Likelihood ratio tests for testing for multiple contaminants in the shocks and labelled slippage models

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Likelihood ratio tests for testing formultiple contaminants in the shocksand labelled slippage modelsUditha Balasooriya a & Veeresh Gadag ba Division of Banking and Finance, Nanyang Business School ,Nanyang Technological University , Singapore , 639798b Division of Community Health and Humanities, Faculty ofMedicine , Memorial University , St. John's, NL, Canada , A1B 3V6Published online: 23 Nov 2010.

To cite this article: Uditha Balasooriya & Veeresh Gadag (2011) Likelihood ratio tests for testingfor multiple contaminants in the shocks and labelled slippage models, Journal of StatisticalComputation and Simulation, 81:5, 607-618, DOI: 10.1080/00949650903449268

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Journal of Statistical Computation and SimulationVol. 81, No. 5, May 2011, 607–618

Likelihood ratio tests for testing for multiplecontaminants in the shocks and labelled

slippage models

Uditha Balasooriyaa* and Veeresh Gadagb

aDivision of Banking and Finance, Nanyang Business School, Nanyang Technological University,Singapore 639798; bDivision of Community Health and Humanities, Faculty of Medicine, Memorial

University, St. John’s, NL, Canada A1B 3V6

(Received 26 May 2008; final version received 29 October 2009 )

In the literature related to the study of lifelengths of experimental units, little attention has been paidto the models where shocks to the units generate outliers. In the present article, we consider a situationwhere n experimental units under investigation receive shocks at several time points. The parameter valuesof the lifelength distribution may change due to each shock, resulting in the generation of outliers. Wederive the likelihood ratio test statistic to investigate if the shocks have significantly altered the param-eter values. We also derive a likelihood ratio test under the labelled slippage alternative with multiplecontaminations. Monte Carlo studies have been carried out to investigate the power of the proposed teststatistics.

Keywords: exponential distribution; shocks model; labelled slippage model; likelihood ratio test

1. Introduction

In the literature related to the detection of outliers/contaminants, several traditional tests have beenproposed for testing the null hypothesis that all lifelengths of experimental units are independentlyand identically distributed (iid) random variables (rvs) against an alternative that a small subsetof observations in the sample has arisen from a distribution which is different from the one underthe null hypothesis. In order to accommodate various types of contaminations, different formsof alternative models have been considered in the literature (see [1]). For example, slippage-model [2], mixture-model [3], exchangeable-model [4,5] and labelled slippage-model [6] aresome of the widely used traditional alternative outlier-generating models (see also [7]). Gatherand Kale [6] considered the problem of testing for multiple outliers under the labelled slippageoutlier-generating models, where a contamination occurs through a single change in the parameterof interest.

*Corresponding author. Email: [email protected]

ISSN 0094-9655 print/ISSN 1563-5163 online© 2011 Taylor & FrancisDOI: 10.1080/00949650903449268http://www.informaworld.com

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608 U. Balasooriya and V. Gadag

Although the literature on outlier detection techniques is abundant, little attention has been paidto the models where contaminants occur as a result of shocks to the units/systems. Consider asituation wherein units/systems, whose lifelengths are iid from a given distribution, are exposedto shocks. Each shock to the surviving units/systems may alter the value(s) of the parameter(s) ofthe lifelength distribution of their residual life. We call such a model a ‘shocks model’. In suchsituations, one may be interested in testing the null hypothesis that the shocks have not altered thevalue(s) of the parameter(s), against the alternative that the shocks have altered the value(s). Forexample, in the stress–strength experiments, it is of interest to the mechanical engineers to find outif the shocks to the identically and independently operating systems have altered the value(s) ofthe parameter(s) of the residual life of the functioning systems. Structural engineers have interestin finding whether the exposure of units such as building or building material to the naturalcalamities have altered the value(s) of parameter(s) of the residual life of the units which are stillfunctioning after the calamity. Public health researchers have interest in determining whether anaccidental release of the toxic/radioactive material into the environment had any effect on the lifespan of the survivors. The contaminations in all these situations occur in a non-traditional way.

The problem considered in the present work is seemingly similar to the change-point problemdiscussed in the literature, in particular to the tests proposed for detecting change-points with epi-demic alternatives [8]. The main difference lies in the way the alternative hypothesis is constructedfor the two situations. Note that in the change-points problem, the epidemic alternative hypothesisconsidered byYao [8] and many others refers to random samples from different distributions. In thepresent work, we are interested in testing whether a sudden shock results in outliers/contaminantsin the data due to the effect of the shock on the remaining lifelengths of the surviving unitsconditional on lifetime at the time of the shock. Among others, Hawkings [9] considered testinga sequence of observations for a shift in location and Hinkley [10] considered inference abouta change-point in a sequence of random variables. The likelihood ratio test for a change-pointproblem for exponential random variables was discussed by Haccou et al. [11]. Worsley [12]studied confidence regions and test for a sequence of exponential family random variables. Sometests proposed for change-points with the epidemic alternative have been studied by Yao [8].The main focus of the present paper is the detection of contaminants due to sudden changes inthe parameter(s) of the underlying distribution due to shocks that affect on the remaining life ofunits.

In this article we assume that the underlying lifelength distribution is exponential with theprobability density function (pdf) given by

f (x; σ) =(

1

σ

)exp

(−x

σ

)for x ≥ 0, (1)

where σ > 0 is the unknown scale parameter. For brevity, we denote the distribution (1) byE(., σ ).

In Section 2, we derive the maximum likelihood estimator of the life-length parameter σ underthe null and multiple shocks alternative hypotheses. We also develop the corresponding likelihoodratio test statistic. It is assumed here that the shocks occur only at the times of failure of a testunit and that the test units do not fail instantaneously due to the occurrence of shocks. The caseof a single shock alternative has been dealt with by Gadag and Balasooriya [13]. In Section 3, weextend the results of Gather and Kale [6] to allow for multiple contaminations. We also derive themaximum likelihood estimators for the parameters under this multiple contamination alternatives.As the null distributions of the proposed test statistics in the above situations are complicated,we provide in Section 4 the cut-off points for selected sample sizes and Type I errors in Table 1.Using these cut-off values, we carry out different power performance studies and the results aregiven in Tables 2–5. Concluding remarks are given in Section 5.

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Journal of Statistical Computation and Simulation 609

2. Likelihood ratio test for the shocks-model with multiple contaminations

As in Section 1, we suppose that a sample of n experimental units to be tested are from E(., σ ) withdensity function given by Equation (1). Let X(1) ≤ X(2) ≤ · · · ≤ X(n) denote the ordered sample.We first consider a model wherein specimens are exposed to only two shocks. We then generalizethe result to k(>2) shocks. In this shocks-model, all the n lifelengths are iid from E(., σ ) until the(n − r1 − r2)th failure at which time the first shock occurs. It is further assumed that the shockmay have altered the scale parameter from σ to σ1 for the residual life of the remaining (r1 + r2)

items. These (r1 + r2) residual lifelengths are iid from E(., σ1) until the (n − r2)th failure, atwhich time the second shock occurs. It is suspected that this second shock has altered the valueof the scale parameter from σ1 to σ2 for the residual life of the existing r2 items, which are iid

from E(., σ2). Our interest is in testing the following:

H0: All the n lifelengths are iid rvs from E(., σ ) versusH1(SM): All the n lifelengths are from E(., σ ) up to the (n − r1 − r2)th failure. Conditional on

x(n−r1−r2), the remaining lifelengths of (r1 + r2) surviving units are iid from E(., σ1)

up to the (n − r2)th failure. The remaining lifelengths of the last r2 surviving itemsconditional on x(n−r2) are iid from E(., σ2) with σ ≤ σ1 ≤ σ2.

The likelihood function under the null hypothesis H0 is given by

L0(σ |x(1), x(2), . . . , x(n)) =(

1

σ

)n [exp

(− ∑ni=1(n − i + 1)(x(i) − x(i−1))

σ

)]n!.

The likelihood function under the alternative hypothesis H1(SM) is given by

L1(SM)(σ, σ1, σ2|x(1), x(2), . . . , x(n)) =(

Pn−r1−r2,n

σ n−r1−r2

)exp

(−

n−r1−r2∑i=1

x(i)

σ

)

×[

exp

(−xn−r1−r2

σ

)]r1+r2(

Pr1,r1+r2

σr11

)

× exp

[−

n−r2∑i=n−r1−r2+1

(x(i) − x(n−r1−r2))

σ1

]

×[

exp − (x(n−r2) − x(n−r1−r2))

σ1

]r2

× (r2!|σ r22 ) exp

[−

n∑i=n−r2+1

(x(i) − x(n−r2))

σ2

],

where Pn−r1−r2,n and Pr1,r1+r2 are the permutations. From the well-known results of Epstein [14],the likelihood function under H1(SM) can be written as

L1(SM)(σ |x(1), x(2), . . . , x(n)) = n!σn−r1−r2σ

r11 σ

r22

exp

[−

n−r1−r2∑i=1

(n − i + 1)(x(i) − x(i−1))

σ

]

× exp

[−

n−r2∑n−r1−r2+1

(n − i + 1)(x(i) − x(i−1))

σ1

]

× exp

[−

n∑n−r2+1

(n − i + 1)(x(i) − x(i−1))

σ2

].

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Let Yi = (n − i + 1)(X(i) − X(i−1)). Then under L1(SM), we have [14,15]

Yi ∼ E(., σ ) for i = 1, 2, . . . , (n − r1 − r2),

∼ E(., σ1) for i = (n − r1 − r2 + 1), . . . , (n − r2),

∼ E(., σ2) for i = (n − r2 + 1), . . . , n.

It can be easily seen that

σ̂ =n−r1−r2∑

i=1

Yi

(n − r1 − r2), σ̂1 =

n−r2∑i=n−r1−r2+1

Yi

r1, σ̂2 =

n∑i=n−r2+1

Yi

r2.

The maximum likelihood estimator for σ under H0 is

σ̂ =n∑

i=1

Yi

n=

n∑i=1

Xi

n.

Further, the likelihood ratio statistic for testing H0 versus H1(SM) is proportional to

[(Ur1)r1(Ur2)

r2(1 − Ur1 − Ur2)n−r1−r2 ]−1,

where

Ur1 =∑n−r2

n−r1−r2+1 Yi∑ni=1 Yi

and Ur2 =∑n

n−r2+1 Yi∑ni=1 Yi

.

Thus, for testing H0 versus H1(SM), we propose the test statistic

U = (Ur1)r1(Ur2)

r2 .

Note also that under H0,

(Ur1 , Ur2) ∼ Dirichlet(r1 + 1, r2 + 1, n − r1 − r2 + 1).

So,

f (ur1 , ur2) = �(n + 3)

�(r1 + 1)�(r2 + 1)�(n − r1 − r2 + 1)ur1

r1ur2

r2(1 − ur1 − ur2)

n−r1−r2

=n−r1−r2∑

k1=0

n−r1−r2−k1∑k2=0

C(r1, r2, k1, k2, n)ur1r1ur2

r2un−r1−r2−k1−k2

r1uk2

r2,

for ur1 , ur2 ≥ 0,

where

C(r1, r2, k1, k2, n) = �(n + 3)

r1�(r1 + 1)�(r2 + 1)�(n − r1 − r2 + 1)

×(

n − r1 − r2

k1

)(n − r1 − r2 − k1

k2

).

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Let W1 = U and W2 = Ur2 . Then the joint distribution of W1 and W2

f (w1, w2) =n−r1−r2∑

k1=0

n−r1−r2−k1∑k2=0

C(r1, r2, k1, k2, n)w(n−r1−r2−k1−k2)/r11

× wk2−r2(n−r1−r2−k1−k2+1)/r12 for 0 < w1 < w2 < 1.

Unfortunately, a closed-form expression for the marginal distribution of W1 does not seem to existand we have to rely on the simulated critical values of the test statistic U .

3. Likelihood ratio test in the labelled slippage model with multiple contaminations

The most common type of life testing models with contaminants in the literature is the slippagealternative model where in its general form, a small number of observations in the data arisesfrom a model that is different from the model specified under H0. In particular in such models,for example, under H0 the lifelengths of all ‘n’ observations are iid with scale parameter, sayσ , while under H1, for a fixed ‘r’ with 0 < r < n, the lifelengths of ‘(n − r)’ observations areiid with scale parameter σ and the lifelengths of the remaining ‘r’ observations are iid withscale parameter bσ (b > 0) and are independent of the (n − r) observations. Further, if such ‘r’observations are specified to be the largest ‘r’observations, then it is termed as a labelled slippagealternative model [6,7]. In what follows, we consider a generalization of this labelled slippagemodel.

Let us suppose a sample of n experimental units is to be tested, wherein the lifelengths are iidE(., σ ) with the density function given by Equation (1). To begin with, for simplicity, we consideralternative hypothesis with two contaminations only as described below. We then generalize theresult to the situation involving k(>2) contaminations.

H0: All the n lifelengths are iid rvs from E(., σ ) versusH1(LS): The smallest (n − r1 − r2) lifelengths are iid observations from E(., σ ), the next smallest

r1 lifelengths are iid observations from E(., σ1) and the largest r2 lifelengths are iid

observations from E(., σ2), with σ < σ1 ≤ σ2.

Note also that 1 ≤ r1 + r2 < n. Let, σ1 = [σ/b1] and σ2 = [σ/(b1b2)], where b1 and b2 are non-negative known constants. With these notations, the above null and alternative hypotheses can berewritten as

H0: b1 = b2 = 1 versusH1(LS): At least one bi for i = 1, 2 is less than 1.

The likelihood under H0, given the ordered observations X(1) ≤ X(2) ≤ · · · ≤ X(n) can bewritten as

L0(σ |x(1), x(2), . . . , x(n)) =(

n!σn

)exp

(n∑

i=1

x(i)

σ

). (2)

Further, the likelihood under the alternative hypothesis is

L1(SM)(σ |x(1), x(2), . . . , x(n)) =(

1

C

)[(n − r1 − r2)!r1!(r2 − 1)!)]

[(1

σ

)(n−r1−r2)]

× exp

(−

n−r1−r2∑i=1

x(i)

σ

)×

[(1

σ1

)r1]

exp

(−

n−r2∑i=n−r1−r2

x(i)

σ1

)

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×[(

1

σ2

)r2]

exp

(−

n∑i=n−r2+1

x(i)

σ2

)

=(

1

C

)[(n − r1 − r2)!r1!(r2 − 1)!)]

[(1

σ

)(n−r1−r2)]

× exp

(−

n−r1−r2∑i=1

x(i)

σ

)×

[(b1

σ

)r1]

exp

(−

n−r2∑i=n−r1−r2

x(i)

σ1

)

×[(

b1b2

σ

)r2]

exp

(−

n∑i=n−r2+1

x(i)

σ2

), (3)

where

C = P [X(n−r1−r2−1) < X(n−r1−r2); X(n−r2−1) < X(n−r2)]

=[(n − r1 − r2)r1(r1 − 1)

(r1 + r2b2)

]β(r2b2b1 + r1b1 + 1, n − r1 − r2)β(r1 − 1, r2b2 + 1)

and β(., .) is the beta function. It is clear from Equation (3) that the maximum likelihood estimatorsfor the parameters under the alternate hypothesis are

σ̂ =n−r1−r2∑

i=1

X(i)

(n − r1 − r2), σ̂1 =

n−r2∑i=n−r1−r2+1

X(i)

r1, σ̂2 =

n∑i=n−r2+1

X(i)

r2.

Also, it is clear that when b1 = b2 = 1, i.e. under H0, C = [(n − r1 − r2)!r1!(r2)!]/n! andEquation (3) reduces to Equation (2) and the maximum likelihood estimator for the parameterunder H0 is

σ̂ =n∑

i=1

X(i)

n.

It follows from Equations (2) and (3) that, the likelihood ratio statistic for testing H0 againstH1(LS) is proportional to

(Vr1)r1(Vr2)

r2(1 − Vr1 − Vr2)n−r1−r2 , (4)

where

Vr1 =∑n−r2

i=n−r1−r2+1 X(i)∑ni=1 X(i)

and

Vr2 =∑n

i=n−r2+1 X(i)∑ni=1 X(i)

.

Thus, for testing H0 versus H1(LS), we propose the statistic

V = (Vr1)r1(Vr2)

r2

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as a likelihood ratio test statistic. In general, if the contaminants are generated with k differentdistributions, the likelihood ratio test statistic that we propose is

V =k∏

i=1

(Vri)ri ,

where

Vri=

∑n−mi+1i=n−mi+1 X(j)∑n

i=1 X(i)

with mi =k∑

l=i

r(l) for i = 1, 2, . . . , k − 1.

4. Power performances

As the null distributions of the test statistics U and V are complicated, we have to depend on thecut-off points based on the Monte Carlo results for carrying out the power performance studies.Since in the present age of high power computing facilities one can easily obtain these criticalvalues, to save the space, we present the cut-off values in Table 1 only for n = 15(1)24 and n = 30for selected values of (r1, r2) by generating 10,000 samples.

We first compare the power performances of U and V by generating 2000 samples for moderatesample size n = 30. Note that the contaminants in each sample are generated under the shocksmodel alternative H1(SM) discussed in Section 2. We observe from the tabulated values in Table 2,that when b1 = b2 = 1, the computed powers of the test statistic are very close to the correspondinglevels of significance α. The situation b1 = 0.75 and b2 = 1 implies that the first shock resultedin a shift in the lifelength parameter upwards, while the second shock was not effective in shiftingthe value of the lifelength parameter any further. Thus, in this situation all five contaminants aregenerated by the first shock. All other situations correspond to both shocks being effective inshifting the parameter values.

The values in Table 2 show that in general the statistic U performs better than V for allcombinations of (b1, b2) and (r1, r2) at both the levels of significance. Since the contaminants aregenerated according to the shocks model such a result is expected. It is interesting to note thatfor the case b1 = b2 = 0.5, the power for the situation (r1, r2) = (1, 4) is higher when comparedto (r1, r2) = (4, 1). This is due to the fact that the generated outliers for the case (r1, r2) = (1, 4)

are more extreme than the ones generated for the case (r1, r2) = (4, 1).We now make the power comparisons under the same setting as above but by generating the

contaminants using the labelled slippage model under H1(LS) in Section 3. As expected, the powerof the statistic V given in Table 3 decreases as r1 increases from 1 to 4, when r1 + r2 = 5. Alsoobserve that, when b1 = b2 = 1 the computed powers of the test statistic are very close to thecorresponding levels of significance α. For all other combinations considered, the statistic V

performs better than the statistic U .However, comparisons made in Tables 2 and 3 may seem to be unfair as under a given outlier-

generating mechanism only one of the two statistics U and V is defined. Thus, we comparethe power performance of the proposed statistics in Table 4, for n = 20(10)40 under the appro-priate labelled slippage and shock-alternative models. For the case of n = 20, we considered(r1, r2) = (1(1)3, 3(1)1) and for the case n = 30, 40, we considered (r1, r2) = (1(1)4, 4(1)1)

at two levels of significance, α = 0.05 and α = 0.01 and for the combinations of b1 andb2, (b1, b2) = (0.5(0.25)1, 0.5(0.25)1). We observe from this table, in general, the statistic U

used for shock model is slightly more powerful than the statistics V used for the labelled slippagemodel.

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Table 1. Simulated critical values of the two test statistics U and V for multiple tests

n R1 R2 u.025 u.05 u.95 u.975 v.025 v.05 v.95 v.975

16 1 1 1.981e−5 5.010e−5 1.412e−2 1.922e−2 1.541e−2 1.697e−2 5.333e−2 5.977e−21 2 4.196e−6 1.036e−5 5.115e−3 7.249e−3 6.502e−3 7.356e−3 2.792e−2 3.137e−22 1 4.240e−6 1.019e−5 5.046e−3 7.368e−3 6.399e−3 7.234e−3 2.782e−2 3.141e−2




20 1 1 1.373e−5 3.267e−5 9.243e−3 1.271e−2 1.138e−2 1.256e−2 3.996e−2 4.496e−21 2 2.119e−6 5.041e−6 2.676e−3 3.884e−3 4.156e−3 4.736e−3 1.901e−2 2.155e−21 3 5.290e−7 1.388e−6 1.233e−3 1.848e−3 2.339e−3 2.718e−3 1.273e−2 1.449e−22 1 2.101e−6 5.045e−6 2.678e−3 3.882e−3 4.114e−3 4.668e−3 1.885e−2 2.159e−22 2 3.426e−7 8.318e−7 7.338e−4 1.097e−3 1.361e−3 1.565e−3 7.151e−3 8.170e−33 1 5.376e−7 1.375e−6 1.220e−3 1.873e−3 2.267e−3 2.608e−3 1.246e−2 1.426e−2





30 1 1 6.004e−6 1.405e−5 4.332e−3 5.994e−3 6.581e−3 7.224e−3 2.309e−2 2.609e−21 2 5.859e−7 1.440e−6 8.534e−4 1.265e−3 1.862e−3 2.106e−3 8.897e−3 1.021e−21 3 9.638e−8 2.451e−7 2.669e−4 4.186e−4 8.173e−4 9.489e−4 4.958e−3 5.781e−31 4 2.250e−8 6.276e−8 1.112e−4 1.857e−4 4.667e−4 5.553e−4 3.384e−3 3.985e−32 1 5.616e−7 1.387e−6 8.674e−4 1.285e−3 1.850e−3 2.087e−3 8.778e−3 1.012e−22 2 5.698e−8 1.523e−7 1.587e−4 2.516e−4 4.767e−4 5.529e−4 2.763e−3 3.256e−32 3 9.574e−9 2.702e−8 4.775e−5 7.963e−5 1.923e−4 2.278e−4 1.318e−3 1.545e−33 1 9.447e−8 2.428e−7 2.681e−4 4.239e−4 7.986e−4 9.248e−4 4.772e−3 5.635e−33 2 1.012e−8 2.767e−8 4.789e−5 7.857e−5 1.891e−4 2.240e−4 1.290e−3 1.522e−34 1 2.215e−8 6.308e−8 1.125e−4 1.850e−4 4.468e−4 5.278e−4 3.204e−3 3.800e−3

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Table 2. Power performance of U and V at α = 0.05, when contaminants are generated according to theshock-model.

N = 30 (r1, r2)

b1 b2 Statistic (1, 4) (2, 3) (3, 2) (4, 1)

1.00 1.00 U 0.049 (0.011)a 0.045 (0.009) 0.047 (0.011) 0.049 (0.009)V 0.050 (0.010) 0.050 (0.010) 0.051 (0.010) 0.050 (0.009)

0.75 1.00 U 0.151 (0.046) 0.145 (0.041) 0.144 (0.041) 0.149 (0.042)V 0.131 (0.041) 0.122 (0.038) 0.120 (0.037) 0.128 (0.037)

0.75 0.75 U 0.275 (0.112) 0.236 (0.085) 0.203 (0.067) 0.176 (0.054)V 0.237 (0.089) 0.169 (0.053) 0.135 (0.038) 0.127 (0.036)

0.75 0.50 U 0.505 (0.272) 0.392 (0.184) 0.300 (0.116) 0.216 (0.068)V 0.449 (0.225) 0.261 (0.088) 0.152 (0.043) 0.123 (0.035)

0.50 0.50 U 0.753 (0.562) 0.674 (0.450) 0.585 (0.354) 0.492 (0.262)V 0.723 (0.500) 0.497 (0.241) 0.333 (0.146) 0.311 (0.134)

aNumbers in parenthesis correspond to α = 0.01.

Table 3. Power performance of U and V at α = 0.05, when contaminants are generated according to the labelledslippage model.

N = 30 (r1, r2)

b1 b2 Statistic (1, 4) (2, 3) (3, 2) (4, 1)

1.00 1.00 U 0.049 (0.011)a 0.045 (0.009) 0.047 (0.011) 0.049 (0.009)V 0.050 (0.010) 0.050 (0.010) 0.051 (0.010) 0.050 (0.009)

0.75 1.00 U 0.105 (0.029) 0.104 (0.026) 0.099 (0.027) 0.108 (0.027)V 0.123 (0.037) 0.121 (0.035) 0.118 (0.032) 0.121 (0.032)

0.75 0.75 U 0.212 (0.075) 0.201 (0.067) 0.185 (0.056) 0.157 (0.045)V 0.248 (0.092) 0.212 (0.074) 0.175 (0.055) 0.150 (0.043)

0.75 0.05 U 0.421 (0.212) 0.387 (0.184) 0.332 (0.136) 0.230 (0.081)V 0.490 (0.260) 0.393 (0.167) 0.266 (0.097) 0.186 (0.046)

0.50 0.50 U 0.646 (0.427) 0.619 (0.385) 0.551 (0.323) 0.438 (0.220)V 0.749 (0.529) 0.648 (0.392) 0.502 (0.257) 0.399 (0.183)


Note that in Tables 2–4, the case b1 = b2 = 1 implies that the two shocks have not altered thescale parameter of the distribution. On the other hand, for i, j = 1, 2 if bi = 1 and bj < 1 wheni �= j , only the j th shock has altered the parameter. This situation, in the shocks-model effectivelyreduces to the situation discussed in Gadag and Balasooriya [13], wherein the test statistic

Ur =∑n

i=n−r+1 Y(i)∑ni=1 Y(i)

with Y(i) = (n − i + 1)(X(i) − X(i−1)) was proposed for detecting r outliers due to a single shock.Thus, when b1 < 1 and b2 = 1, only the first shock is effective and the appropriate statistic isUr1+r2 , while when b1 = 1 and b2 < 1, only the second shock is effective and the appropriatestatistic is Ur2 . This fact is clearly exemplified in Table 5.

The tabulated values in Table 5 have the following explanation, for example, consider thesituation α = 0.05, b1 = 1 and b2 = 0.25, r1 = 1 and r2 = 4. While the value refers to the powerof detecting four contaminants as a result of one shock at the time of failure of the 26th test unitout of 30, the value Ur1+r2 = 0.785 refers to the power of detecting five contaminants when infact there are only four contaminants described earlier. Further, U = 0.655 refers to the power

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Table 4. Power performance of U and V at α = 0.05(0.10), when contaminants aregenerated according to the shock and labelled slippage models.

(r1, r2)

b1 b2 Statistic (1,3) (2, 2) (3,1)

n = 201.00 1.00 U 0.051 (0.009) 0.054 (0.009) 0.057 (0.009)

V 0.049 (0.008) 0.048 (0.009) 0.050 (0.009)

0.75 1.00 U 0.131 (0.040) 0.134 (0.043) 0.139 (0.043)V 0.103 (0.024) 0.101 (0.025) 0.104 (0.026)

0.75 0.75 U 0.229 (0.091) 0.195 (0.070) 0.166 (0.060)V 0.190 (0.058) 0.151 (0.040) 0.129 (0.035)

0.75 0.50 U 0.394 (0.200) 0.295 (0.119) 0.204 (0.076)V 0.356 (0.143) 0.234 (0.075) 0.155 (0.047)

0.50 0.50 U 0.636 (0.417) 0.535 (0.315) 0.440 (0.224)V 0.588 (0.312) 0.408 (0.184) 0.315 (0.125)

(r1, r2)

b1 b2 Statistic (1, 4) (2, 3) (3, 2) (4, 1)

n = 301.00 1.00 U 0.049 (0.011) 0.045 (0.009) 0.047 (0.011) 0.049 (0.009)

V 0.050 (0.010) 0.050 (0.010) 0.051 (0.010) 0.050 (0.009)

0.75 1.00 U 0.151 (0.046) 0.145 (0.041) 0.144 (0.041) 0.149 (0.042)V 0.123 (0.037) 0.121 (0.035) 0.118 (0.032) 0.121 (0.032)

0.75 0.75 U 0.275 (0.112) 0.236 (0.085) 0.203 (0.067) 0.176 (0.054)V 0.248 (0.092) 0.212 (0.074) 0.175 (0.055) 0.150 (0.043)

0.75 0.50 U 0.505 (0.272) 0.392 (0.184) 0.300 (0.116) 0.216 (0.068)V 0.490 (0.260) 0.393 (0.167) 0.266 (0.097) 0.186 (0.046)

0.50 0.50 U 0.753 (0.562) 0.674 (0.450) 0.585 (0.354) 0.492 (0.262)V 0.749 (0.529) 0.648 (0.392) 0.502 (0.257) 0.399 (0.183)

n = 401.00 1.00 U 0.055 (0.011) 0.050 (0.011) 0.053 (0.010) 0.053 (0.011)

V 0.052 (0.010) 0.054 (0.011) 0.057 (0.011) 0.054 (0.012)

0.75 1.00 U 0.168 (0.053) 0.165 (0.052) 0.161 (0.054) 0.165 (0.054)V 0.156 (0.042) 0.152 (0.042) 0.146 (0.043) 0.147 (0.041)

0.75 0.75 U 0.313 (0.132) 0.272 (0.109) 0.234 (0.089) 0.196 (0.067)V 0.302 (0.130) 0.266 (0.105) 0.221 (0.082) 0.184 (0.061)

0.75 0.50 U 0.541 (0.328) 0.435 (0.231) 0.333 (0.150) 0.245 (0.092)V 0.570 (0.338) 0.483 (0.247) 0.353 (0.152) 0.236 (0.086)

0.50 0.50 U 0.788 (0.610) 0.720 (0.525) 0.629 (0.418) 0.536 (0.322)V 0.824 (0.637) 0.762 (0.525) 0.636 (0.385) 0.501 (0.267)

of detecting five contaminants as a result of the first ineffective shock (b1 = 1) and a further fouroutliers as a result of the second effective shock. Thus, as expected in this situation Ur2 performsbetter than U .

Similarly, consider the situation α = 0.05, b1 = 0.25 and b2 = 1, r1 = 1 and r2 = 4. Here,while the value refers to the power of detecting five contaminants as a result of one shock atthe time of failure of the 26th test unit out of 30, the value Ur2 = 0.733 refers to the power ofdetecting four contaminants when in fact there are five contaminants. Also, U = 0.824 refers tothe power of detecting five contaminants as a result of the first effective shock (b1 = 0.25) and afurther four outlier as a result of the second ineffective shock. Again as expected Ur1+r2 performsbetter than U .

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Table 5. Power performances of U , Ur1+r2 and Ur2 at α = 0.05.

n = 30 (r1, r2)

b1 b2 Statistic (1, 4) (2, 3) (3, 2) (4, 1)

When both shocks are not effective1.00 1.00 U 0.049 (0.011)a 0.045 (0.009) 0.047 (0.011) 0.049 (0.009)

Ur1+r2 0.047 (0.012) 0.047 (0.012) 0.047 (0.012) 0.047 (0.012)Ur2 0.045 (0.013) 0.050 (0.011) 0.054 (0.013) 0.051 (0.011)

When only the first shock is effective0.75 1.00 U 0.151 (0.046) 0.145 (0.041) 0.144 (0.041) 0.149 (0.042)

Ur1+r2 0.154 (0.047) 0.154 (0.047) 0.154 (0.047) 0.154 (0.047)Ur2 0.139 (0.040) 0.130 (0.038) 0.112 (0.036) 0.097 (0.028)

0.50 1.00 U 0.406 (0.197) 0.404 (0.193) 0.403 (0.192) 0.403 (0.194)Ur1+r2 0.439 (0.222) 0.439 (0.222) 0.439 (0.222) 0.439 (0.222)Ur2 0.372 (0.176) 0.310 (0.146) 0.254 (0.105) 0.175 (0.071)

0.25 1.00 U 0.824 (0.666) 0.830 (0.666) 0.835 (0.664) 0.824 (0.652)Ur1+r2 0.871 (0.733) 0.871 (0.733) 0.871 (0.733) 0.871 (0.742)Ur2 0.733 (0.598) 0.657 (0.461) 0.517 (0.328) 0.337 (0.186)

When only the second shock is effective1.00 0.75 U 0.120 (0.034) 0.095 (0.023) 0.073 (0.019) 0.061 (0.012)

Ur1+r2 0.132 (0.038) 0.107 (0.029) 0.085 (0.022) 0.064 (0.015)Ur2 0.145 (0.042) 0.139 (0.040) 0.120 (0.042) 0.105 (0.030)

1.00 0.50 U 0.295 (0.121) 0.202 (0.066) 0.127 (0.037) 0.079 (0.018)Ur1+r2 0.355 (0.161) 0.268 (0.114) 0.183 (0.070) 0.108 (0.034)Ur2 0.397 (0.195) 0.349 (0.171) 0.298 (0.135) 0.214 (0.095)

1.00 0.25 U 0.655 (0.424) 0.431 (0.206) 0.238 (0.080) 0.115 (0.025)Ur1+r2 0.785 (0.615) 0.662 (0.467) 0.488 (0.298) 0.266 (0.129)Ur2 0.819 (0.667) 0.756 (0.580) 0.654 (0.462) 0.464 (0.294)


5. Concluding remarks

We have studied an outlier-generating technique through multiple shocks model generalizing theresults of our earlier work [13]. We have also provided maximum likelihood estimator and havederived likelihood ratio test statistic U for testing the effectiveness of the shocks in altering thevalues of the lifelength parameter. Our findings are particularly useful to researchers in the area ofmechanical and structural engineering among others. Further, we have generalized the results ofGather and Kale [6] to obtain the likelihood ratio statistic V for detecting multiple contaminantsunder the labelled slippage model. Our simulation studies show that, in general, the statistic U

performs better than V in terms of power. The results of this article are easily extendable to twoparameter exponential lifelength distribution case.

Acknowledgements

We thank the referee for suggestions and comments which improved this article. This research was partially supported bythe Nanyang Technological University AcRF grant, Singapore and NSERC grant, Canada.

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Technometrics 15(4) (1973), pp. 723–738.[3] G.E.P. Box and G.C. Tiao, A Bayesian approach to some outlier problems, Biometrika 55(1) (1968), pp. 119–129.

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[4] B.K. Kale and S.K. Sinha, Estimation of expected life in the presence of an outlier observation, Technometrics 13(4)(1971), pp. 755–759.

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[10] D.V. Hinkley, Inference about the change-point in a sequence of random variables, Biometrika 57 (1970), pp. 1–17.[11] P. Haccou, E. Meelis, and S. Van de Geer, The likelihood ratio test for the change point problem for exponentially

distributed random variables, Stochast. Process. Appl. 27 (1988), pp. 121–139.[12] K.J. Worsley, Confidence regions and tests for a change-point in a sequence of exponential random variables,

Biometrika 73 (1986), pp. 91–104.[13] V. Gadag and U. Balasooriya, Likelihood ratio test for a shift in the lifelength distribution, Commun. Statist. Theory

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25 (1954), pp. 373–381.[15] A.P. Basu, On some tests of hypotheses relating to the exponential distribution when outliers are present, J. Am.

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Likelihood ratio tests for testing for multiple contaminants in the shocks and labelled slippage models