Power of Global Test in Deformation AnalysisCüneyt Aydin1

Abstract: There are two kinds of global test procedures in deformation analysis; χ2-test (CT) and F-test (FT). This study discusses theirpower functions. The CT is more powerful than the other one in an analytical point of view. However, it requires an accurate knowledge on thea priori variance of unit weight. Therefore, in practice, the FT is mostly chosen. Despite its common usage, a χ2-power function is consideredin the sensitivity design of deformation networks. It is claimed in this study that the F-distribution’s power function should be taken intoaccount, if, in reality, the FT will be applied. Thereby, some boundary values deduced from the noncentral F-distribution to be used insensitivity analysis are computed and tabulated. Furthermore, a simulation for a monitoring network is designed, and it is shown thatthe mean success rates of the two testing procedures are identical with their own powers known beforehand. This numerical experimentdepicts that one should consider the related power function in the design stage, and that each power function gives a realistic probabilityof how the corresponding test procedure is successful. DOI: 10.1061/(ASCE)SU.1943-5428.0000064. © 2012 American Society of CivilEngineers.

CE Database subject headings: Deformation; Tests; Surveys; Sensitivity analysis.

Author keywords: Deformation analysis; Power of the test; Global test; Sensitivity; Mean success rate.

Introduction

Since the late 1970s, the geodetic community has been focused onmonitoring displacements of engineering constructions and theEarth’s crust to determine the movements and/or the deformationsof those bodies (Welsch and Heunecke 2001). During this threedecades, many statistical methods and approaches as well as geo-detic sensors have been applied and developed. Geodetic measure-ment techniques today may yield a few parts per million accuracyin positioning, however, some problems in deformation analysisstill exist. They are mostly because of lack of prior informationon the dynamic structure of the studied object, lack of heteregenousdata required to remove some uncertainties, insufficient point dis-tribution over the object, and observation periods in time. The restof the problems may result from intrinsic problems of the estima-tion approaches and statistical methods used in deformation analy-sis; for instance, improper combination of heteregenous data inadjustment of deformation networks (Aydin and Demirel 2007;Even-Tzur 2004), insufficient datum definition for the correspond-ing deformation parameters, and datum dependency of geodeticallyderived geometrical changes (Even-Tzur 2011; Proszynski 2003;Neitzel 2001), and some testing problems of datum independentquantities, such as strain tensors (Cai et al. 2005; Marjetic et al.2010). On the one hand, every statistical test procedure has itsown capacity to verify the occurance of a random event. Niemeier(1985), therefore, proposes the sensitivity concept to investigatesuch a capacity in our deformation networks by using a power func-tion of the global test. There have been many studies on the sensi-tivity concept in geodesy, such as Kuang (1991), Chen andChrzanowski (1994), Kutterer (1998), Even-Tzur (2002, 2010),

and Wu and Chen (2002). Furthermore, Hekimoglu et al. (2002,2010) define such a capacity in a different way: They measurethe reliabilities of some conventional deformation test proceduresby using the concept of mean success rate proposed by Hekimogluand Koch (1999) for outlier test methods. Their numerical findingsshow that the deformation test procedures may produce misleadingresults depending on the network and object size and on thedisplacement field under consideration.

Before any realization, the power function of the global test tellsus about its statistical behaviors against a deformation model in thedesigned network. For example, it yields the power of the test,i.e., the probability of correctly accepting the alternative hypothesispostulating the existence of the corresponding nonrandom changes.For this, a cumulative distribution function (CDF) in accordancewith the used test statistic’s distribution function is to be used.In sensitivity analysis, the CDF of the noncentral χ2-distributionis taken into account (Aydin and Demirel 2005). However, theglobal test procedure may be realized in two ways; by using a teststatistic following χ2-distribution (χ2-test: CT) and a test statistichaving F-distribution (F-test: FT). For the latter test procedure,which is the one commonly applied in deformation analysis,one should consider the CDF of the noncentral F-distribution in-stead of χ2-distribution for the power function of the global test.Otherwise, sensitivity analysis produces optimistic results, becausethe CT is more powerful than the FT, as will be shown sub-sequently. It is, therefore, important to consider the relevantCDF in power derivation for the deformation studies. For this,we tabulate some boundary values to be used in sensitivity analysisfrom the CDF of the noncentral F-distribution by using the algo-rithm of Gaida and Koch (1985).

On the other hand, it is mostly sensed that the power of the testis only a design parameter used for optimizing deformationnetworks. According to the author, if the network points’ covari-ance matrix may be defined realistically beforehand, it gives a real-istic result about how the considered changes, or their someintervals, will be discriminated against the random errors. For this,a simulation study is considered and the mean success rates ofthe above-mentioned global test procedures are computed for some

1Yildiz Technical Univ., Dept. of Geomatic Engineering, Istanbul,Turkey. E-mail: caydin78@gmail.com

Note. This manuscript was submitted on May 5, 2011; approved onAugust 4, 2011; published online on August 6, 2011. Discussion periodopen until October 1, 2012; separate discussions must be submitted forindividual papers. This paper is part of the Journal of Surveying Engineer-ing, Vol. 138, No. 2, May 1, 2012. ©ASCE, ISSN 0733-9453/2012/2-51–56/$25.00.

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displacements produced according to the corresponding powerfunction. Eventually, it is shown that these estimated mean successrates approximate to the considered probabilities of the global test.

Global Test

The global test is a testing procedure to verify whether the wholenetwork or some part is deformed between two periods of time ornot. For this, the u × 1 estimated displacement vector d, where udenotes the number of coordinates, is considered in two hypothesesas follows:

H0 : EðdÞ ¼ 0; H1 : EðdÞ ≠ 0 ¼ Δ ð1Þwhere H0 and H1 = null-hypothesis and alternative hypothesis,respectively; and Δ = the u × 1 expected displacement vectorderived from a prior information on the dynamic structure of thestudied object. Two kinds of testing procedures may be applieddepending on the chosen variance factor: They are briefly explainedsubsequently.

Test with a priori Variance Factor (CT)

Let the a priori variance of unit weight σ20 be known accurately from

the experiences. In this case, the test statistic θðσÞ is set as follows:

θðσÞ ¼dTQþ

dddσ20

ð2Þ

where Qdd ¼ u × u cofactor matrix of the displacement vector.The distribution of θðσÞ depends on which hypothesis is true:

If the null-hypothesis H0 is true, the test statistic follows a centralχ2-distribution; θðσÞ∼χ2ðhÞ, where h = rank of Qdd . Otherwise, itfollows a noncentral χ2-distribution; θðσÞ ∼ χ02ðh; λÞ with the non-centrality parameter λ. The parameter, the so-called statisticaldistance between the hypotheses, is derived by the expecteddisplacement vector Δ on the alternative space as (Niemeier 1985)

λ ¼ ΔTQþddΔ

σ20

ð3Þ

In a real case, the central χ2-distribution is considered because itis not known whether H1 is actually true. Therefore, θðσÞ is com-pared with the upper α-percentage point of the χ2-distributionχ21�α;h: If θðσÞ > χ2

1�α;h, the null-hypothesis H0 is rejected withthe significance level α. Hence, its alternative is accepted.

Test with Estimated Variance Factor (FT)

In practice, the a priori variance factor σ20 is generally estimated as

the pooled variance s20 from the estimation results of the comparedperiods. One should, therefore, take into account the F-distributioninstead of the χ2-distribution in the global test. For the samehypotheses given in Eq. (1), the test statistic θðsÞ is set as follows(see e.g., Cooper 1987):

θðsÞ ¼dTQþ

dddhs20

ð4Þ

The test statistic θðsÞ follows a central or noncentralF-distributiondepending on which hypothesis is true: When H0 is true, itfollows a central F-distribution, θðsÞ ∼ Fðh; f Þ, where f = totaldegrees of freedom used in estimating of unbiased estimator s20.On thealternativespace, ithasanoncentralone;θðsÞ ∼ F0ðh; f ;λÞwiththe same noncentrality parameterλ as in Eq. (3) (Koch 1999). A stat-istical decision is made having compared the test statisticθðsÞ with the upper α-percentage point of F-distribution F1�α;h;f

assuming that H0 is true: With the significance level α, thenull-hypothesis is rejected if θðsÞ > F1�α;h;f .

Power of the Test

There are two main error types in hypothesis testing; (1) Type-Ierror and (2) Type-II error. The first one, denoted generally signifi-cance level α, is the probability of wrongly rejecting a null-hypothesis. On the other hand, Type-II error shows the probabilityof rejecting the alternative hypothesis although it is actually true.Their complements give a confidence level and power of the test,respectively (see e.g., Teunissen 2000). Hence, the probability ofcorrectly accepting the null-hypothesis expresses the confidencelevel; whereas, the power of the test is the probability of dealingwith correctly accepting the alternative one.

The powers of the global test procedures explained in theprevious section, denoted γðσÞ and γðsÞ, respectively, may be givenas follows (see e.g., Gaida and Koch 1985; Hahn et al. 1989; Koch1999; Aydin and Demirel 2005):

γðσÞ ¼ 1� Fðχ21�α;h; h;λÞ ð5a Þ

γðsÞ ¼ 1� Fðχ21�α;h;f ; h; f ;λÞ ð5b Þ

where Fðχ21�α;h; h;λÞ and FðF1�α;h;f ; h; f ;λÞ = the CDF of the

noncentral χ2-distribution and the noncentral F-distribution, i.e.,Type-II errors, respectively. From the analytical and some geodeticpoints of view, the following properties may be written:

Property 1: Eqs. (5a) and (5b) go to 1 with increasing λ. In otherwords, the more precise network and/or the bigger displacementmagnitude, the bigger the power of the test.

Property 2: Type -II error increases with decreasing Type-I error.Therefore, when the significance level α decreases, the power ofthe test decreases monotonically. For that reason, the α-probabilityis chosen small enough in the test procedures not to cause unde-sirably risen statistical errors.

Property 3: The power of the CT γðσÞ is always bigger thanthe power of the FT γðsÞ for the same displacement vector andnetwork design. The proof may be given as: With increasing thedegrees of freedom f , the Type-II error [FðF1�α;h;f ; h; f ;λÞ]decreases, so the power of the test γðsÞ increases monotonically(see Fig. 1). When f ¼ ∞; γðsÞ becomes identical to γðσÞ becauseFðF1�α;h;∞; h;∞;λÞ ¼ Fðχ2

1�α;h; h;λÞ holds (Abramowitz andStegun 1968, p.947). Hence, with Eqs. (5a) and (5b), the con-clusion follows.

h

Pow

er o

f th

e T

est (

%)

0 10 20 30 40 500

20

40

60

80

100

f=100f=10

f=∞

Fig. 1. Some power curves of F-distribution for the constant values ofα ¼ 5% and λ ¼ 10

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It is also essential to notice that both power functions aremonotonic with changing the rank h: They both decrease withincreasing h (see Fig. 1) when the others, i.e., λ and/or f , are con-stant parameters. This is an analytical interpretation rather thangeodetic one. Because, h is related to the number of points in adeformation network and when h is changed, it is obvious thatthe other parameters may not be constants. Therefore, it is moresuitable to consider how the design of a network affects the powerof the test and/or to derive which magnitude of displacement maybe just detectable with a given power of the test and significancelevel. This leads to apply sensitivity analysis in deformationnetworks.

Sensitivity Concept and Lower Bound ofNon-Centrality Parameter

In geodesy, the noncentrality parameter λ in Eq. (3) is comparedwith its boundary value λ0. If the following inequality:

λ ≥ λ0 ð6Þ

is fullfilled, the network is called sensitive against the expecteddisplacements. Sensitivity optimization of a deformation networkand derivation of minimal detectable displacements, movements, ordeformation parameters depend on the above inequality (Niemeier1985; Kuang 1991; Chen and Chrzanowski 1994; Wu and Chen2002; Even-Tzur 2002, 2010).

The boundary value λ0 is a noncentrality parameter what yieldsa desired power of the test (γ0) in the corresponding power func-tion. In geodesy, because of the lack of real observations in thedesign stage, the degrees of freedom are attained as f ¼ ∞. Hence,the power function in Eq. (5a) is considered to obtain the boundaryvalue as (Baarda 1968; Aydin and Demirel 2005)

λ0;ðσÞ ¼ f ðα; γ0; hÞ ð7Þ

On the other hand, Eq. (5a) and Eq. (7) are valid for the casesin which the CT is applied. Now suppose two assumptions:(1) The monitored displacement vector d has the following nor-mal distribution d∼NðΔ;σ2

0QddÞ with Δ expectation and σ20Qdd

covariance matrix. (2) The expected noncentrality parameter λis equal to the boundary value λ0;ðσÞ in Eq. (7). Thereby, if thetest statistic θðσÞ in Eq. (2) is considered for testing d, it is expectedthat the power of the global test becomes γ0 as desired in Eq. (7).However, when the FT is applied, the power will be less than γ0according to the afore mentioned Property 3. Therefore, a differ-ent definition is needed for the boundary value if one considers theFT: For this, Eq. (5b) is to be solved such that it yields the desiredpower of the test γ0 as

λ0;ðsÞ ¼ f ðα; γ0; h; f Þ ð8Þ

for the attained probability α, the known numerator degrees offreedom h, and denominator degrees of freedom f . For this, Gaidaand Koch (1985) and Hahn et al. (1989) give a bisection algorithmand a Newton-Raphson algorithm, respectively. The algorithm ofGaida and Koch (1985) was chosen to compute the boundary val-ues λ0;ðsÞ for α ¼ 5% and γ0 ¼ 80% as well as for 2 ≤ f ≤ ∞ and1 ≤ h ≤ 30 by using a code written in SciLab [5.3.0]. They aregiven in Table 1 (the parameters for f ¼ 1 are not provided be-cause this case is not so realistic). Although the degrees of free-dom increase, the noncentrality parameters decrease, and, at last,when f ¼ ∞, they become identical to the ones of the noncentralχ2-distribution (see e.g., Aydin and Demirel 2005). Hence, it is

clear that more statistical distance between the hypotheses isneeded to reach the same power of the CT.

Mean Success Rate of Global Test

It is important to know the statistical behaviors of some testingprocedures not only for designing the network properly but alsofor deciding which method is more adaptable and effective. Oneway of doing this is adapting the concept of mean success rate(MSR), which is proposed by Hekimoglu and Koch (1999), to mea-sure the reliability of some outlier test methods applied in geodeticnetworks.

For obtaining the MSR of a test procedure for a definite case, asimulation technique is used. In this technique, some experimentsare simulated properly in accordance with the studied case. Havingapplied the corresponding test procedure for each experiment, thosepassing the corresponding threshold value, the so-called successfulresults, are counted. Let m and p be the number of experiments andthe number of successful results, respectively. Then, the MSR isobtained as follows:

MSR ¼ p∕m ð9Þ

To find the MSR of a global test procedure, the vector d ofdisplacements should be simulated first: Let the network designbe known and be identical in two periods and let Δ be the vectorof displacements whose magnitudes and directions are selectedrandomly from some intervals. Then, considering that the secondperiod observations include the effect of the vectorΔ of the knowndisplacements, two periods’ observations may be created as(Hekimoglu et al. 2002)

y1 ¼ yþ ε1; y2 ¼ yþ ε2 þ XΔ ð10Þ

where y ¼ n × 1 observation vector with no-errors; X ¼ n × uknown coefficient matrix of the network; and εi ¼ ith period’s ran-dom error vector following a common standard normal distributionwith zero expectation and a known covariance matrix. Afterward,the standard procedure in deformation analysis may be realized:Applying free-adjustment for each period independently, the vectorof displacements is obtained as d ¼ β2 − β1 from two estimatedcoordinate vectors (β1 and β2) of the periods. Hence, one of them experiments is created. Performing the corresponding global testprocedure given in the second section to each experiment, the MSRof the test is then easily computed by Eq. (9).

Table 1. Some Noncentrality Parameters Yielding 80% Power of the Testin Noncentral F-Distribution

f h ¼ 1 h ¼ 2 h ¼ 3 h ¼ 5 h ¼ 10 h ¼ 20 h ¼ 30

2 32.0 62.3 92.7 153.4 305.3 608.9 912.4

5 12.4 18.7 24.5 35.7 62.9 116.8 170.5

10 9.7 13.2 16.2 21.7 34.5 59.1 83.4

20 8.7 11.2 13.3 16.8 24.4 38.4 51.9

30 8.4 10.7 12.4 15.4 21.5 32.3 42.5

50 8.2 10.2 11.8 14.3 19.3 27.7 35.2

100 8.0 9.9 11.3 13.6 17.8 24.3 29.9

∞ 7.9 9.6 10.9 12.8 16.2 21.0 24.6

Note: [λ0;ðsÞ ¼ fðα ¼ 5%; γ0 ¼ 80%; h; f Þ].

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Numerical Example

Our aim in this study is to investigate whether the power of thecorresponding global test is compatible with the concept ofMSR. Moreover, it is important to show that one should take intoaccount the boundary value λ0;ðsÞ in Eq. (8) for the sensitivity analy-sis, if actually the FT will be applied after a realization. For that,a leveling network with seven points and 13 height differences (seeFig. 2) is used. Table 2 shows the observation equations in the formof EðyÞ ¼ Xβ, where β ¼ uð¼ 7Þ × 1 vector of unknown heights(β ¼ ½δH1…δH7�T ), and the height differences with no-errors(yi; i ¼ 1;…; 13), and the assumed standard deviations for thedifferences (σi; i ¼ 1;…; 13).

From the definition in Eq. (9), it is clear that the MSR of amethod lies between 0 and 1, or, as a percentage between 0 and100%. Moreover, in this study it is an estimation approach tothe probability of accepting the alternative hypothesis in Eq. (1).Therefore, when it is known that the periods include only randomerrors, it is expected that the MSR goes to the significance level αbecause this error type already defines the probability of wronglyrejecting the null-hypothesis. By using this information, thesimulation model may first be tested: For each experiment inthe simulations, the periods’ observations are produced by usingEq. (10) with Δ ¼ 0. The number of experiments for each trialhas taken as 5,000. The MSRs of the global test procedures (here-after, MSRðσÞ and MSRðsÞdenote the MSRs for the global test with

variance factor σ20 ¼ 4 mm2 and the global test with the estimated

variance factor s20, respectively) are computed for α ¼ 1, 5, and10% considered in the corresponding threshold values (theupper-α percentage points for h ¼ 6 and f ¼ f 1 þ f 2 ¼ 14). Thecomputed MSRs are given in Table 3. As evident from the table,the computed MSRs of the global test procedures are compatiblewith the corresponding significance level α as expected.

Now, the real displacement vector Δ will be created as theminimal detectable displacement (MDD) vector Δmin accordingto the sensitivity concept. From Niemeier (1985)’s definitionthe MDD is considered for a given power of the test γ0 asfollows:

Δmin ¼ τming with τmin ¼ σ0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiλ0∕ðgTQþ

ddgÞq

λ0 ¼�λ0;ðσÞ; for the test with σ2

0λ0;ðsÞ; for the test with s20

ð11Þ

where g = known form vector including �1 for the subsidedpoints, þ1 for the uplifted points, and 0 for the stable points.Furthermore, the form vector may be taken as the eigenvectorbelonging to the maximum eigenvalue of the matrix Qdd to re-present the worst case in the corresponding network (Niemeier1982, 1985).

It is expected in this study that the MSR of the global test goes tothe power of the test, if the MDD vector produced by using Eq. (11)is taken as the expected displacement vector in Eq. (10) for thesimulations. To verify this expectation, we compute someMSRðσÞ and MSRðsÞ having created the MDDs for the probabilitiesγ0 ¼ 80; 70; 60, and 50% and α ¼ 5%. Table 4 shows the MSRsfor the MDDs created for the worst case (Column I) and randomlychosen three displaced points with random signs (Column II) in thenetwork, respectively. It is evident that the MSRs are compatiblewith the corresponding power of the test as expected (the maximumdifference between the MSR and the power of the test is 0.7%).Although the MSRs for two global test procedures are identicalin that table, the reader should notice that the magnitudes of theMDDs incorporated into the observations for the FT are bigger thanthe other one.

This numerical example clarifies that the power function inEq. (5b) is to be used in sensitivity analysis if the FT is appliedfor testing of the estimated displacements. Because this kind ofglobal test is mostly used, it is recommended that the sensitivityanalysis should be performed by using the boundary values givenin Table 1. However, as mentioned previously, this means thatbigger displacement magnitudes or more precise networks areneeded to reach the same power of the CT. For example, supposethat just first four points in the network are displaced betweentwo periods, and the displacement vector is expected asΔ ¼ ½5 3 4 5 0 0 0�T ðmmÞ. The noncentrality parameter λ inEq. (3) then becomes 8.0859. For the same statistical distanceλ, the powers of the CT and the FT are computed as 53.6 and36.3% from Eqs. (5a) and (5b) for the constant Type-I error

1

3

4

6

7

2

5

Fig. 2. Used leveling network configuration

Table 2. Coefficients between Unknown Heights and Observations,Height Differences (yi), and their Standard Deviations (σi)

Observation δH1 δH2 δH3 δH4 δH5 δH6 δH7 yi (m) σi (mm)

1 �1 1 0 0 0 0 0 1 2

2 0 �1 1 0 0 0 0 1 2

3 0 �1 0 1 0 0 0 3 2

4 0 0 �1 1 0 0 0 2 2

5 0 0 0 �1 1 0 0 2 2

6 0 0 0 0 1 �1 0 7 2

7 1 0 0 0 0 �1 0 1 2

8 1 0 0 0 0 0 �1 2 8

9 0 0 0 0 1 0 �1 8 8

10 0 1 0 0 0 0 �1 3 8

11 0 0 0 1 0 0 �1 6 8

12 0 0 1 0 0 0 �1 4 8

13 0 0 0 0 0 1 �1 1 8

Table 3. Computed MSRs in Global Test While the Periods Include onlyRandom Errors

Significancelevel (%)

MSRðσÞ(%)

MSRðsÞ(%) χ2

1�α;h F1�α;h; f

α ¼ 1 0.94 1.18 16.81 4.46

α ¼ 5 4.94 5.08 12.59 2.85

α ¼ 10 9.70 10.18 10.64 2.24

Note: Each MSR is a result of 5,000 experiments.

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α ¼ 5%, respectively. To increase the second power of the test to53.6%, the statistical distance λ is to be 12.1937 from Eq. (8).

Conclusion

To verify whether a deformation monitoring network is deformed,a testing procedure, the so-called global test, is applied. The testmay be realized in two ways; (1) by using a test statistic havinga χ2-distribution (CT) and (2) a test statistic following anF-distribution (FT). In this study, their power functions were stud-ied, and so, the power of the global test in deformation analysis.For simplification, only verification of displacements was consid-ered throughout the paper. The other kind of changes, such asmovements and deformation parameters, may be easily adaptedinto the power functions with the relevant models. But this wasthe out of the scope of the study.

The power of the global test is used in sensitivity analysis toderive the capacity of a monitoring network against the expecteddisplacements, movements, or deformations. It is indicated that thispower of the test should be from the corresponding distributionfunction of the test statistics, which will be applied after a realiza-tion. For instance, if one will use the FT in a real application, thesensitivity analysis should be realized by using the power of the FT.However, the tabulated values (lower bound of the noncentralityparameters) in geodesy are the functions of the power of theCT. The results of the sensitivity analysis obtained by using thesetabulated values become optimistic for the global test procedureperformed by the FT because the CT is more powerful than theFT as shown also in this study. Some boundary values are, there-fore, given from the power function of the F-distribution to be usedin sensitivity analysis. The corresponding program computing thevalues for any parameter is available from the author upon request.

Moreover, the aim of the study was to determine how the powerof the global test is a realistic probability in deformation networks.For this, a leveling network is simulated, and the MSR of each testis computed for some displacements produced according to thepower of the corresponding test. The obtained MSR results forthe CT and the FT are compatible with their own powers of the testas expected. The numerical experiment supports the idea about us-ing the relevant power of the test in geodetic networks and showsthat the power of the global test may be used also to derive a real-istic capacity of our networks in verification of deformations.

Acknowledgments

The mentioned code in this study has been written with Scilab 5.3.1,free and open source software (distributed under CeCILL license-GPL compatible) developed by the Scilab Consortium—Digiteo.The author thanks the editor, two anonymous reviewers, and SimayAtayer for helpful comments.

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Table 4. Computed MSRs for the MDDs in the Worst Case (I) and for the Randomly Chosen Three Points (II)

Power of the test

Global test with known variance factor (χ2-test) Global test with estimated variance factor (F-test)

MSRðσÞ (I) MSRðσÞ (II) λ0;ðσÞ τmina(mm) MSRðsÞ(I) MSRðsÞ(II) λ0;ðsÞ τmin

a (mm)

γ0 ¼ 80% 80.64 79.46 13.62 15.78 80.24 80.56 20.79 19.50

γ0 ¼ 70% 69.54 70.66 11.14 14.27 70.02 70.62 16.91 17.58

γ0 ¼ 60% 60.14 60.30 9.19 12.96 60.74 59.62 13.89 15.94

γ0 ¼ 50% 50.08 50.08 7.50 11.71 50.22 49.80 11.30 14.37

Note: Each MSR (%) is computed from 5,000 experiments; α ¼ 5%.aτmin values denote the scale factors for the worst case.

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