SHAPIRO-WILK TEST FOR NORMALITY WITH KNOWN MEAN

Transcription

1 SHAPIRO-WILK TEST FOR NORMALITY WITH KNOWN MEAN Wojcech Zelńsk Departmet of Ecoometrcs ad Statstcs Warsaw Uversty of Lfe Sceces Nowoursyowska 66, -787 Warszawa e-mal: Zofa Hausz, Joaa Tarasńska Departmet of Appled Mathematcs ad Computer Scece Uversty of Lfe Sceces Lubl Akademcka, -95 Lubl e-mals: Summary Adaptato of Shapro-Wlk W test to the case of ormalty wth a kow mea s cosdered The table wth crtcal values for dfferet sample szes ad several sgfcace levels s gve The power of ths test s vestgated ad compared wth Kolmogorov ad the two-step procedure of Shapro-Wlk W ad t-tests Addtoally, the ormalzg coeffcets for test statstc are gve The advatage of ths test over the classc Shapro-Wlk W test s llustrated by a example Keywords ad phrases: Shapro-Wlk W test, ormalty Classfcato AMS : 6G Itroducto Testg ormalty o a bass of a radom sample X, X, X plays a mportat role classcal statstcal aalyss I lterature, there exst may dfferet tests for the ull hypothess that dstrbuto of radom varable X s ormal wth a ukow expectato µ ad varace a ombus oe σ However, the Shapro-Wlk W statstc (Shapro, Wlk, 965) s regarded as

2 I practce, frequetly we are terested testg ull hypothess that dstrbuto of X s ormal wth a kow expectato µ I the paper we focus o testg ths partcular ull hypothess We propose a modfcato W of the Shapro-Wlk W statstc I Secto we defe W statstc ad descrbe ts propertes I Secto we preset smulato results o the power of the test Applcato of the test for the chose regresso problem s preseted Secto Some cocludg remarks are eclosed Secto 5 Dervato of W statstc ad ts propertes Suppose that we observe a radom varable X wth dstrbuto F ad we are terested testg the hypothess o a bass of a sample X, X, X (, ) H : F s N µ σ Shapro ad Wlk (965) proposed W test based o the statstc W = ( ) ( X X ) a X, () where X ( ) X () X ( ) are the ordered values of the sample, ad a are tabulated coeffcets Now, let us assume that we kow the expected value, say µ Thus we are terested testg the ull hypothess ( µ ) H : σ () F s N, Applcato of Shapro ad Wlk s techque to the problem of testg () gves the statstc W = ( X µ ) a X ( ) The ull hypothess () s rejected whe W < W ( ; ), where W ( α ;) s the crtcal value at a sgfcace level α α The statstc W has propertes smlar to the W statstc, amely, W s scale varat ad the maxmum value of W s oe The mmum value of W s ε = a (Shapro ad Wlk, 965)

3 Lemma The mmum value of W s zero Proof Sce subject to = large x W s scale varat t suffces to cosder the maxmzato of ( µ ) a The lemma follows from the fact that ( µ ) x x may be arbtrarly Shapro ad Wlk (965) gave the aalytc form of the probablty desty fucto for W statstc the case of sample sze whch s equal to It s of the form g( w) = ( w) w for w < π They also stated that W s depedet of radom varables X ad ( X X ) Thus, t s easy to obta the probablty desty fucto of W for samples of sze = Let us otce that W = W C, where C = ( X X ) ( X µ ) = ( X X ) ( X X ) + ( X µ ) s a radom varable dstrbuted as = we have the probablty desty fucto of C Takg the ew varable Beta,, depedet of W Thus the case of ( 5) ( ) Γ f ( c) = c for < c < π w = w c the jot probablty desty fucto g ( w) f ( c) ad tegratg ths fucto over c, we get the probablty desty fucto for W the followg form ϕ ( w ) Γ = π Γ π ( 5) π ( 5) π w w w w w ( c) ( c w ) ( c) ( c w ) dc dc for for < w w < Fally, after tegratg, we get

4 ϕ ( w ) ( 5) Γ π π = Γ π ( 5) w w 5w arcs ( w ) π + for for < w w < The plot of ϕ ( w ) s show Fgure here Fgure For sample sze > the aalytcal form of the ull dstrbuto of W s ot avalable Hece, to obta ay formato about the dstrbuto a Mote Carlo expermet was performed I smulatos for each =,,, 5, N =,, samples from the dstrbuto (, ) sample w,, N were draw ad for each sample the value W was calculated, so the w N of values of the was take as the α-th quatle of W statstc were obtaed The crtcal value W ( ;) N α w,, w All calculatos were doe R program usg the procedure shaprotest whch Roysto s procedure s used (Roysto, 99) The results are gve Table here Table Shapro ad Wlk (968) approxmated the dstrbuto of the W statstc by a Johso curve For each they made the least squares regresso of the emprcal samplg value of o p W ( p) ε u( p) = l W ( p) z, where ε was the mmum value of the W statstc, W ( p) was the p-th emprcal samplg quatle, z p was the p-th quatle of the stadard ormal dstrbuto They took the followg values of p ad gave the tables for dstrbuto p =,, 5 ε, γ, δ such that ( 5) 5 ( 5) 75 ( 5) 95, 98, 99 W ε Z = γ + δl has approxmately stadard ormal W I ths paper, a smlar approach was appled for the W statstc for sample szes =,,, 5 The least squares regresso of W ( p) l o z p was based o,, W ( p),

5 pseudoradom samples from N (, ) The values of γ ad δ such that Z W = γ + δl has W approxmately stadard ormal dstrbuto are eclosed Table The lower tal of Z dcates oormalty here Table To check the goodess of approxmato aother N=,, pseudoradom samples from (, ) N were geerated ad for each of them W ad = The ratos calculated (,,, N ) # { Z : Z z } < N p wth p =,, 5,, 5, 9, 95, 98, 99 Z W = γ + δl were W are gve Table here Table Power comparsos Suppose that the hypothess : F s N ( µ σ ) H s verfed wth the ad of the W test It, s terestg to kow the power of the W test Three kds of alteratves are cosdered Namely: (a) F s ( µ,σ ) N wth µ µ ; (b) F s ot ormal wth µ = µ ; (c) F s ot ormal wth µ µ The Shapro-Wlk W test was vestgated agast dfferet oormal alteratves Very exhaustve research was doe by Shapro et al (968) ad Che (97) Those researches showed that the W test s very powerful comparso to other ormalty tests such as Kolmogorov, ch-square, β, β ad agast very dfferet dstrbutos lke Studet s t, Gamma, Beta or Uform Because the costructo of W s smlar to the W test, t may be expected that the W test wll also be powerful agast alteratves of kd (b) ad (c) Hece our studes we cofe ourselves to (a) alteratve, e whe the true dstrbuto s ormal wth a mea other tha µ The W test was compared wth two other procedures The frst oe s the stadard Kolmogorov test The test statstc of the Kolmogorov test s gve by 5

6 where F X ( ) X µ s ( ) ( ) = Φ ormal dstrbuto max F( X ( ) ), F( X ( ) ) X, s = ( µ ), ad Φ s the cdf of the stadard The secod procedure s a two step oe I the frst step the ormalty s verfed by the classcal W test If ormalty s ot rejected, the the hypothess of equalty of the mea to a gve umber µ s verfed by the t test All tests were calculated o the sgfcace level α I the two step procedure there s a eed of applyg two sgfcace levels α w ad chose such a way that the overall sgfcace level s α, e α t for both used tests Those umbers were { accepts ormalty ad accepts mea µ } ( α + α ) = α P H t W w t Because there are o prefereces to W or t test hece α w = αt = α were take The power comparso of three tests was performed by the Mote Carlo method A sample of sze from the ormal dstrbuto wth a gve µ was geerated ad ths sample was used all tests The sample was the shfted to dfferet values of µ ad each of the tests were the appled to shfted samples Ths procedure was repeated, tmes The umber of rejectos of the hypothess () was calculated I the smulatos the hypothess : F s N ( σ ) H was verfed for samples of szes,,,,, 5 ad sgfcace levels α =, 5, The varace σ = was used all cases The smulated powers are gve the Table Here Table The relatve powers of W wth respect to Kolmogorov ad W+t tests are show Fgure O the x axs there are values of µ ad o the y axs there are gve values of power of W test power of W test power of Kolmogorov test power of W = t test ( sold le) ad ( dotted le) Oe may see that geerally les are above oe whch shows that W s more powerful tha the other two tests Here Fgure 6

7 Example Cosder a problem of fttg a regresso le I the aalyss of the model Y = f (x) + ε oe has to check whether ε s dstrbuted as N (, σ ) for each x I the expermet the radom varable Y was geerated accordg to the model wth f ( x) = x + 7x +, σ = ad x =,,, 6, 8,, te tmes at each pot Two regresso fuctos ( x) = β + x ad f ( x) = β + β x + β were ftted Note that f β x the secod model s the true oe Classcal aalyss of varace the F test showed that both models are acceptable e f ( x) as well as ( x) regresso fucto Results are preseted Table 5 Here Table 5 f may be cosdered as a approprate The ext step of the aalyss of fttg s to check whether the resduals are ormally dstrbuted wth zero mea e for each x ad regresso le the hypothess that resduals are dstrbuted as (, σ ) N should be verfed Results, rouded to the fourth decmal place, are show Table 6 I the W colum, values of a approprate test statstc are gve The crtcal value for = ad α = 5 s equal to 585 (see Table ) I the last colum of Table 6 the p-values of the Shapro-Wlk W test are gve here Table 6 I the case of lear fucto, the hypothess of ormalty wth zero mea was rejected at four x pots, whle the case of quadratc fucto the hypothess was ever rejected Hece, fucto f ( ) s ot acceptable as a regresso fucto whereas f ( ) s acceptable Let us x otce that the Shapro-Wlk W test ever rejected the ormalty of resduals, ether a lear or a quadratc case 5 Cocludg remarks I may statstcal models t s assumed that errors are ormally dstrbuted wth zero mea Thus the W test s more adequate ad should be used stead of the classcal Shapro- Wlk W test I the paper t s show va smulato study that the W test s geerally more powerful tha the Kolmogorov, ad W ad Studet t tests combed x 7

8 Refereces Che, E H (97) The Power of the Shapro-Wlk W Test for Normalty Samples from Cotamated Normal Dstrbutos Joural of the Amerca Statstcal Assocato 66, Roysto P (99) Approxmatg the Shapro-Wlk W-test for o-ormalty Statstcs ad Computg,, 7-9 R Developmet Core Team (8) R: A laguage ad evromet for statstcal computg R Foudato for Statstcal Computg Vea, Austra ISBN , URL Shapro SS, Wlk MB (965) A aalyss of varace test for ormalty (complete samples) Bometrka 5,, 59-6 Shapro SS, Wlk MB (968) Approxmatos for the ull dstrbuto of the W statstc Techometrcs, Shapro, S S, Wlk, M B, Che, H J (968) A Comparatve Study of Varous Tests for Normalty Joural of the Amerca Statstcal Assocato 6, 7 8

9 Table Crtcal values of W statstc for sample szes ad sgfcace level α α α

10 Table The ormalzg costats for W for sample szes γ δ γ δ γ δ -,7, ,56, ,679,78 -, ,9586,85 -, ,99,98 -, ,778, ,695, ,896, ,79, ,7, ,9, ,55, ,68, ,8, ,9, ,, ,55,

11 Table The smulated probabltes W P γ + δl < z p for sample szes W Probablty

12 Table Power of W, Kolmogorov ad W + t tests α = α = 5 α = µ W K W+t W K W+t W K W+t

13 Table 5 Estmated coeffcets regresso fuctos ad p values Fucto ˆβ ˆβ ˆβ p-value of F test f ( x) f ( x)