics, 35, 446–448.

Cox, D. R. (1955), “Some Statistical Methods Connected With Series ofEvents,” Journal of the Royal Statistical Society, Ser. B, 17, 129–164.

Crow, L. H. (1974), “Reliability Analysis for Complex Repairable Systems,” inReliability and Biometry, eds. F. Proschan and R. J. Serfling, Philadelphia:Society for Industrial and Applied Mathematics, pp. 379–410.

Deshpande, J. V., and Sengupta, D. (1995), “Testing the Hypothesis of Propor-tional Hazards in Two Populations,” Biometrika, 82, 251–261.

Johnson, N. L., and Kotz, S. (1970), Continuous Univariate Distributions,Vol. I, New York: Wiley.

Karr, A. (1991), Point Processes and Their Statistical Inference, New York:Marcel Dekker.

Lee, L. (1980), “Testing Adequacy of the Weibull and Log-Linear Rate Modelsfor a Poisson Process,” Technometrics, 22, 195–199.

Majumdar, S. K. (1993), “An Optimum Maintainance Strategy for a VerticalBoring Machine System,” Opsearch, 30, 4.

Mitrinovic, D. C. (1970), Analytic Inequalities, New York: Springer-Verlag.Peña, E. (1998), “Smooth Goodness-of-Fit Tests for the Baseline Hazard in

Cox’s Proportional Hazards Model,” Journal of the American Statistical As-sociation, 93, 673–692.

Randles, R. H., and Wolfe, D. A. (1979), Introduction to the Theory of Non-parametric Statistics, New York: Wiley.

Sengupta, D., and Deshpande, J. V. (1994), “Some Results on the RelativeAging of Two Life Distributions,” Journal of Applied Pobabability, 31,991–1003.

Thompson, W. A., Jr. (1981), “On the Foundations of Reliability,” Technomet-rics, 23, 1–13.

TECHNOMETRICS, AUGUST 2004, VOL. 46, NO. 3

Dow

nloa

ded

by [

Nov

a So

uthe

aste

rn U

nive

rsity

] at

12:

08 0

7 O

ctob

er 2

014