The performance of univariate goodness-of-fit tests for normality based on the empirical characteristic function in large samples

Size: px

Start display at page:

Download "The performance of univariate goodness-of-fit tests for normality based on the empirical characteristic function in large samples"

Hillary Arnold
5 years ago
Views:

1 The performace of uivariate goodess-of-fit tests for ormality based o the empirical characteristic fuctio i large samples By J. M. VAN ZYL Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State, Bloemfotei, South Africa wwjvz@ufs.ac.za SUMMARY A empirical power compariso is made betwee two tests based o the empirical characteristic fuctio ad some of the best performig tests for ormality. A simple ormality test based o the empirical characteristic fuctio calculated i a sigle poit is show to outperform the more complicated Epps-Pulley test ad the frequetist tests icluded i the study i large samples. Key words: Normality test; Empirical characteristic fuctio; Cumulat; Goodess-offit 1. INTRODUCTION 1

2 Several goodess-of-fit tests based o the empirical characteristic fuctio (ecf) are available. Feuerverger ad Mureika (1977) developed a test for symmetry ad this was exteded by Epps ad Pulley (1983) to test uivariate ormality. Heze ad Barighaus (1988) exteded the Epps-Pulley test to test multivariate ormality ad this is called the BHEP test. A review paper with commets of procedures based o the ecf is the paper by Meitais (016) ad also the book by Ushakov (1999). The Epps-Pulley approach is still the mai approach for testig ormality of tests based o the ecf ad is based o a itegral over the weighted squared distaces betwee the ecf ad the expected characteristic fuctio. A weakess of the test is that the asymptotic ull-distributio is ot very accurate ad otherwise itractable ad complicated i fiite samples (Taufer (016), Swaepoel ad Alisso (016)). I this work a test is proposed ad asymptotic distributioal results derived usig the work by Murota ad Takeuchi (1981). They derived a locatio ad scale ivariat test usig studetized observatios ad showed that the use of a sigle value whe calculatig the ecf is sufficiet to get good results with respect to power whe testig hypotheses. Csörgő (1986) derived a multivariate extesio of the asymptotic results derived by Murota ad Takeuchi (1981). A simulatio study is coducted to compare the power of the proposed test agaist the Epps-Pulley test ad five of the most recogized goodess-of-fit tests for ormality. The proposed test with asymptotic properties similar to those of the test of Murota ad Takeuchi (1981) performs reasoably i small sample, but excellet i large samples with respect to power. The test statistic is a simple ormal test which will perform better

3 as the sample icreases ad it was foud to domiate the much more complicated Epps- Pulley test. Murota ad Takeuchi (1981) compared their test agaist a test based o the sample kurtosis ad coducted a small simulatio study. Various overview simulatio studies were coducted to ivestigate the performace of tests for ormality. Oe of the most cited papers is the work by Yap ad Sim (011), but they did ot iclude a goodess-offit test based o the empirical characteristic fuctio. A paper which icluded a very large selectio of tests for ormality is the work by Romao et al. (013), but the test of Murota ad Takeuchi (1981) was ot icluded i this study. The tests icluded are the Jarque Bera, Shapiro-Wilk, Lilliefors, Aderso-Darlig ad D Agostio ad Pearso tests. The focus will be o uimodal symmetric distributios ad large sample sizes, that is sample sizes larger tha fifty. Murota ad Takeuchi (1981) proved that the square of the modulus of the empirical characteristic fuctio coverges weakly to a complex Gaussia process where the observatios are stadardized usig affie ivariat estimators of locatio ad scale ad they derived a expressio for the asymptotic variace. Let X,..., 1 X be a i.i.d. sample itx of size, from a distributio F. The characteristic fuctio is E( e ) = φ( t) ad it is estimated by the ecf ˆ 1 itx ( ) j φf t = e, (1) j= 1 3

4 The studetized sample is Z,..., 1 Z, where Z ( ˆ ) / ˆ j = X j µ σ, j = 1,...,, with µ = ad σ = S. Deote the ecf, based o the studetized data by ˆ X ˆ ˆ itz ( ) (1/ ) j φ t = e. The statistic proposed to test ormality is S j= 1 ν (1) = log( ˆ φ (1)/exp( 1/ ) ), () S where ( ν (1)) N(0,0.0431) asymptotically. Absolute value deotes the modulus of a complex umber if the argumet is complex. The expressio is ˆ ˆ I = φs ( t) φ0( t) w( t) dt, with φ ( t) 0 deotig the ecf of a stadard ormal, w( t) a weight fuctio which is of the same form as a ormal desity with mea zero ad variace the sample estimate of the variace. Of the may variatios usig the ecf to test goodess-of-fit tests, this expressio attracted the most iterest ad is still used Meitais (016). Epps ad Pulley (1983) used this expressio ad derived a test for ormality usig a weight fuctio which has the form of a stadard ormal desity. They gave a exact expressio for the characteristic fuctio of the ormal distributio. Heze (1990) derived a large sample approximatio for this test ad used Pearso curves to approximate the distributio. It is show i the simulatio study that the proposed test with similar properties as that of Murota ad Takeuchi (1981) ad calculated i a sigle 4

5 poit without usig a weight outperforms the Epps-Pulley test i the cases cosidered with respect to power.. MOTIVATION AND ASYMPTOTIC VARIANCE OF THE TEST STATISTIC A motivatio will be give i terms of the cumulat geeratig fuctio. The ormal distributio has the uique property that the cumulat geeratig fuctio caot be a fiite-order polyomial of degree larger tha two, ad the ormal distributio is the oly distributio for which all cumulats of order larger tha 3 are zero (Cramér (1946), Lukacs, (197)). The motivatio for the test will be show by usig the momet geeratig fuctio, but experimetatio showed that the use of the characteristic fuctio rather tha the momet geeratig fuctio gives much better results whe used to test for ormality. Cosider a radom variable X with distributio F, mea µ ad variace σ. The cumulat geeratig fuctio Κ F ( t ) of F ca be writte as r Κ F ( t) = κ rt / r!, where r= 1 κ r is the r-th cumulat. The first two cumulats are κ µ κ σ 1 = E( X ) =, = Var( X ) =. Sice Κ ( t ) is the logarithm of the momet geeratig fuctio, the momet geeratig F fuctio ca be writte as F M ( t) E( e ) e Κ t F tx ( ) = =. It follows that Κ ( t ) = tx F log( E ( e )) 5

6 = r= 1 κ t r r / r! =( 1 ) r κ t + κ t / + ( κrt / r!) = ( ) r= 3 r µ t + σ t / + ( κ t / r!). r= 3 r Let F N deote a ormal distributio with a mea µ ad variace M σ. N ( ) t deotes the momet geeratig fuctio of the ormal distributio with cumulat geeratig fuctio 1 Κ ( t N ) = µ t + σ t. The logarithm of the ratio of the momet geeratig fuctios of F ad F N is give by log( M ( t)/ M ( t)) = log(exp( Κ ( t) Κ ( t))) F N F N = 1 r [( µ t + σ t ) + κ t / r!] ( µ t + 1 σ t ) r r = 3 r = Κ ( t) + κ t / r! Κ ( t)) N r N r= 3 r = κrt / r!. (3) r= 3 If F is a ormal distributio, the sum give i (1) is zero. Replacig Κ ( t) Κ ( t) by Κ ( it) Κ ( it) it follows that F N F N log( log( φ ( t)) log( φ ( t)) ) = log( Κ ( it) Κ ( it) ) F N F N 4+ r = κ4+ rt /(4 + r)!, r= 0 6

7 which would be equal to zero whe the distributio F is a ormal distributio ad this expressio ca be used to test for ormality. Murota ad Takeuchi (1981) used the fact that the square root of the log of the modulus of the characteristic fuctio of a ormal distributio is liear i terms of t, i other words ( log( φ ( t N ) )) 1/ is a liear fuctio of t. A asymptotic variace of ν ( t) = log( ˆ φ ( t) / exp( t / ) ) ca be foud by usig the NS delta method ad the results of Murota ad Takeuchi (1981). Let ˆ φ ( t ) deote the ecf calculated i the poit t usig studetized ormally distributed observatios. They showed that the process defied by NS Z ɶ t ˆ t t, (4) ( ) = ( φns ( ) exp( )) coverges weakly to a zero mea Gaussia process ad variace 4 E( Zɶ ( t)) = 4exp( t )(cosh( t ) 1 t / ). (5) Note that ˆ φ t / NS ( t ) e = ad by applyig the delta method it follows that Var( ˆ φ ( t) ) Var( ˆ φ ( t) )( ˆ φ ( t) ), NS NS NS thus Var( ˆ φ ( t) ) Var( ˆ φ ( t) ) /( ˆ φ ( t) ). NS NS NS By applyig the delta method agai it follows that 7

8 Var ν t Var ˆ φ t e 1/ ( ( )) = (log( NS ( ) / )) (1/ ˆ φ ( t) ) Var( ˆ φ ( t) ) NS NS = Var( ˆ φ ( t) ) / 4( ˆ φ ( t) ) NS 4 NS 4 = (cosh( t ) 1 t / ) /. (6) The statistic ν ( t ) coverges weakly to a Gaussia distributio with mea zero ad 4 variace Var( ν ( t)) = (cosh( t ) 1 t / ) /, where ( ν (1)) N(0,0.0431) asymptotically. Var( ν (1)) = /, t = 1. (7) Reject ormality if ν (1) / ( / = ν (1) > z 1 α /. (8) I the followig figure the average of the log the modulus calculated i the poit, t = 1, usig stadard ormally distributed samples, for various samples sizes is show. The solid lie is where studetized observatios were used ad the dashed lie where the ecf is calculated usig the origial sample. It ca be see that there is a large bias i small samples ad the studetized ecf has less variatio. 8

9 average log of the modulus Fig. 1. Plot of the average log of the modulus of the ecf for various sample sizes calculated usig m = 5000 calculated usig samples form a stadard ormal distributio. The solid lie is where studetized observatios are used ad the dashed lie usig the origial sample. Calculated i the poit t=1, ad the expected value is I the followig histogram 5000 simulated values of ν (1) /( / = ν (1) are show, where the ν (1)' s are calculated usig simulated samples of size = 1000 from a stadard ormal distributio. 9

10 frequecy ν Fig.. Histogram of 5000 m = simulated values of v (1) / ( / ), with = Calculated usig ormally distributed samples, data stadardized usig estimated parameters. I Figure 3 the variace of ν (1) is estimated for various sample sizes, based o 1000 estimated values of ν (1) for each sample size cosidered. The estimated variaces is plotted agaist the asymptotic variace Var( v ) = /. 10

11 1.8 x asymptotic ad estimated variace sample size Fig. 3. Estimated variace of v ad asymptotic variace for various values of. Dashed lie the estimated variace. Estimated variace calculated usig 1000 ormally distributed samples, data stadardized usig estimated parameters. There are a few variatios of the parameters used whe choosig the weight fuctio for the Epps-Pulley test, but a versio suggested by Epps ad Pulley to be used as a omibus test is whe choosig the weigh fuctio a ormal desity with mea zero ad variace the sample estimate of the variace, that is t /( ˆ σ ) = πσˆ e. w( t) (1/ ) I = ˆ t ˆ t w t dt φs ( ) φ0( ) ( ) = ˆ ˆ X j X k σ X j X σ j= 1 k= 1 j= 1 exp{ ( ) / } exp{ ( ) / } + 1/ 3. (9) 11

12 Heze (1990) derived a large sample approximatio for T = I ad used Pearso curves to approximate the distributio. A simulatio study was coducted to calculate the 1 α percetiles of T, α = 0.05, based o m = simulated values of T. It was foud that eve with this large sample there is still variatio i the 4 th decimal ad the first 3 decimals where used i the simulatio study to estimate the power of the test. Table 1. Simulated percetiles to test for ormality usig the Epps-Pulley test at the 5% level. Calculated from m = simulated samples of size.95 percetile mea variace Table 1 Simulated percetiles to test for ormality usig the Epps-Pulley test at the 5% level. Calculated from m = simulated samples of size. Heze (1990) used the value 1 istead of σ ad foud for example for = 100, that the.95 th percetile is which is approximately equal to foud i this study. ˆ They calculated the asymptotic expected value of T as ad variace

13 3. SIMULATION STUDY The paper of Yap ad Sim (011) is used as a guidelie to decide which tests to iclude. The proposed test will be deoted by ECFT ad EP deotes the Epps-Pulley test i the tables. The power of the test will be compared agaist several tests for ormality: The Lilliefors test (LL), Lilliefors (1967) which is a slight modificatio of the Kolmogorov-Smirov test for where parameters are estimated. The Jarque-Bera test (JB), Jarque ad Bera (1987), where the skewess ad kurtosis is combied to form a test statistics. The Shapiro-Wilks test (SW), Shapiro ad Wilk (1965). This test makes use of properties of order statistics ad were later developed to be used for large samples too by Roysto (199). The Aderso-Darlig test (AD), Aderso ad Darlig. The D Agostio ad Pearso test (DP), D Agostio ad Pearso (1973). This statistic combies the skewess ad kurtosis to check for deviatios from ormality. Samples are geerated from a few symmetric uimodal symmetric distributios with sizes = 50,100, 50,500, 750,1000. The proportio rejectios are reported based o m=5000 repetitios. The test are coducted at the 5% level ad for the ecf, the ormal approximatio is used. Sice the sample sizes are large, o-ormality with respect to multi-modal ad skewed distributios ca easily be picked up by usig graphical methods. 13

14 The followig symmetric distributios are cosidered, uiform o the iterval [0,1], the logistic distributio with mea zero the stadard t-distributio ad the Laplace distributio with mea zero ad scale parameter oe. The stadard t -distributio with 4, 10 ad 15 degrees of freedom. Skewed distributios ad multimodal distributios would ot be ivestigated, sice i large samples such samples ca be already excluded with certaity as beig ot from a ormal distributio by lookig at the histograms. All the tests performed for the ecf test were coducted usig the ormal approximatio, but percetiles ca also easily be simulated. Simulated estimates of the Type I error for = 30,50,100, 50,500, 750,1000, are give i Table 1 based o m = 5000 simulated samples. The simulated samples are stadard ormally distributed ad studetized to calculate the Type I error. Table Simulated percetiles to test for ormality at the 5% level. Calculated from m = 5000 simulated samples of size each i the poit t=1. Type I error ECFT EP LL JB SW AD DP

15 I Table the rejectio rates, whe testig at the 5% level ad symmetric distributios, are show for various sample sizes based o samples each time. I table 1 the t-distributio where ot all momets exist is cosidered. The JB, DP ad ECFT tests performs best, ad i large samples the ECFT test performs the best. Table 3. Simulated power of ormality tests. Rejectio proportios whe testig for ormality at the 5% level. ECFT EP LL JB SW AD DP t(4) t(10) t(15)

16 Fig. 4. Plot of the three best performig tests with respect to power, testig for ormality, data t- distributed with 10 degrees of freedom. Solid lie, ecf test, dashed lie JB test ad dash-dot the DP test power - ECFT, JB, DP - t(10)

17 Table 4. Simulated power of ormality tests. Rejectio proportios whe testig for ormality at the 5% level. ECFT EP LL JB SW AD DP U(0,1) Laplace Logistic It ca be see the ECFT outperforms the other tests with respect to power i large samples, especially whe testig data from a logistic distributio. Samples were simulated from a mixture of two ormal distributios, with a proportio α from a stadard ormal ad a proportio 1 α from a ormal with variace σ. This ca also be cosidered as a cotamiated distributio. The results are show i Table 4. 17

18 The proposed test yielded good results. Table 5. Simulated power of ormality tests. Rejectio proportios whe testig for ormality at the 5% level. Mixture of two ormal distributios (cotamiated data). ( σ, α ) ECFT EP LL JB SW AD DP (.0,0.) (0.5,0.) (.0,0.5)

19 1 0.9 power - ECFT,JB,AD - mixture ormal Fig. 5. Plot of the three best performig tests with respect to power, testig for ormality, data mixture of ormal distributios with two compoets. Mixture.5N(0,1)+0.5N(0,4). Solid lie, ecf test, dashed lie JB test ad dash-dot the AD test. 4. CONCLUSIONS The proposed test performs better with respect to power i large samples tha the other tests for ormality for the distributios cosidered i the simulatio study. I small samples of say less tha = 50, it was foud that the test of D Agostio ad Pearso (1973) was ofte either the best performig or close to the best performig test. 19

20 I practice oe would ot test data from a skewed distributio for ormality i large samples. The simple ormal test approximatio will perform better, the larger the sample is. The asymptotic ormality ad variace properties, which is of a very simple form, ca be used i large samples. This test ca be recommeded as probably the test of choice i terms of power ad easy of applicatio i large samples ad shows that the empirical characteristic fuctio has the potetial to outperform the usual frequetist methods. Refereces ANDERSON, T.W., DARLING, D.W. (1954). A test of goodess of fit. J. Amer. Statist. Assoc., 49 (68), BARINGHAUS, L., HENZE, N. (1988). A cosistet test for multivariate ormality based o the empirical characteristic fuctio. Metrika 35 (1): CRAMÉR, H. (1946). Mathematical Methods of Statistics. NJ: Priceto Uiversity Press, Priceto, NJ. CSÖRGŐ, S. (1986). Testig for ormality i arbitrary dimesio. Aals of Statistics, 14 (), D AGOSTINO, R., PEAESON, R. (1973). Testig for departures from ormality. Biometrika, 60, EPPS, T.W., PULLEY, L.B. (1983). A test for ormality based o the empirical characteristic fuctio. Biometrika, 70, FEUERVERGER, A., MUREIKA, R.A. (1977). The Empirical Characteristic Fuctio ad Its Applicatios. The Aals of Statistics, 5 (1), HENZE, N. (1990), A Approximatio to the Limit Distributio of the Epps-Pulley Test Statistic for Normality. Metrika, 37,

21 JARQUE, C.M., BERA, A.K. (1987). A test for ormality of observatios ad regressio residuals. It. Stat. Rev., 55 (), LILLIEFORS, H.W. (1967). O the Kolmogorov-Smirov test for ormality with mea ad variace ukow. J. America. Statist. Assoc., 6, LUKACS, E. (197). A Survey of the Theory of Characteristic Fuctios. Advaces i Applied Probability, 4 (1), MEINTANIS, S. G. (016). A review of testig procedures based o the empirical characteristic fuctio. South Africa Statist. J., 50, MUROTA, K., TAKEUCHI, K. (1981). The studetized empirical characteristic fuctio ad its applicatio to test for the shape of distributio. Biometrika; 68, ROMÃO, X, DELGADO, R, COSTA, A. (010). A empirical power compariso of uivariate goodess-of-fit tests for ormality. J. of Statistical Computatio ad Simulatio.; 80:5, ROYSTON, J.P. (199). Approximatig the Shapiro-Wilk W-test for o-ormality. Stat. Comput.,, SHAPIRO, S.S., WILK, M.B. (1965). A aalysis of variace test for ormality (complete samples). Biometrika 5, SWANEPOEL, J.W.H., ALLISON, J. (016). Commets: A review of testig procedures based o the empirical characteristic fuctio. South Africa Statist. J., 50, TAUFER, E. (016). Commets: A review of testig procedures based o the empirical characteristic fuctio. South Africa Statist. J., 50, USHAKOV, N.G. (1999). Selected Topics i Characteristic Fuctios. VSP, Utrecht. YAP, B.W., SIM, C.H. (011). Comparisos of various types of ormality tests. J. of Statistical Computatio ad Simulatio, 81 (1),

A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality

A goodness-of-fit test based on the empirical characteristic function and a comparison of tests for normality A goodess-of-fit test based o the empirical characteristic fuctio ad a compariso of tests for ormality J. Marti va Zyl Departmet of Mathematical Statistics ad Actuarial Sciece, Uiversity of the Free State,