Holmquist, Björn

PirouziFard, MirNabi

Lund University Publications

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00Goodness of fit tests for the extreme value distribution based on regression, EDF andthe stabilized probability plotPirouziFard, MirNabi; Holmquist, Björn2008Link to publicationCitation for published version (APA):PirouziFard, M., & Holmquist, B. (2008). Goodness of fit tests for the extreme value distribution based onregression, EDF and the stabilized probability plot. 152-162. Paper presented at 7th International Conference onSocial Science MEthodology.Total number of authors:2General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portalRead more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.           Goodness of fit tests for the extreme value distribution based on regression, EDF  and the stabilized probability plot   Mir Nabi Pirouzi Fard  and  Björn Holmquist                                                                                      2007:1                                                                                    DEPARTMENT OF  STATISTICS   S-220 07 LUND SWEDEN Goodness of fit tests for the extreme value distribution based onregression, EDF and the stabilized probability plotBy MIR NABI PIROUZI FARD and BJÖRN HOLMQUISTDepartment of Statistics, Lund University, Box 743, S-220 07 Lund, Swedenmirnabi.pirouzifard@stat.lu.se, bjorn.holmquist@stat.lu.seSummaryProcedures for goodness-of-fit test of the extreme value distribution are inves-tigated. The different procedures are all based on order statistics and take intoaccount the dependence structure within the ordered sample. The power of the teststatistics are examined, and shows that two tests that outperform the others can befound.Some key words: Empirical Distribution Function; Extreme value distribution; Generalized LeastSquares; Means, variances and covariances of order statistics; Regression test.1. IntroductionThe extreme value distribution is widely used in lifetime testing, in the study ofsize effects on material strengths, the reliability of systems made up of a large numberof components, in assessing the level of air pollution and in the flood frequencyanalysis. This distribution has an important role in modelling lifetime data andhence considerable efforts have been dedicated to testing the hypothesis of extremevalue distribuion. For reviews of the subject the reader is referred to D’Agostinoand Stephens (1986) and Balakrishnan and Rao (1998).Let Y have an extreme value distribution with cumulative distribution functionF (y) = 1 − exp(− exp(y − αβ)) −∞ < y < ∞ (1)where the parameters β > 0 and −∞ < α < ∞. The mean and variance of thisdistribution, sometimes referred to as the Gumbel distribution, are respectively,E(Y ) = α − γβ var(Y ) = β2π2/6where γ ≈ 0.57721 is Euler’s constant.In this paper we consider tests of fit based either on regression, the EmpiricalDistribution Function (EDF) or the stabilized probability plot. A test of fit is a testof H0: a random sample of Y -values comes from an extreme value distribution withunknown parameters α and β. In section 2 we discuss the test statistics for randomsamples from an extreme value distribution. The results of the power comparisonsand tables of significance points are given in section 3.2. Test statisticsGoodness of fit tests mostly require estimation of location and scale parametersin the tested distribution F (y) which is the cumulative distribution function in (1).Let x1 ≤ · · · ≤ xn denote an ordered random sample of size n from equation (1)with α = 0 and β = 1, andmi = E(xi) (i = 1, . . . , n) and σij = cov(xi, xj) (i, j = 1, . . . , n)Define m to be the (n × 1) vector of the expected values mi, Σ the (n × n)matrix of variances and covariances σij . If we let Z be a vector of ordered random2observations from equation (1) for general α and β, then the elements zi of Z maybe expressed aszi = µ + θxi, i = 1, . . . , n (2)where µ = α is a location parameter and θ = β a scale parameter.In a regression modelzi = µ + θmi + εi, i = 1, . . . , n (3)the points of the n pairs (mi, zi) should be approximately a straight line with inter-cept µ on the vertical axis and slope θ. The parameters µ and θ, in equation (3)can be estimated by a suitable method.The Ordinary Least Squares (OLS) is a method for obtaining estimates of pa-rameters in equation (3). If the variances of the dependent variable in equation (3)are constant and the covariances are equal to 0 the estimated parameters by thismethod are the minimum variance linear unbiased estimators of µ and θ.But since the observed values in equation (3) are order statistics with var(zi) =θ2var(xi) and cov(zi, zj) = θ2cov(xi, xj), and var(xi) depend on i and also cov(xi, xj)depends on i and j. Thus the OLS estimators will not be the minimum varianceestimators.The best linear unbiased estimates of µ and θ can be obtained from the general-ized least-squares (GLS) regression of the order statistic (Aitken, 1935; Lloyd, 1952)and are given byµ̂ =m′Σ−1(m1′ − 1m′)Σ−1Z1′Σ−11m′Σ−1m − (1′Σ−1m)2 (4)andθ̂ =1′Σ−1(1m′ − m1′)Σ−1Z1′Σ−11m′Σ−1m − (1′Σ−1m)2 (5)3where 1 is a n-dimensional vector of ones.Thus the ith fitted value ẑi is given by the equationẑi = µ̂ + θ̂mi i = 1, . . . , n (6)The use of GLS requires information on the expected values and the variance-covariance matrix of the order statistics from the standard extreme value distribu-tion.Lieblein and Zelen (1956) presented the expected values, variances and covari-ances of the order statistics from the standard extreme value distribution for n =1(1)6. Lieblein and Salzer (1957) presented a table of expected values of order statis-tics for n = 1(1)10(5)25 and the first 26 largest values for n = 30(5)60(10)100. White(1967, 1969), tabulated means and variances of order statistics for n = 1(1)50(5)100.For sample sizes n = 1(1)15(5)30, Balakrishnan and Chan (1992) presented tablesof means, variances and covariances of the order statistics.To calculate µ̂ and θ̂ in equations (4) and (5) we use the approximate values ofmeans of order statistics, suggested by Pirouzi Fard and Holmquist (2007)mi ≈− log n − γ for i = 1,log(− log(1 − ((i − 0.4866)/(n + 0.1840)))) for i = 2, ..., n,(7)We also apply the approximate variances and covariances of the order statistics,given by Pirouzi Fard and Holmquist (2006) in an unpublished statistical researchreport as4σij ≈π2/6 for i = j = 1(i−0.469)[(n+0.831−i)(n+0.073)]−1log(n+0.831−in+0.356) log(n+0.779−jn+0.356)1 ≤ i ≤ j ≤ n(8)and σji = σij is the covariance of the ith and jth order statistics of standard extremevalue distribution.2.1 Regression tests based on residualsWe are interested in using residuals to test how well the data fit {ẑi}. Theresiduals can be expressed as zi − ẑi i.e. the differences between the observed valuesand the values given by the model. We will examine two methods to measurelinearity of data. The first method of measure linearity is the error (or unexplained)sum of squares (ESS) divided by the total sum of squares (TSS) given byT1 =ESSTSS=(Z − Ẑ)′(Z − Ẑ)(Z − Z̄1)′(Z − Z̄1)where Ẑ is the vector of estimated values (by using the GLS regression) with elementsgiven in equation (6) and Z̄ = (1′Z)/(1′1).The other method to measure linearity of data isT2 =GESSGTSS=(Z − Ẑ)′Σ−1(Z − Ẑ)(Z − Z̄∗1)′Σ−1(Z − Z∗1)where GESS is the generalized error (or unexplained) sum of squares, GTSS is thegeneralized total sum of squares and Z̄∗ is a constant which Buse (1973) has definedas (1′Σ−1Z)/(1′Σ−11). Both measures of linearity are location and scale invariant.52.2 Test of fit based on EDFLet x1, . . . , xn be the order statistics from a continuous distribution functionF (y) of size n. The empirical distribution function for the sample, given byFn(y) =number of observations ≤ yn; −∞ < y < ∞is a step function with a step size 1/n at the order statistics x1 < · · · < xn. Thedistance between the EDF and the hypothesized distribution, F (y) = F̂ (y) can beconsidered as a way of testing for H0. Large values of the test statistics indicatethat H0 should be rejected. In our case, F̂ (y) is given byF̂ (y) = 1 − exp(−e(y−µ̂)/θ̂)i.e. the estimated cumulative distribution function of the extreme value distribution,where the parameters are estimated by GLS according to equations (4) and (5).In this study we discuss three such tests. The first of the EDF-based test weconsider is the Dn statistic:Dn = supy|Fn(y) − F̂ (y)| = max(D+n , D−n )introduced by Kolmogorov-Smirnov, where D+n and D−n is obtained byD+n = max1≤i≤n(in− F̂ (xi)), D−n = max1≤i≤n(F̂ (xi) − i − 1n)The second EDF test statistic is the Cramer-von Mises statistic, W 2n :W 2n = n∫ ∞−∞[Fn(y) − F̂ (y)]2dF̂ (y)6which can also be expressed by the formulaW 2n =112n+n∑i=1(F̂ (xi) − i − 0.5n)2The third EDF test is the Anderson-Darling statistic, A2n:A2n = n∫ ∞−∞(Fn(y) − F̂ (y))2F̂ (y)(1 − F̂ (y)) dF̂ (y)This test statistic can be calculated by the formulaA2n = −n −n∑i=12i − 1n[log(F̂ (xi)) + log(1 − F̂ (x(n+1−i)))]2.3 Test of fit based on stabilized probability plotLet x1, . . . , xn be the ordered observations in a random sample of size n from thedistribution of the form F (x) = F0((x−µ)/θ), where µ is a location parameter andθ is a scale parameter. A probability plot is a plot of the xi versus a correspondingtheoretical quantities ui = F−10 (di), where di is an estimate of F0((xi − µ)/θ). Insuch a plot the points should lie fairly near the line xi = µ+θui, and it indicates thatthe hypothesized distribution is a resonable model for the data. The interpretationof the plot can be complicated due to the existence of outliers and the unequalvariances of the plotted points. The stabilized probability plot is introduced byMichael (1983) to handle the problem. The plot is formed by plottingsi = (2/π)sin−1[F 0.50 {(xi − µ)/θ}] (9)againstri = (2/π)sin−1[{(i − 0.5)/n}0.5] (10)7where according to Michael (1983), s follows the sine distribution and all its orderstatistics have the same asymptotic variance. Hence by this transformation, thevariance of the plotted points are approximately equal over the range of probabilityvalues.A goodness-of-fit statistic based on stabilized probability plot is also suggestedby Michael (1983) asDsp = max |ri − si| (11)Kimber (1985) used the statistic Dsp for testing of the extreme value distributionof maxima. He applied Downton’s estimates (1966) of µ and θ, to obtain the criticalvalues. Coles (1989) investigated the statistic Dsp for testing the extreme valuedistribution of minima and denoted it by D∗sp. He estimated the parameters αand β in equation (1), by using Blom’s procedure (1958) and showed that due tothe improved estimation procedure, the test statistic D∗sp had higher power thanKimber’s proposed test statistic.The best linear unbiased estimates of the parameters whose distribution functionis of a location - scale form have been considered by Lloyd (1952). Therefore weestimate the parameters in equation (1) by using equations (4) and (5) to get thetest statisticD+sp = max |ri − ŝi| (12)where ŝi is obtained from equation (9) by replacing µ and θ by the estimates from(4) and (5).A power comparison of the D∗sp, D+sp and some other test statistics are given insection 3.83 The results of the monte carlo study3.1 Power of the testsThe power of test statistics have been examined for samples from equation (1)against a range of alternative distributions based on 40000 replicates. As mentioned,the purpose of this study is to apply the expected values, variances and covariancesof order statistics of the standard extreme value distribution in order to estimatethe unknown parameters in equation (1) by GLS regression.In Tables 1 and 2 the power of the test statistics by using approximations (equa-tions (7) and (8)) are compared with the power when exact values are used forsample size 20. These tables show that the differences are negligible, most of thedifferences being in the second decimal place. The power also reveals that the teststatistic T1 has higher power in comparison with the other test statistics for manyof the alternatives.In the case of the extreme value distribution, Stephens (1977) determined ap-proximate critical values of the Cramer-von Mises statistic when using the maximumlikelihood estimates of µ and θ. The W 2n and A2n tests have also been studied byLittel et al. (1979), in which case again the maximum likelihood estimates wereused in substitution for µ and θ.Spinelli (1980), in an unpublished M.Sc. Thesis at Simon Fraser University, stud-ied the test statistics based on regression and EDF for the extreme value distribution,of maxima. He pointed out problems arising when using the GLS method for theextreme value distribution due to the unavailability of the variances and covariancesof the order statistics for large sample sizes.9We have also performed a power study for the extreme value distribution, ofmaxima. Comparison of the power of the tests for some different distributions withthe results presented by Spinelli show that our results seems to be of the same orderas Spinelli’s results except for T2 statistic. Our result mostly give lower power forthe T2 statistic than is indicated in Spinelli’s study.The test based on the T1 statistic is in terms of power, superior to most otherstest statistics in this study. The test based on D+sp is almost of the same power asthe T1 test.Table 1 also shows that due to the improved parameter estimation, the teststatistic D+sp is generally more powerful than D∗sp. The procedure for parameterestimation used in D+sp is simple to implement which is another advantage of thistest statistic.3.2 Percentage PointsThe critical values of the T1 statistic for n = 10(10)100 are given in Table 3. Inthis table for n = 10(10)30, the percentage points of the statistics based on boththe exact and approximate values of means and covariances. Table 3 reveals thatthe approximation yield an error less than 6 % for all sample sizes. This encouragesus to trust the use of approximations of means and variances in calculating the teststatistics.ReferencesAITKEN, A.C. (1935). On least squares and linear combination of observations. Proc. Roy. Soc.Edinb., 55, 42.10BLOM, G. (1958). Statistical Estimates and Transformed Beta-Variables. Almqvist and Wiksell,Uppsala.BALAKRISHNAN, N. & CHAN, P.S. (1992). Order statistics from extreme value distribution I:Tables of means, variances and covariances, Commun. Stat. Sim. Comp. 21, 1199-1217.BALAKRISHNAN, N. & Rao, C.R. (1998). Handbook of Statistics 17, Elsevier, Amsterdam.BUSE, A. (1973). Goodness of fit in Generalized Least Squares Estimation. Amer. Statistician27, 106-108.COLES, S.G (1989). On Goodness-of-Fit Tests for the Two-Parameter Weibull Distribution De-rived from Stabilized Probability Plot. Biometrika 76, 593-598.D’AGOSTINO, R.B. & STEPHENS, M.A. (1986). Goodness-of-fit techniques, Marcel Dekker, NewYork.DOWNTON, F. (1966). Linear Estimations of Parameters in the Extreme Value Distribution.Technometrics 8 3-17.KIMBER, A.C. (1985). Tests for Exponential, Weibull and Gumbel Distributions based on theStabilized Probability Plot. Biometrika 72, 661-663.LIEBLEIN, J. & SALZAER, H.E. (1957). Table of the first moment of ranked extremes. J. Res.Nat. Bur. Stand. 59, 203-206.LIEBLEIN, J. & ZELEN, M. (1956). Statistical investigation of the fatigue life of deep groove ballbearing. J. Res. Nat. Bur. Stand. 57, 273-316.LITTEL, R.C., McCLAVE, J.T. & OFFEN, W.W. (1979). Goodness of fit tests for two-parameterWeibull distribution. Commun. Stat. B8, 257-269.LLOYD, E.H. (1952). Least-Squares Estimation of Location and Scale Parameters Using OrderStatistics, Biometrika 39, 88-95.MICHAEL, J.R. (1983). The stabilized probability plot. Biometrika 70, 11-17.11PIROUZI FARD, M.N. & HOLMQUIST, B. (2007). First moment approximations for order statis-tics from the extreme value distribution. Statistical Methodology, 4, 196-203.PIROUZI FARD, M.N. & HOLMQUIST, B. (2006). Approximations of variances and covariancesfor order statistics from the standard extreme value distribution, Statistical research report, No.2006:1, Lund University, Sweden.SPINELLI, J. (1980). Contributions to goodness-of-fit. M.Sc. Thesis, Department of Mathemat-ics, Simon Fraser University. STEPHENS M.A. (1977). Goodness of Fit for the Extreme ValueDistribution. Biometrika, 64 583-588.WHITE, J.S. (1967). The moments of log-Weibull order statistics, Research Publication GMR-717,Research Laboratories, General Motors Corp., Warren, Mich.WHITE, J.S. (1969). Moments of Log-Weibull Order Statistics, Technometrics 11, 373-386.12Table 1: Empirical power for tests on level 0.10, for selected alternative distribu-tions based on 40000 replicates by using the exact values of means, variances andcovariances of order statistics for n=20. 1Density 4(1 − x)3, 0 < x < 1. 2Density116xe−x/4, x ≥ 0. 3Density 10x4e−2x5 , x ≥ 0.Distribution T1 T2 W2n A2n Dn D+sp D∗spU(0, 1) 0.46 0.22 0.34 0.38 0.27 0.47 0.40Beta(1,4)1 0.98 0.41 0.88 0.92 0.81 0.98 0.96N(0, 1) 0.43 0.13 0.32 0.33 0.26 0.38 0.33Student t(6) 0.48 0.21 0.42 0.44 0.35 0.45 0.42Gamma(2,4)2 0.97 0.46 0.87 0.90 0.78 0.96 0.93χ2(5) 0.95 0.42 0.83 0.86 0.73 0.93 0.90Weibull(2,5)3 0.26 0.10 0.19 0.20 0.17 0.24 0.20Empirical CriticalValues, level= 0.10 0.14 0.47 0.10 0.61 0.17 0.11 0.1113Table 2: Empirical power for tests on level 0.10, for selected alternative distributionsbased on 40000 replicates by using the estimated values of means, variances andcovariances of order statistics for n=20.Distribution T1 T2 W2n A2n Dn D+sp D∗spU(0, 1) 0.46 0.23 0.35 0.39 0.28 0.47 0.40Beta(1,4) 0.98 0.42 0.88 0.92 0.81 0.98 0.97N(0, 1) 0.44 0.14 0.32 0.33 0.26 0.38 0.33Student t(6) 0.48 0.21 0.42 0.44 0.35 0.45 0.41Gamma(2,4) 0.97 0.47 0.87 0.90 0.78 0.96 0.94χ2(5) 0.95 0.42 0.83 0.87 0.73 0.94 0.90Weibull(2,5) 0.27 0.10 0.19 0.20 0.17 0.24 0.20Empirical CriticalValues, level= 0.10 0.13 0.46 0.10 0.61 0.17 0.11 0.1114Table 3: Quantiles of the test statistic T1 based on 40000 replicates.n 0.80 0.85 0.90 0.95 0.975 0.9910 0.15 0.17 0.21 0.27 0.33 0.41 exact0.15 0.17 0.20 0.26 0.32 0.41 approx.20 0.10 0.11 0.14 0.17 0.22 0.28 exact0.10 0.11 0.14 0.18 0.22 0.28 approx.30 0.07 0.09 0.10 0.13 0.17 0.21 exact0.07 0.09 0.10 0.13 0.17 0.21 approx.40 0.06 0.07 0.09 0.11 0.14 0.1850 0.05 0.06 0.07 0.10 0.12 0.1560 0.05 0.05 0.06 0.08 0.10 0.1370 0.04 0.05 0.06 0.07 0.09 0.1280 0.04 0.04 0.05 0.07 0.08 0.1190 0.03 0.04 0.05 0.06 0.08 0.10100 0.03 0.04 0.04 0.06 0.07 0.0915

English

Goodness of fit tests for the extreme value distribution based on regression, EDF and the stabilized probability plot

LUND UNIVERSITYPO Box 117221 00 Lund+46 46-222 00 00Goodness of Fit Tests for the Extreme Value Distribution based on Regression, EDFand the Stabilized Probability PlotPirouziFard, MirNabi; Holmquist, Björn2008Link to publicationCitation for published version (APA):PirouziFard, M., & Holmquist, B. (2008). Goodness of Fit Tests for the Extreme Value Distribution based onRegression, EDF and the Stabilized Probability Plot.Total number of authors:2General rightsUnless other specific re-use rights are stated the following general rights apply:Copyright and moral rights for the publications made accessible in the public portal are retained by the authorsand/or other copyright owners and it is a condition of accessing publications that users recognise and abide by thelegal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private studyor research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portalRead more about Creative commons licenses: https://creativecommons.org/licenses/Take down policyIf you believe that this document breaches copyright please contact us providing details, and we will removeaccess to the work immediately and investigate your claim.           Goodness of fit tests for the extreme value distribution based on regression, EDF  and the stabilized probability plot   Mir Nabi Pirouzi Fard  and  Björn Holmquist                                                                                      2007:1                                                                                    DEPARTMENT OF  STATISTICS   S-220 07 LUND SWEDEN Goodness of fit tests for the extreme value distribution based onregression, EDF and the stabilized probability plotBy MIR NABI PIROUZI FARD and BJÖRN HOLMQUISTDepartment of Statistics, Lund University, Box 743, S-220 07 Lund, Swedenmirnabi.pirouzifard@stat.lu.se, bjorn.holmquist@stat.lu.seSummaryProcedures for goodness-of-fit test of the extreme value distribution are inves-tigated. The different procedures are all based on order statistics and take intoaccount the dependence structure within the ordered sample. The power of the teststatistics are examined, and shows that two tests that outperform the others can befound.Some key words: Empirical Distribution Function; Extreme value distribution; Generalized LeastSquares; Means, variances and covariances of order statistics; Regression test.1. IntroductionThe extreme value distribution is widely used in lifetime testing, in the study ofsize effects on material strengths, the reliability of systems made up of a large numberof components, in assessing the level of air pollution and in the flood frequencyanalysis. This distribution has an important role in modelling lifetime data andhence considerable efforts have been dedicated to testing the hypothesis of extremevalue distribuion. For reviews of the subject the reader is referred to D’Agostinoand Stephens (1986) and Balakrishnan and Rao (1998).Let Y have an extreme value distribution with cumulative distribution functionF (y) = 1 − exp(− exp(y − αβ)) −∞ < y < ∞ (1)where the parameters β > 0 and −∞ < α < ∞. The mean and variance of thisdistribution, sometimes referred to as the Gumbel distribution, are respectively,E(Y ) = α − γβ var(Y ) = β2π2/6where γ ≈ 0.57721 is Euler’s constant.In this paper we consider tests of fit based either on regression, the EmpiricalDistribution Function (EDF) or the stabilized probability plot. A test of fit is a testof H0: a random sample of Y -values comes from an extreme value distribution withunknown parameters α and β. In section 2 we discuss the test statistics for randomsamples from an extreme value distribution. The results of the power comparisonsand tables of significance points are given in section 3.2. Test statisticsGoodness of fit tests mostly require estimation of location and scale parametersin the tested distribution F (y) which is the cumulative distribution function in (1).Let x1 ≤ · · · ≤ xn denote an ordered random sample of size n from equation (1)with α = 0 and β = 1, andmi = E(xi) (i = 1, . . . , n) and σij = cov(xi, xj) (i, j = 1, . . . , n)Define m to be the (n × 1) vector of the expected values mi, Σ the (n × n)matrix of variances and covariances σij . If we let Z be a vector of ordered random2observations from equation (1) for general α and β, then the elements zi of Z maybe expressed aszi = µ + θxi, i = 1, . . . , n (2)where µ = α is a location parameter and θ = β a scale parameter.In a regression modelzi = µ + θmi + εi, i = 1, . . . , n (3)the points of the n pairs (mi, zi) should be approximately a straight line with inter-cept µ on the vertical axis and slope θ. The parameters µ and θ, in equation (3)can be estimated by a suitable method.The Ordinary Least Squares (OLS) is a method for obtaining estimates of pa-rameters in equation (3). If the variances of the dependent variable in equation (3)are constant and the covariances are equal to 0 the estimated parameters by thismethod are the minimum variance linear unbiased estimators of µ and θ.But since the observed values in equation (3) are order statistics with var(zi) =θ2var(xi) and cov(zi, zj) = θ2cov(xi, xj), and var(xi) depend on i and also cov(xi, xj)depends on i and j. Thus the OLS estimators will not be the minimum varianceestimators.The best linear unbiased estimates of µ and θ can be obtained from the general-ized least-squares (GLS) regression of the order statistic (Aitken, 1935; Lloyd, 1952)and are given byµ̂ =m′Σ−1(m1′ − 1m′)Σ−1Z1′Σ−11m′Σ−1m − (1′Σ−1m)2 (4)andθ̂ =1′Σ−1(1m′ − m1′)Σ−1Z1′Σ−11m′Σ−1m − (1′Σ−1m)2 (5)3where 1 is a n-dimensional vector of ones.Thus the ith fitted value ẑi is given by the equationẑi = µ̂ + θ̂mi i = 1, . . . , n (6)The use of GLS requires information on the expected values and the variance-covariance matrix of the order statistics from the standard extreme value distribu-tion.Lieblein and Zelen (1956) presented the expected values, variances and covari-ances of the order statistics from the standard extreme value distribution for n =1(1)6. Lieblein and Salzer (1957) presented a table of expected values of order statis-tics for n = 1(1)10(5)25 and the first 26 largest values for n = 30(5)60(10)100. White(1967, 1969), tabulated means and variances of order statistics for n = 1(1)50(5)100.For sample sizes n = 1(1)15(5)30, Balakrishnan and Chan (1992) presented tablesof means, variances and covariances of the order statistics.To calculate µ̂ and θ̂ in equations (4) and (5) we use the approximate values ofmeans of order statistics, suggested by Pirouzi Fard and Holmquist (2007)mi ≈− log n − γ for i = 1,log(− log(1 − ((i − 0.4866)/(n + 0.1840)))) for i = 2, ..., n,(7)We also apply the approximate variances and covariances of the order statistics,given by Pirouzi Fard and Holmquist (2006) in an unpublished statistical researchreport as4σij ≈π2/6 for i = j = 1(i−0.469)[(n+0.831−i)(n+0.073)]−1log(n+0.831−in+0.356) log(n+0.779−jn+0.356)1 ≤ i ≤ j ≤ n(8)and σji = σij is the covariance of the ith and jth order statistics of standard extremevalue distribution.2.1 Regression tests based on residualsWe are interested in using residuals to test how well the data fit {ẑi}. Theresiduals can be expressed as zi − ẑi i.e. the differences between the observed valuesand the values given by the model. We will examine two methods to measurelinearity of data. The first method of measure linearity is the error (or unexplained)sum of squares (ESS) divided by the total sum of squares (TSS) given byT1 =ESSTSS=(Z − Ẑ)′(Z − Ẑ)(Z − Z̄1)′(Z − Z̄1)where Ẑ is the vector of estimated values (by using the GLS regression) with elementsgiven in equation (6) and Z̄ = (1′Z)/(1′1).The other method to measure linearity of data isT2 =GESSGTSS=(Z − Ẑ)′Σ−1(Z − Ẑ)(Z − Z̄∗1)′Σ−1(Z − Z∗1)where GESS is the generalized error (or unexplained) sum of squares, GTSS is thegeneralized total sum of squares and Z̄∗ is a constant which Buse (1973) has definedas (1′Σ−1Z)/(1′Σ−11). Both measures of linearity are location and scale invariant.52.2 Test of fit based on EDFLet x1, . . . , xn be the order statistics from a continuous distribution functionF (y) of size n. The empirical distribution function for the sample, given byFn(y) =number of observations ≤ yn; −∞ < y < ∞is a step function with a step size 1/n at the order statistics x1 < · · · < xn. Thedistance between the EDF and the hypothesized distribution, F (y) = F̂ (y) can beconsidered as a way of testing for H0. Large values of the test statistics indicatethat H0 should be rejected. In our case, F̂ (y) is given byF̂ (y) = 1 − exp(−e(y−µ̂)/θ̂)i.e. the estimated cumulative distribution function of the extreme value distribution,where the parameters are estimated by GLS according to equations (4) and (5).In this study we discuss three such tests. The first of the EDF-based test weconsider is the Dn statistic:Dn = supy|Fn(y) − F̂ (y)| = max(D+n , D−n )introduced by Kolmogorov-Smirnov, where D+n and D−n is obtained byD+n = max1≤i≤n(in− F̂ (xi)), D−n = max1≤i≤n(F̂ (xi) − i − 1n)The second EDF test statistic is the Cramer-von Mises statistic, W 2n :W 2n = n∫ ∞−∞[Fn(y) − F̂ (y)]2dF̂ (y)6which can also be expressed by the formulaW 2n =112n+n∑i=1(F̂ (xi) − i − 0.5n)2The third EDF test is the Anderson-Darling statistic, A2n:A2n = n∫ ∞−∞(Fn(y) − F̂ (y))2F̂ (y)(1 − F̂ (y)) dF̂ (y)This test statistic can be calculated by the formulaA2n = −n −n∑i=12i − 1n[log(F̂ (xi)) + log(1 − F̂ (x(n+1−i)))]2.3 Test of fit based on stabilized probability plotLet x1, . . . , xn be the ordered observations in a random sample of size n from thedistribution of the form F (x) = F0((x−µ)/θ), where µ is a location parameter andθ is a scale parameter. A probability plot is a plot of the xi versus a correspondingtheoretical quantities ui = F−10 (di), where di is an estimate of F0((xi − µ)/θ). Insuch a plot the points should lie fairly near the line xi = µ+θui, and it indicates thatthe hypothesized distribution is a resonable model for the data. The interpretationof the plot can be complicated due to the existence of outliers and the unequalvariances of the plotted points. The stabilized probability plot is introduced byMichael (1983) to handle the problem. The plot is formed by plottingsi = (2/π)sin−1[F 0.50 {(xi − µ)/θ}] (9)againstri = (2/π)sin−1[{(i − 0.5)/n}0.5] (10)7where according to Michael (1983), s follows the sine distribution and all its orderstatistics have the same asymptotic variance. Hence by this transformation, thevariance of the plotted points are approximately equal over the range of probabilityvalues.A goodness-of-fit statistic based on stabilized probability plot is also suggestedby Michael (1983) asDsp = max |ri − si| (11)Kimber (1985) used the statistic Dsp for testing of the extreme value distributionof maxima. He applied Downton’s estimates (1966) of µ and θ, to obtain the criticalvalues. Coles (1989) investigated the statistic Dsp for testing the extreme valuedistribution of minima and denoted it by D∗sp. He estimated the parameters αand β in equation (1), by using Blom’s procedure (1958) and showed that due tothe improved estimation procedure, the test statistic D∗sp had higher power thanKimber’s proposed test statistic.The best linear unbiased estimates of the parameters whose distribution functionis of a location - scale form have been considered by Lloyd (1952). Therefore weestimate the parameters in equation (1) by using equations (4) and (5) to get thetest statisticD+sp = max |ri − ŝi| (12)where ŝi is obtained from equation (9) by replacing µ and θ by the estimates from(4) and (5).A power comparison of the D∗sp, D+sp and some other test statistics are given insection 3.83 The results of the monte carlo study3.1 Power of the testsThe power of test statistics have been examined for samples from equation (1)against a range of alternative distributions based on 40000 replicates. As mentioned,the purpose of this study is to apply the expected values, variances and covariancesof order statistics of the standard extreme value distribution in order to estimatethe unknown parameters in equation (1) by GLS regression.In Tables 1 and 2 the power of the test statistics by using approximations (equa-tions (7) and (8)) are compared with the power when exact values are used forsample size 20. These tables show that the differences are negligible, most of thedifferences being in the second decimal place. The power also reveals that the teststatistic T1 has higher power in comparison with the other test statistics for manyof the alternatives.In the case of the extreme value distribution, Stephens (1977) determined ap-proximate critical values of the Cramer-von Mises statistic when using the maximumlikelihood estimates of µ and θ. The W 2n and A2n tests have also been studied byLittel et al. (1979), in which case again the maximum likelihood estimates wereused in substitution for µ and θ.Spinelli (1980), in an unpublished M.Sc. Thesis at Simon Fraser University, stud-ied the test statistics based on regression and EDF for the extreme value distribution,of maxima. He pointed out problems arising when using the GLS method for theextreme value distribution due to the unavailability of the variances and covariancesof the order statistics for large sample sizes.9We have also performed a power study for the extreme value distribution, ofmaxima. Comparison of the power of the tests for some different distributions withthe results presented by Spinelli show that our results seems to be of the same orderas Spinelli’s results except for T2 statistic. Our result mostly give lower power forthe T2 statistic than is indicated in Spinelli’s study.The test based on the T1 statistic is in terms of power, superior to most otherstest statistics in this study. The test based on D+sp is almost of the same power asthe T1 test.Table 1 also shows that due to the improved parameter estimation, the teststatistic D+sp is generally more powerful than D∗sp. The procedure for parameterestimation used in D+sp is simple to implement which is another advantage of thistest statistic.3.2 Percentage PointsThe critical values of the T1 statistic for n = 10(10)100 are given in Table 3. Inthis table for n = 10(10)30, the percentage points of the statistics based on boththe exact and approximate values of means and covariances. Table 3 reveals thatthe approximation yield an error less than 6 % for all sample sizes. This encouragesus to trust the use of approximations of means and variances in calculating the teststatistics.ReferencesAITKEN, A.C. (1935). On least squares and linear combination of observations. Proc. Roy. Soc.Edinb., 55, 42.10BLOM, G. (1958). Statistical Estimates and Transformed Beta-Variables. Almqvist and Wiksell,Uppsala.BALAKRISHNAN, N. & CHAN, P.S. (1992). Order statistics from extreme value distribution I:Tables of means, variances and covariances, Commun. Stat. Sim. Comp. 21, 1199-1217.BALAKRISHNAN, N. & Rao, C.R. (1998). Handbook of Statistics 17, Elsevier, Amsterdam.BUSE, A. (1973). Goodness of fit in Generalized Least Squares Estimation. Amer. Statistician27, 106-108.COLES, S.G (1989). On Goodness-of-Fit Tests for the Two-Parameter Weibull Distribution De-rived from Stabilized Probability Plot. Biometrika 76, 593-598.D’AGOSTINO, R.B. & STEPHENS, M.A. (1986). Goodness-of-fit techniques, Marcel Dekker, NewYork.DOWNTON, F. (1966). Linear Estimations of Parameters in the Extreme Value Distribution.Technometrics 8 3-17.KIMBER, A.C. (1985). Tests for Exponential, Weibull and Gumbel Distributions based on theStabilized Probability Plot. Biometrika 72, 661-663.LIEBLEIN, J. & SALZAER, H.E. (1957). Table of the first moment of ranked extremes. J. Res.Nat. Bur. Stand. 59, 203-206.LIEBLEIN, J. & ZELEN, M. (1956). Statistical investigation of the fatigue life of deep groove ballbearing. J. Res. Nat. Bur. Stand. 57, 273-316.LITTEL, R.C., McCLAVE, J.T. & OFFEN, W.W. (1979). Goodness of fit tests for two-parameterWeibull distribution. Commun. Stat. B8, 257-269.LLOYD, E.H. (1952). Least-Squares Estimation of Location and Scale Parameters Using OrderStatistics, Biometrika 39, 88-95.MICHAEL, J.R. (1983). The stabilized probability plot. Biometrika 70, 11-17.11PIROUZI FARD, M.N. & HOLMQUIST, B. (2007). First moment approximations for order statis-tics from the extreme value distribution. Statistical Methodology, 4, 196-203.PIROUZI FARD, M.N. & HOLMQUIST, B. (2006). Approximations of variances and covariancesfor order statistics from the standard extreme value distribution, Statistical research report, No.2006:1, Lund University, Sweden.SPINELLI, J. (1980). Contributions to goodness-of-fit. M.Sc. Thesis, Department of Mathemat-ics, Simon Fraser University. STEPHENS M.A. (1977). Goodness of Fit for the Extreme ValueDistribution. Biometrika, 64 583-588.WHITE, J.S. (1967). The moments of log-Weibull order statistics, Research Publication GMR-717,Research Laboratories, General Motors Corp., Warren, Mich.WHITE, J.S. (1969). Moments of Log-Weibull Order Statistics, Technometrics 11, 373-386.12Table 1: Empirical power for tests on level 0.10, for selected alternative distribu-tions based on 40000 replicates by using the exact values of means, variances andcovariances of order statistics for n=20. 1Density 4(1 − x)3, 0 < x < 1. 2Density116xe−x/4, x ≥ 0. 3Density 10x4e−2x5 , x ≥ 0.Distribution T1 T2 W2n A2n Dn D+sp D∗spU(0, 1) 0.46 0.22 0.34 0.38 0.27 0.47 0.40Beta(1,4)1 0.98 0.41 0.88 0.92 0.81 0.98 0.96N(0, 1) 0.43 0.13 0.32 0.33 0.26 0.38 0.33Student t(6) 0.48 0.21 0.42 0.44 0.35 0.45 0.42Gamma(2,4)2 0.97 0.46 0.87 0.90 0.78 0.96 0.93χ2(5) 0.95 0.42 0.83 0.86 0.73 0.93 0.90Weibull(2,5)3 0.26 0.10 0.19 0.20 0.17 0.24 0.20Empirical CriticalValues, level= 0.10 0.14 0.47 0.10 0.61 0.17 0.11 0.1113Table 2: Empirical power for tests on level 0.10, for selected alternative distributionsbased on 40000 replicates by using the estimated values of means, variances andcovariances of order statistics for n=20.Distribution T1 T2 W2n A2n Dn D+sp D∗spU(0, 1) 0.46 0.23 0.35 0.39 0.28 0.47 0.40Beta(1,4) 0.98 0.42 0.88 0.92 0.81 0.98 0.97N(0, 1) 0.44 0.14 0.32 0.33 0.26 0.38 0.33Student t(6) 0.48 0.21 0.42 0.44 0.35 0.45 0.41Gamma(2,4) 0.97 0.47 0.87 0.90 0.78 0.96 0.94χ2(5) 0.95 0.42 0.83 0.87 0.73 0.94 0.90Weibull(2,5) 0.27 0.10 0.19 0.20 0.17 0.24 0.20Empirical CriticalValues, level= 0.10 0.13 0.46 0.10 0.61 0.17 0.11 0.1114Table 3: Quantiles of the test statistic T1 based on 40000 replicates.n 0.80 0.85 0.90 0.95 0.975 0.9910 0.15 0.17 0.21 0.27 0.33 0.41 exact0.15 0.17 0.20 0.26 0.32 0.41 approx.20 0.10 0.11 0.14 0.17 0.22 0.28 exact0.10 0.11 0.14 0.18 0.22 0.28 approx.30 0.07 0.09 0.10 0.13 0.17 0.21 exact0.07 0.09 0.10 0.13 0.17 0.21 approx.40 0.06 0.07 0.09 0.11 0.14 0.1850 0.05 0.06 0.07 0.10 0.12 0.1560 0.05 0.05 0.06 0.08 0.10 0.1370 0.04 0.05 0.06 0.07 0.09 0.1280 0.04 0.04 0.05 0.07 0.08 0.1190 0.03 0.04 0.05 0.06 0.08 0.10100 0.03 0.04 0.04 0.06 0.07 0.0915

Goodness of Fit Tests for the Extreme Value Distribution based on Regression, EDF and the Stabilized Probability Plot

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Goodness of fit tests for the extreme value distribution based on regression, EDF and the stabilized probability plot

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