CONDITIONAL PROBABILITY INTEGRAL TRANSFORMATIONS FOR MULTIVARIATE NORMAL DISTRIBUTIONS Satiago Rico Gallardo, C. P. Queseberry, F. J. O'Reilly Istitute of Statistics Mimeograph Series No. 1148 Raleigh, November 1977
CONDITIONAL PROBABILITY mtegral TRANSFORMATIONS FOR MULTIVARIATE NORMAL DISTRIBUTIONS Satiago Ricm-Gallardo l ad C. P. Q.ueseberry2 North Carolia State Uiversity ad Federico J. O'ReillY limas, Uiversidad Nacioal Aut6oma de M~xico Let Xl' '" X be a radom sample from a full-rak multivariate ormal distributio N(~,~). The two cases (i) ~ ukow :d ~ = a2~o'~o kow, ad (ii) ~ ad ~ completely ukow are cosidered here. Trasformatios are give that trasform the observatio vectors to a (smaller) set of i.i.d. uiform rv's. These trasformatios ca be used to costruct goodess-of-fit tests for these multivariate ormal distributios. 1.. ItroductioI) ad S1JX!lmau. There is a large literature that cosiders the multivariate ormal distributio. However, there is very little available i the way of goodess-of-fit tests for multivariate ormality, ad othig whatever that is based upo exact distributio theory ad therefore is applicable for small ad moderate size samples. Recetly, Moore (1976) has commeted upo the eed for goodess-of-fit tests for multivariate ormality. I this work we give trasformatios which ca be used to costruct exact level goodess-of-fit tests for multivariate ormality. O'Reilly ad Queseberry (1973), O-Q, itroduced the coditioal probability itegral trasformatios, CPIT's. Trasformatios were give i that paper for a multivariate ormal paret N(~,~) for the case whe ~ is ukow ad 1 Supported by limas, UNAM. 2Research supported by Natioal Sciece Foudatio Grat MCS76-82652.
2 t = to is kow. Here we give trasformatios for the two cases: (i) ~ ukow, ad t = a 2t o with EO kow, ad (ii) ~ ad E ukow. I both of these cases the compoets of the observatio vectors are trasformed usig certai Studet-t distributio fuctios. The trasformatios ad model testig techiques cosidered here ca be carried out o high-speed computers. g. Notatio~ ~ Prelimiaries. Let P deote the class of k-variate full-rak ormal distributios with mea vector ~ ad variace-covariace matrix E. For X X i.i.d. (colum) vector rv's from PEP, with l', correspodig probability desity fuctio f ad distributio f~~ctio F, both defied o ~, a complete sufficiet statistic for P is T = (X, s ), where X = (li).2:_ X. ad S =.tlx.x' - X X' ~=1 ~ ~= ~ i readily verified that T is doubly trasitive (cf. O-Q), i.e., -- It is a(t, X ) =a(t l' X ), where a(w) deotes the a - algebra iduced by a - statistic W. Cosider the coditioal distributio fuctio -F of a sigle observatio give the statistic T. For > k+l ~ is absolutely cotiuous ad possesses - a desity fuctio f which is the miimum variace ubiased, MVU, estimator of the paret desity fuctio. These fuctios were obtaied by Ghurye ad Olki (1969), p. 1265, cases 3.2 ad 3.4, for the cases (i) ad (ii) above. Case (ii) will be developed i detail, but we will oly summarize the results for case (i). The ext lemma gives the desity f i a form that will be - coveiet i this work. The idicator fuctio of the set satisfyig coditio [.] is deoted by I[.] LEMMA 2.1. If Xl"'" X are i.i.d. rv's with a commo multivariate ormal distributio PEP, the MVU estimator f of the correspodig ormal
3 probability desity fuctio is (2.1).I[(x-X )' S-l(x_X ) ~ (-l)/], > k + 1 PROOF. This result is immediate from Ghurye ad Olki (1969) ad the two facts: (a) For B(k x k) osigular ad x(k.x 1), IB-xx'\ = IBI(l-x'B-lx) (b) If B is p.d. the B-xx' is p.d. iff x'bx < 1. LEMMA 2.2. Suppose Y is a TV which has for fixed T the coditioal desity fuctio f of (2.1). The for Z = A (Y-X )/[((-l)/) - (Y-X )'s-l(y_x )}t, where A'A = s-l, the coditioal desity fuctio of Z give T is 1 PROOF. If z = A (y-x )/[((-l)/) - (y-x )' s-l(y_x )}2, 1 1 the y = A- l z((_l)/)2 (1+ Z'Z)-2 + X The Jacobia is +! [((-l)/)/(l+ z'z)} 2 IA I-III - zz'/(l+ z'z)1, ad usig the relatios I '/( ')-11 (,)-1 (- )' -l( -) I - zz 1+ z z = 1+ z z ad y-x S y-x = the result follows from Lemma 2.1. ((-l)/)z'z/(l+ z'z), The desity fuctio g of (2.2) has the form of a geeralized multivariate t distributio. Dickey (1967), Theorems 3.2 ad 3.3, gives coditioal ad
4 margial distributios for geeralized multivariate t distributios from which the coditioal ad margial distributios of g of (2.2) ca be obtaied. Let G deote the distributio fuctio of a uivariate Studet-t \.l distributio with \.l from results give by Dickey. degrees of freedom. The the followig ca be obtaied LEMMA 2.3. Let Z' = (Zl' ' \) deote a vector rv with (coditioal) probability desity fuctio g(z) of equatio (2.2). The for i = 1,, k. Cosider agai the origial sample Xl' " X ' ad put 1 Z. = A.(X. - X.)/([(j-l)/j] _ (X. _ X.)'S:l(X. _ X.)}2", J JJ J J J J J J ad deote Z~ = (Zl.,.", ~.) for j = k+2,..,. The the ext theorem J ~J ~,J follows from Lemma 2.3 ad a slight extesio of Theorem 5.1 of O-Q. THEOREM: 2.1. The k(-k-l) radom variables give by (2.4) u.. = G. k. 2 (Z..[(j-k+i-2)/(1 + Z2 l. + + i 1.)J~, ~,J J- +~- ~,J,J ~-,J for j = k+2,, ad i =1,", k; are i.i.d. U(O,l) rv's. We ow summarize the results for case (i) whe IJ. is ukow ad 2 E =a Eo for EO kow. For - X 'ad S defied above put here ad
5 THEOREM 2.2. (-2)k radom variables give by For Xl' ' X LLd. from N(I-L,ciE o )' EO kow, the for j = 3,, ad i = 1,, k; are i.i.d. U(O,l) rv's. ~. Discuss~. After the multivariate sample Xl' ' X has bee trasformed by usig either (2.4) or (2.5), the a level ~ goodess-of-fit test for the correspodig composite multivariate ormal ull hypothesis class (case (i) or case (ii)) ca be made by testig the surrogate simple ull hypothesis that the trasformed values are i.i.d. U(O,l). Queseberry ad Miller (1977) ad Miller ad Queseberry (1977) have studied power properties of omibus tests for uiformity ad recommed either the Watso ~ test (Watso (1962)) or the Neyma smooth test (Neyma (1937)) for testig simple uiformity. REFERENCES Dickey, J. M. (1967). Matricvariate geeralizatios of the multivariate t distributio ad the iverted multivariate t distributio. A. Math. Statist. j, 511-18. Ghurye, S. G. ad Olki, I. (1969). Ubiased estimatio of some multivariate probability desities ad related fuctios. A. Math. Statist.~, 1261-71. Miller, F. L., Jr. ad Queseberry, C. P. (1977). Power studies of tests for uiformity, II. Commu. Statist. submitted. Moore, D. D. (1976). Recet developmets i chi-square tests for goodess-of-fit. Departmet of Statistics, Purdue Uiversity, Mimeograph Series #459.
6 Neyma, Jerzy (1937). "Smooth" test for goodess-of-fit. Skadiavisk Aktuarietidskrift 20, 149-99...;;;...;.;.;,--"~---..-...;...;;;- -- O'Reilly, F. J. ad Queseberry, C. P. (1973). The coditioal probability itegral trasformatio ad applicatios to obtai composite chi-square goodess-of-fit tests. A. Statist. 1, 74-83. Queseberry, C. P. ad Miller, F. L., Jr. (1977). Power studies of some tests for uiformity. J. Statist. Comput. Simul. 2, 169-91. Watso, G. S. (1962). Goodess-of-fit tests o a circle. II. Biometrika ~, 57-63.