tests for positive jumps in the intensity of a poisson process: a power study

356 IEEE TRANSACTIONS ON RELIABILITY, VOL. 39, NO. 3, 1990 AUGUST

Tests for Positive Jumps in the Intensity of a Poisson Process: A Power Study

Max Engelhardt

James M. Guffey

F. T. Wright

University of Missouri, Rolla

Northeast Missouri State University, Kirksville

University of Missouri, Columbia

Key Words - Order restricted likelihood ratio test, Nonhomogeneous Poisson process, Step-intensity, Test of homogeneity

Reader Aids - Purpose: Evaluate existing statistical procedures Special math needed for explanations: Statistical inference Special math needed to use results: Same Results useful to: Reliability analysts and practicing statisticians

Summary & Conclusions - Tests are considered for the hypothesis of a constant intensity against the alternative of an intensity which increases with time in a nonhomogeneous Poisson p m cess. Attention is focused on stepfunction alternatives and tests designed for such alternatives. One application is testing for abrupt changes in equipment following scheduled overhauls. We recommend the order-restricted likelihood-do test over an ordered chi- square test for such situations - provided the points at which jumps can occur are known, otherwise we recommend the test based on the Laplace statistic. The performance of these tests is evaluated for smooth alternatives - with the result that the smallest relative power of the order-restricted likelihood-ratio test is 73% and for the Laplace test is 82%. Section 4 includes a numerical example based on failure times for a main-propulsion diesel engine. The result is that the order-restricted likelihood-ratio test corresponds to the lowest statistical significance level.

1. INTRODUCTION

Nonhomogeneous Poisson processes (NHPPs) frequently are used to model the number of occurrences of a particular event in time; for examples, see Ascher & Feingold [l]. Sometimes the increases in intensity of the phenomenon being modeled are quite pronounced at certain known times, but are negligible at other time. For example, Ascher & Feingold 11, p 751 provide a plot of cumulative failures vs cumulative operating hours of a marine diesel engine; the plot strongly sug- gests that a marked increase in the failure intensity has occur- red after certain scheduled overhauls. A test which is designed to differentiate between a constant intensity and intensities which are non-decreasing step-functions with steps at known times would also be useful in other examples.

The procedures considered here are easily modified to test for a constant intensity with alternatives that are non-increasing step functions. In the present context, an omnibus test such as the ordinary chi-square test would be appropriate for testing

the null hypothesis of a constant intensity vs the alternative hypothesis of all possible non-constant functions of intensity. However, if the alternative hypothesis only includes functions with a certain type of monotonicity (eg, increasing functions) it is often possible to adapt an omnibus test in order to improve its sensitivity for detecting this special type of alternative. Typically, tests which assume monotonicity are more powerful than the corresponding omnibus tests - provided the intensity is monotonic or nearly so.

Section 3 contains the results of a power study to compare several tests of a constant vs an increasing intensity for time truncated sampling. Bain, et al[2] describe a similar investiga- tion in which smooth alternatives were emphasized. In our study, tests which were developed especially for step function alternatives are included as well as several such smooth alternatives. In some of the cases in section 3, the times at which jumps occur in the intensity are incorrectly specified. We found that such an error in the assumptions substantially reduces the power of tests developed for step-function alternatives.

Notation & Acronyms

NHPP nonhomogeneous Poisson process Z" truncation time of experiment N number of occurrences in (0, Z"], a r.v. n observed value of N

ordered time of occurrence i m number of intensity-increases cj ordered time of intensity-increase j ; co =0, c,,, = Z" Cj cellj, ( C ~ - ~ , C ~ ]

p ( t ) intensity (sometimes called an intensity function) p j intensity over cel l j M( t ) mean number of occurrences in (0, Zl Oj observed number of occurrences in Cj, a r.v. x2 statistic for omnibus chi-square test lj length of cellj: cj-cj- l 1 Ej aj Oj/4: estimate of pi MLE maximum likelihood estimator p j MLE under order restriction PAVA pool-adjacent-violators algorithm LRT likelihood-ratio test W statistic for LRT V P(j ,m; l ) weights for approximate distribution of V UMPU uniformly most powerful unbiased P 6 Z Xi pj L statistic for Laplace test

( l,>l*,. . . 2 4 n ) mean number of occurrences in cell j

statistic for LRT under order restriction

shape parameter of power-law model scale parameter of power-law model statistic for UMPU test of /3 number of occurrences in cell j , a r.v. probability of occurrence in cell j

0018-9529/90/0800-356$1.0001990 IEEE

ENGELHARDT ET AL.: TESTS FOR POSITIVE JUMPS IN THE INTENSITY OF A POISSON PROCESS: A POWER STUDY 351

Other, standard notation is given in “Information for Readers & Authors” at the rear of each issue.

2. TESTS FOR AN INCREASING INTENSITY

Suppose a NHPP is observed over the interval (0, PI. The intensity of a NHPP is the derivative of the mean number of Occurrences in (O,t], p (t) = M‘ (t) . Occasionally there is con- fusion between the intensity and a hazard rate, which may have the same functional form. The latter is a relative rate of failure for a nonrepairable system, whereas the former is usually in- terpreted as an absolute rate of failure for a repairable system. Further discussion on this point is provided by Ascher & Feingold [ l , p 331.

We consider intensities of the form:

p(t) = pi. for cj-1 5 t < cj, and j = 1,2 ,..., m; (2-1)

and study tests of the hypothesis Ho: p1 = p2 = ... = pm vs Hl: p1 I p2 I ... I pm with p1 < pm. Each of the tests is conditional on N = n > 0. We compare their powers for n = 10, 20, 40. Because our conclusions do not depend greatly on n, similar conclusions will hold for the tests viewed as uncondi- tional tests.

Given n occurrences of events modeled by an NHPP, it is well-known that the occurrence times are distributed as the order statistics of a random sample from the distribution with pdf{x} = p ( x ) / M ( F ) for0 < x I F , andpdf{x} = Ofor x > P.

One could use a chi-square goodness-of-fit test to determine if the came from a uniform distribution on the interval (0, F ), but this test does not use the monotonicity assumption. We consider a related 1-sided test discussed in Magel & Wright [8] which can be used whether the cells are of equal size or not. (This test generalizes those proposed by Chacko [5] and Robertson [ 101 which would require equal cell sizes in this setting.) We use the cells cj = ( cj- l,cj] for j = 1,2,. . . ,m.

The omnibus chi-square test rejects Ho if - m

x 2 = ( 0 , - E j ) 2 / q L xT-, ( m - 1 ) (2-2) j = 1

Notation

Ej n$/F x: -, ( m - 1 ) ( 1 -a) quantile of the chi-square distribution

with m - 1 degrees of freedom

Under H1, one would anticipate the intensity estimates j j = Oj/$ to be nondecreasing, but this might not occur due to chance variation. Magel & Wright [9] obtained the MLEs under the restriction that either Ho or Hl is true.

These ordered estimates can readily be computed using the PAVA applied to {Oj/b}y=l with weights {b}y!l [3, p 131. The LRT of HO versus H I , conditional on n occurrences, rejects H, for large values of -

m

V = 2 Oj log($ pj/Ej). j = 1

(2-3)

We also considered an ordered chi-square test as studied by Lee [7]. However, for the particular cells, with 1 I j I m, used in this study, Vmaintained the correct s-significance level better than the ordered chi-square test in those cases with mean cell counts at least five. Also, for the majority of the intensities, Vwas more powerful than the ordered chi-square test. Thus, attention is restricted to the test based on V.

Magel & Wright [8] show that under Ho, for v > 0,

rn

lim {Pr{V L v}} = P(j,m;Z)Pr{x~-l 1 v } . (2-4)

Barlow, et a1 - define the P ( j , m ; l ) [3, p 1261 give the values of the P(j,m;Z) for the equal-weights case

tabulate the associated critical values in [3, table A.31 for arbitrary 1 and m I 5, give formulas for P(j,m;Z) [3,

Robertson & Wright [ 111 found that the tail probability (4) is robust to moderate changes in the $ and that if -

n-m j = 2

( 1 1 = 1 2 = ... Im) in [3, table AS]

pp 137, 140-1421.

max{$}/min{$} I 1.5, (2-5)

then the equal-weights value provides a reasonable approximation.

If one wishes to test Ho vs Hi: p12p22.. . 2 p m with p1 > p m , then the MLE of p , under the restriction that either Ho or H i is true, which is denoted by p i for 1 I j r m , can be computed just by applying the PAVA to { -Oj/ l j}r! with weights {$}y=l, and then negating the solutions. The test statistic I/ is computed as in (3) with p j replaced by p i and the critical values for V are the same for these hypotheses as for Ho vs HI.

Tests for a constant intensity vs an increasing alternative have been developed for other families of intensities. We have included some of them in this study as well as some “smooth” alternatives to indicate the relative powers of all of these tests over a broad range of alternatives.

For intensities of the form p(t) = CY exp(Pt), Cox [6] studied a test attributed to Laplace which rejects Ho if -

n

L T / P 2 n/2 + ~ ~ - , ( n / 1 2 ) ” ~ . i = l

Notation

zI-, (1 -a) quantile of gauf(z).

Bain, et al[2] found that for the tests and alternatives they considered, the L test is preferred.

The UMPU test of /3 = 1 vs /3 > 1 for the family of intensities,


is discussed in Mil-Hdbk-189 [9]. It rejects Ho if - n

2 = 2 log(T*/Ti) I x', (2n). i = l

If attention is restricted to smooth alternatives, then Bain, et al[2] recommended the 2 test. They give a detailed bibliography concerning inferences for power-law Poisson processes.

Considering an NHPP with an arbitrary intensity that is not restricted to be a step function, Boswell [4] developed the LRT, based on a statistic, W, conditional on N = n > 0, of a constant vs an increasing intensity. In our study, critical values for the sample sizes considered were estimated by Monte Carlo techniques with io4 iterations.

Before considering some specific step alternatives, we note some facts established in [23.

If p,(1) = r] p ( t ) with p ( t ) fixed and r ] a positive parameter, then N is a s-sufficient statistic for 1, and thus multiplying the intensity by a positive constant does not change the powers of these conditional tests. The conditional power for an intensity p ( t ) observed over the interval [0, T*] is the same as for the intensity Y (t) = P p (2" t) observed over the interval [0, 13. Thus, we set T* = 1 and vary the scale parameter. If ordered standard uniform r.v.'s, V u ) , j = 1 , ..., n are generated, then for Monte Carlo simulation the II: can be obtained from

critical values of W, the test statistic for the LRT, were estimated using Monte Carlo techniques with lo4 iterations.

We studied the s-significance levels of Y for a nominal level of a = 0.05, n = 10,20,40 and m = 2,3,4 with a variety of cell lengths including equal lengths. Monte Carlo techniques with lo4 iterations were employed. However, for small m, one can easily compute the exact power of these tests, conditional on N = n > 0. The conditional distribution of (Xl,X2,. . . ,Xm) is multinomial with n trials and probability vector (p1,p2, ..., p,) wherepi E [ M ( c j ) - M ( C ~ - ~ ) ] / M ( ~ " ) . The power is the sum of the multinomial probabilities of those points (xl,x2,. . . , x,) , for which the test statistic exceeds the critical value. These calculations give exact powers which we can compare with the Monte Carlo estimates.

For the cases with m = 2, the largest discrepancy between the exact and estimated values is 0.0024. With lo4 iterations, Monte Carlo estimates ofp=0.05 have a standard deviation of 0.0022.

The s-significance levels do not behave monotonically as n increases. This is due primarily to the discreteness of the underlying multinomial distribution. Also, some of the exact levels are quite different from the nominal levels. This is not surprising where, under Ho, the mean number of observations in the cells is below 5. For instance, with a nominal level of a = 0.05, m = 2, n = 10, and c1 = 1/3 the Monte Carlo estimates of the s-significance level is 0.103 and the exact value is 0.104. However, for all cases with mean cell counts at least 5 , the s-significance levels for V range from 0.037 to 0.059.

3.2 Power of the Tests

Table 1 contains estimates of the powers of these tests at intensities which are in the alternative region, HI, and are step functions of the form p (t) = 1 for 0 s t < d and p ( t ) = 4 ford 5 t s 1 = P. (Throughout this power study we take F = 1; see section 2 and (2-6) for an explanation.) The estimates were obtained by Monte Carlo techniques based on lo4 iterations. To determine the consequences of an incorrect specification of the ci, the power of I/ was estimated with c1 # d. Clearly, if the jump point d is known, then Vis the preferred test. In some cases, the increase in power over the next most

M ( T , ) / M ( P ) = U(i) . (2-6)

3. RESULTS

3.1 s-Significance Levels

Bain, et al [2] studied the s-significance levels of many of these tests. The Z test is exact and L is based on a s-normal approximation which works quite well even for small n. The

TABLE 1 Estimated' Powers for Two Step Alternatives with (u=0.05

p ( t ) = 1, 0 5 f < d a n d p ( t ) = 4, d 5 t 5 1.

d=1/3 d = 1/2 d=2/3

Test\n 10 20 40 10 20 40 10 20 40

Z 0.381 0.618 0.865 0.499 0.720 0.918 0.479 0.677 0.885 L 0.307 0.543 0.828 0.513 0.789 0.%9 0.591 0.842 0.981 W 0.287 0.504 0.805 0418 0.677 0.929 0.492 0.747 0.957 V( C, =d) 0.6% 0.821 0.971 0.676 0.915 0.992 0.788 0.910 0.997 V(cl#d)+ 0.295 0.478 0.660 0.461 0.531 0.819 0.522 0.789 0.948

*The standard deviation of Monte Carlo estimates of power based on 10,oCNl iterations is no larger than 0.005. +For d= 1/3 and 2/3, the powers are for cI = 1/2. For d= 1/2, they are for c1 =2/3 (this choice of cI pro- duced smaller powers than c, = 1/3).

ENGELHARDT ET AL.: TESTS FOR POSITIVE JUMPS IN THE INTENSITY OF A POISSON PROCESS: A POWER STUDY 359

powerful test is more than 30%. However, incorrect specification of the jump point can take this test from being the most powerful to one of the least powerful. Consequently, the use of V is not recommended if d is not known.

If the jump point is not known, which test should be used? In the cases considered here, L dominates Wand thus we need to choose between Z and L. If d < 112 then use Z, otherwise use L. However, if one test is desired, it should be L. The greatest loss in power due to using L or Z rather than Vis about 20%, but L performs better over a larger range of d .

Table 2 contains the estimated powers of these tests at intensities in the alternative region which are step functions with jumps at dl and d2. Again, if the jump points are known then use V, otherwise use L.

To determine if our basic conclusions hold for larger m, step functions with three jumps were considered. Table 3 contains the estimates of the powers of these tests at intensities in the alternative region with equally spaced jump points. With the first set of jump heights, viz, p = ( 1,2,3,4), irregular jump points were also considered. In each case as well as those treated in table 3, the same conclusions were reached: If the jump points are known then use V, otherwise use L.

For researcher considering the use of V, it would be helpful to know how it performs if the assumption of a step-function intensity were incorrect. To provide some information of this type, we considered the smooth alternatives in Bain, et al [2]:

p ( t ) = /3tS-' with /3 = 2,4 p ( t ) = exp(pr) with = 1,2,3 (their table 2 erroneously gives the values 1,2,4)

p ( t ) = log(Pt+l) with /3 = 10, 15, 25.

The powers of these tests were estimated for n = 10,2090, and for V with 2,3,4 regular cells considered.

For a fixed intensity and a fixed sample size, the power of each test was divided by the maximum of the powers of all the tests considered for that intensity and that sample size. We refer to this ratio as the relative power of the test. For V , the smallest relative power is 58.3% - occurring when m =2 . For m=3 & 4 , thesmallestrelativepowers were72.7% and78.3%, respectively. Bain, et al[2] reported that for these smooth alternatives, the smallest relative power of Z was 87% and for L it was 82%.

4. NUMERICAL EXAMPLE

These tests are illustrated on the data in Ascher & Feingold [ 1, p 751, viz, the failure times for the #3 main propulsion diesel engine of the USS Halfbeak. Ref [2, figure 5-31 is a plot of cumulative failures vs operating time, and shows that there is a difference in the failure intensity before and after 20 k hours. (The test statistic V which tests homogeneity with a non- decreasing alternative, was computed for the entire data set. As one would anticipate from examining the plot, its value is highly s-significant.) However, it is not as clear from the plot whether the intensity is constant over the first 20 k hours, but one would anticipate that it has a non-decreasing trend. One could attempt to model these failure times using (1) with the cj chosen to be the times at which scheduled maintenance was

TABLE 2 Estimated' Powers for Three-Step Alternatives with (Y =0.05

p ( t ) = l , 0 5 t < d , ; p ( t ) = 2, d , < t < d2 and p ( t ) = 4, d2 5 t 5 1. ~ ___ ~ ~

d = (1/4, 1/2) d = (1/4, 3/4) d = (1/2, 3/4)

10 20 40 10 20 40 10 20 40 Test\n

Z 0.414 0.641 0.875 0.365 0.560 0.799 0.451 0.650 0.866 L 0.383 0.640 0.895 0.392 0.627 0.874 0.544 0.798 0.967 W 0.315 0.526 0.806 0.324 0.517 0.779 0.440 0.674 0.911 V 0.588 0.711 0.943 0.441 0.601 0.903 0.601 0.830 0.977 V(equal cells) 0.363 0.598 0.852 0.345 0.587 0.815 0.481 0.775 0.945

'The standard deviation of Monte Carlo estimates of power based on 10,OOO iterations is no larger than 0.005.

TABLE 3 Estimated' Powers for Four-Step Alternatives with a=0.05

p ( t ) = pi for ( i - 1)/4 5 r < i/4, i = 1 , 2, 3, 4.

p = ( 1 , 2, 3, 4) p = (1, A, A, 2) p = (1, 4, 9, 16)

Testbi 10 20 40 10 20 40 10 20 40

Z 0.392 0.603 0.843 0.177 0.258 0.405 0.801 0.960 0.999 L 0.388 0.632 0.884 0.171 0.264 0.439 0.800 0.977 1.OOO W 0.312 0.512 0.775 0.145 0.202 0.332 0.702 0.937 0.999 V 0.455 0.635 0.890 0.223 0.272 0.438 0.839 0.980 1.OOO

'The standard deviation of Monte Carlo estimates of power based on 10,OOO iterations is no larger than 0.005.


performed. Scheduled maintenance was performed at 9.453, 11.528, 11.993,15.058, 17.315khours. Toobtainareasonable number of failures in each of the cells, we combine intervals 2 & 3, as well as intervbals 4 & 5. Thus, c1 = 9 453, c2 = 11 993, c3 = 17 315, and of course c4 = 20 OOO. The model (1) assumes that the intensity is constant on each interval ( C ~ - ~ , C ~ ] . To check this assumption, the Laplace test statistic was computed for each interval. That is, Lj was obtained by summing -

[ q - cj- J [ C j - Cj- 11

over those i with q in ( cj- 1, cj] , and

submitted in partial fulfillment for the PhD degree at the Univer- sity of Missouri-Rolla.

REFERENCES

[l] H. Ascher, H. Feingold, Repairable Systems Reliability, 1984; Marcel Dekker.

[2] L. J. Bain, M. E. Engelhardt, F. T. Wright, “Tests for an increasing trend in the intensity of a Poisson process: A power study”, J. Amer. Statistical Assoc., vol 80, 1985 Jun, pp 419-422.

[3] R. E. Barlow, D. J. Bartholomew, J. M. Bremner, H. D. B&, Srclrisrical Inferences Under Order Restrictions, 1972; John Wiley & Sons.

[4] M. T. Boswell, “Estimating and testing trend in a stochastic process of the Poisson type”, Ann. Mathemorical Statistics, vol 37, 1966 Dec, pp 1564-1573.

[5] V. J. Chacko, “Modified chi-square tests for ordered alternatives”,

[6] D. R. Cox, “Some statistical methods connected with series of events”, J. Royal Statistical SOC. B, vol 17, 1955, pp 129-164.

171 c. I. C. Lee, “Chi-square tests for and against an order restriction on multinomial parameters”, J. Amer. Statistical Assoc., ~0182, 1987 Jun,

[8] R. C. Magel, F. T. Wright, “Tests for and against trends in Poisson intensities”, in Inequalities in Statistics and Probnbility, (Y. L. Tong, ed) 1984, pp 236-243; Institute of Mathematical Statistics.

[9] US Mil-Hdbk-189, Reliability Growth Manugement, 1981.

4 = ( ~ ~ - n ~ / 2 ) / ( n ~ / 1 2 ) ” * sanwlya B, VOI 28, 1966 DK, pp 185-190.

was computed with nj the number of failures in ( C ~ - ~ , C ~ ] . The four zj are 1,405, 0.826, 0.211, 1.224 and the s-significance levels for these ,,dues are 0.08, o . ~ ~ , 0.42, o. 11, respective-

this is not strong evidence against the assumption of homogeneity within the cells. We consider models of the form (2-1).

,&&led -emce was performd, there were 29 failures in (o, 2o 0001, Straightforward calculations yield,

pp 611-618. On the

hcluding the failure times at [lo] T. Robertson, “Testing for and against an order restriction on multinomial

parameters”, J. Amer. Statistical Assoc., vol73, 1978 Mar, pp 197-202. [ 1 I] T. Robertson, F. T. Wright, “On approximation of level probabilities

and associated distributions in order restricted inference”, Biometrika, Z = 32.632, L = 19.018 VOI 70, 1983 Ds, pp 597-606.

Pr{X& I 32.632) = 0.0029,

ga~fc[(19.018-14.5)/(29/12)”~] = 0.0018.

Next, we compute the value of V and the associated p-value. The numbers of failures in the four cells are 8,5,5, 1 1 respectively, and the unrestricted estimates of the pi are 819453, 512540, 515322, 1112685. In applying the PAVA to obtain (p1$2,p3$4), we note that b2 > &. Hence, the estimates over these two intervals are pooled to obtain 1017862. Because this new sequence of estimates is non-decreasing, we have p 1 = jl, p2 = j3 = 10/7862, p 4 = j4. From (3), we find that V = 11.615.

Using [3,(3.17),(3.26)-(3.28)], we find that P(4,4;1) = 0.0331, P(3,4;1) ~ 0 . 2 3 4 2 , P(2,4;1) =0.4669 mdP( 1,4;1) = 0.2658. Applying (4), the approximate s-significance level for V is 0.0013, viz, smaller than the s-significance levels for 2 and L.

ACKNOWLEDGMENT

AUTHORS

Dr. Max Engelhardt; Department of Mathematics and Statistics; University of Missouri; Rolla, Missouri 65401 USA.

Max Engelhardt is Professor of Mathematics and Statistics at the Universi- ty of Missouri-Rolla. He received his PhD from the University of Missouri- Columbia. His research interests are life-testing and reliability. He is a reviewer for zentralbla&r Muthemdk and a f o m r Associate Editor for Technomehics, and a Fellow of the American Statistical Association.

James M. Guffey; Division of Mathematics and Computer Science; Northeast Missouri State University; Kirksville, Missouri 63501 USA. James Guffey was born 1961 Oct 16 in Tennessee. He received a BS in

Mathematics from Centre College of Kentucky, an MS in Applied Mathematics from the University of Missouri-Rolla, and is a PhD candidate in Mathematics at the University of Missouri-Rolla. He is an Assistant Professor in the Division of Mathematics and Computer Science at Northeast Missouri State University.

Dr. Farroll T. Wright; Department of Mathematics and Statistics; University of Missouri; Rolla, Missouri 65401 USA.

Farroll T. Wright is Professor of Mathematics and Statistics at the Universi- ty of Missouri-Rolla. His research interests are in order-restricted inference, and he is coauthor of Order Restricted Sraristicallnference, 1988 (John Wiley & Sons). He is a Fellow of the American Statistical Association and the Institute of Mathematical Statistics.

The research for this was supported by us Office of Naval Research Contract NOOO14-80-C-0322 and US Air Force Office of Scientific Research Grant AFOSR-84-0164.

Manuscript TR88-130 received 1988 July 22; revised 1989 June 9; revised 1990 March 14.

IEEE Log Number 36342 Parts of this work are taken from the J. M. Guffey dissertation *TR,

tests for positive jumps in the intensity of a poisson process: a power study

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