Near-exact distributios for the ideedece ad shericy lielihood ratio test statistics Carlos A. Coelho cmac@fct.ul.t Filie J. Marques fjm@fct.ul.t Mathematics Deartmet, Faculty of Scieces ad Techology The New Uiversy of Lisbo, Portugal Abstract I this aer we will show how a suably develoed decomosio of the characteristic fuctio of the logarhm of the lielihood ratio test statistic to test ideedece i a set of variates may be used to obtai extremely well-t ear-exact distributios both for this test statistic as well as for the lielihood ratio test statistic for shericy, based o a decomosio of this latter test i two ideedet tests. For the ideedece test statistic, umerical studies ad comarisos wh asymtotic distributios roosed by other authors show the extremely high closeess of the ear-exact distributios develoed to the exact distributio. Cocerig the shericy test statistic, comarisos wh the ear-exact distributios develoed i [9] show the advatages of these ew ear-exact distributios. Key words: Wils Lambda statistic, ideedece test, shericy test, Geeralized Near-Iteger Gamma distributio, mixtures. Itroductio Let X be a vector wh a -multivariate Normal distributio wh exected value ad variace-covariace matrix, that is, let X N (; ) : () Corresodig author. Address: The New Uiversy of Lisbo, Faculty of Scieces ad Techology, Mathematics Deartmet, Qua da Torre, 89-56 Caarica, Portugal; Tel:(35) 94 8388; Fax:(35) 94 839. Prer submted to Elsevier Sciece 9 November 007
The, the ower ( + ) of the lielihood ratio test statistic to test the ull hyothesis H 0 : diag( ; ; : : : ; ) () based o a samle of size +, is the statistic jv j Q j V j (3) where the matrix V is eher the MLE (Maximum Lielihood Estimator) of, the samle matrix of sum of squares ad roducts of deviatios from the samle mea or the samle variace-covariace matrix of the variables i X ad V j is the j-th diagoal elemet of V. The statistic i (3) is a articular case of the geeralized Wils statistic used to test the ideedece of grous of variables, each wh oe oly variable (see [7],[8],Ch. 9 i [3], Ch. 0 i [7], Ch. i []). Near-exact distributios for this statistic are thus readily available from the results i [9,0,,5]. However, give the secicy of the case uder cosideratio, some further develomets may be sought, amely a simler way to obtai ad a simler formulatio for the shae arameters of the Gamma distributios ivolved i the art of the distributio of left utouched. These details will be addressed i Sectio. O the other had, the ower ( + ) of the lielihood ratio test statistic to test the shericy hyothesis o, based o a samle of size +, that is, to test the ull hyothesis H 0 : I ( usecied) (4) is the statistic (see Ch. 0 i [3], Ch. 0 i [7], Ch. 8 i []) jv j (trv ) ; (5) where V is the matrix i (3). Well-t ear-exact distributios have alread bee develoed for this statistic by Marques ad Coelho i [9]. However, ad somehow uexectedly, i this aer we will show that eve better ear-exact distributios may be obtaied for this statistic by taig as a basis the ear-exact distributios develoed for the statistic i (3) ad the decomosio erformed o s characteristic fuctio. These ear-exact distributios for will be obtaied from a
decomosio of the statistic i (5), which may be wrte as ; (6) where is the statistic i (3) ad Q j V j (trv ) ; (7) is the ower (+) of the lielihood ratio test statistic to test the hyothesis H 0j0 : : : : (give that, or, assumig that the variables i X are ideedet) (8) based o ideedet estimates of the variaces of the variables i X, oe for each j, based o samles of size +. I (7), V ad V j are the same as i (3). We may ote that V j is eher the MLE of j (j ; : : : ; ), the samle variace of the j-th variable i X i (), or the sum of squares of the deviatios from the samle mea for the j-th variable i X, accordig to the choice of V. The statistic i (7) may be derived from the lielihood ratio test statistic for the equaly of variace-covariace matrices (see Ch. 0 i [3], Ch. 0 i [7], Ch. 8 i []), taig each matrix to have dimesios (there they are agai, the grous of oe variable each). We may wre for H 0 i (4), H 0 i () ad H 0 i (8), H 0 H 0j0 o H 0 ; (9) to be read as "H 0j0 after H 0 ", meaig that we may test H 0 i two stes: (i) testig rst H 0, that is, if the variables i X are ideedet ad (ii) oce the hyothesis of ideedece of the variables is ot rejected, testig the if they all have the same variace. Uder H 0 i (4) the two test statistics ad i (6) are ideedet (see Ch. 0, subsec. 0.7.3 i [3]). This way to loo at this test will eable us to obtai eve better ear-exact distributios tha the already much well-t oes i [9]. As a side ote we may stress that the test statistic i (7) may be used, uder a slightly dieret settig, to test the ull hyothesis of equaly of variaces i (8), whout ay codioig if the estimators V j are based o ideedet samles, i which case those samles may have dieret sizes. 3
Ideed i the test statistic i (7) the requiremet is that the estimators V j have to be ideedet (amog other reasos, because oe has to be able to easily derive the distributio of trv, their sum). We may ote that this will be the case eve if the estimators V j come from a multivariate samle of size + of the variables i X, oce the ull hyothesis of ideedece of the variables is ot rejected, sice the the matrix V will have a Wishart distributio wh degrees of freedom ad arameter matrix the matrix i (), so that the diagoal elemets of V are ideedet (through a simle extesio of Theorem 3..7 i []). Near-exact distributios for the lielihood ratio test statistic of ideedece I order to obtai the c.f. (characteristic fuctio) of W log we may cosider Theorems 9.3. ad 9.3.3 of [3] which state that, for a samle of size +, Y j Y j ; (0) (where '' is to be read as 'is distributed as') wh + j Y j B ; j (j ; : : : ; ) () where is the statistic i (3), ad where, uder H 0 i (), the radom variables Y j i (0) ad () are ideedet. The, sice we ow that +j + h E Y h j + h +j ; h > + j we have, for i ( ), W (t) E e W Y j E Y j Y j +j : () +j Now, i order to be able to obtai a suable decomosio of the c.f. of W we may eher cosider the results ad develomets i sectio 5 of [0], taig for ; : : : ; m ad bc, or we may tae a dieret aroach which will ideed eable us to obtai simler exressios for the 4
shae arameters of the Gamma distributios ivolved i the art of the distributio of W which will be left uchaged. We will tae this secod aroach. The followig Lemma gives the c.f. of W, for both eve ad odd, uder a form that is suable for the develomet of ear-exact distributios for both W ad. Lemma Uder H 0 i (), taig bc ad "a?b", wh a; b; IN, as reresetig the remaider of the eger ratio of a by b, the c.f. of W log (where is the statistic i (3), used to test the ideedece amog the variables i X), may be wrte uder the form W (t) 0 @ A Y (t)! + (?) b c! + (?) +b c (t) (3) ; where (t) is the c.f. of the sum of ideedet Logbeta r.v.'s wh arameters ( ) ad ad (t) is the c.f. of a GIG (Geeralized Iteger Gamma) distributio [8] j of deth wh rate arameters ( ) ad shae arameters + (?) ( ; : : : ; ), that is the distributio of the sum of ideedet Gamma r.v.'s wh the give rate ad eger shae arameters. Proof: We will eed to cosider searately the two cases of eve ad odd. For eve, from () we may wre W (t) Y +j +j j ste (t) (t) ; (t) Y j ste +j +j (t) 5
where, for eve j, ( j) IN, so that, usig for z / Cf0; ; ; : : :g ad IN, we may wre (z + ) (z) Y 0 (z + ) ; (t) Y j ste Y j ste Y j ste Y j 0 Y j 0 j Y + j +j + + j + j + +j Y + j ( ) ad (t) Y j ste 0 @ 0 @ 0 @ +j +j Y A j ste Y A j ste Y A Y j 0 Y j 0 +j +j +j + + + j +j + j ( ) ; (4) so that we may ally wre, for eve, W (t) 0 @ Y A! b + c! + +b c : 6
For odd, we may wre W (t) Y j ste +j +j (t) (t) (t) ; Y j ste +j +j (t) where ow is for odd j that ( j) IN, so that followig similar stes to the oes used above to hadle (t), we may wre (t) Y j ste Y j 0 + j + + j + Y j ste Y j Y ( ) ad (t) Y j ste 0 @ 0 @ +j +j Y j ste 3 Y ( ) A Y j 0 ( ) A +j +j + + ( ) ; (5) 7
so that we may ally wre, for odd, W (t) 0 @ Y ( ) A!! t +b c ; ad so that, taig bc, we may wre W (t) uder the form i (3), for ay eve or odd. The, taig o accout that sice a sigle Logbeta distributio may be rereseted uder the form of a ie mixture of eher Exoetial or GIG distributios ([]), a sum of ideedet Logbeta r.v.'s wh eher the same or dieret arameters may thus be rereseted uder the form of a ie mixture of sums of ideedet Exoetials or GIG distributios, which are all GIG distributios ad that the GIG distributio self may be see as a mixture of Gamma distributios ([]), the relacemet of the sum of ideedet Logbeta r.v.'s by a sigle Gamma distributio or by a (e) mixture of Gamma distributios seems to be most adequate. Thus, ear-exact distributios for W may the be obtaied uder the form of a (Geeralized Near-Iteger Gamma) distributio ([0]) or mixtures of GNIG distributios by relacig (t) by the c.f. of a Gamma distributio or the c.f. of a mixture of Gamma distributios (see Aedix A for details o the GNIG distributio). These ear-exact distributios will match, by costructio, the rst two, four ad six exact momets of W. Theorem Usig for (t) i (3), the aroximatios: - s ( ) s wh s; > 0, such that @ h @t h s ( ) s t0 @h (t) @t h for h ; ; (6) t0 - X s ( ) s, where wh ; s ; > 0, such that @ h @t h X s ( ) s @h (t) t0 @t h for h ; : : : ; 4 ; (7) t0 8
- 3X s ( ) s, where 3 wh ; s ; > 0, such that @ h @t h 3X s ( ) s @h (t) t0 @t h for h ; : : : ; 6 ; (8) t0 we obtai as ear-exact distributios for W, resectively, i) a GNIG distributio of deth wh cdf (cumulative distributio fuctio) (usig the otatio i (4) i Aedix A) F (wjr ; : : : ; r ; s; ; : : : ; ; ) ; (9) where r j $ % j+ (?) ; j j ; (j ; ::: ; ); (0) ad m m m ad s m m m () wh m h i h @h (t) @t h t0 ; h ; ; ii) a mixture of two GNIG distributios of deth otatio i (4) i Aedix A), wh cdf (usig the X F (wjr ; : : : ; r ; s ; ; : : : ; ; ) ; () where r j ad j (j ; : : : ; ) are give by (0) above ad,, r ad r are obtaied from the umerical solutio of the system of four equatios X (r + h) (r ) h i h @h (t) @t h for these arameters, wh ; t0 (h ; : : : ; 4) (3) iii) or a mixture of three GNIG distributios of deth, wh cdf (usig the otatio i (4) i Aedix A) 3X F (wjr ; : : : ; r ; s ; ; : : : ; ; ) ; (4) 9
wh r j ad j (j ; : : : ; ) give by (0) above ad,,, s, s ad s 3 3X j obtaied from the umerical solutio of the system of six equatios (r + h) (r ) h i h @h (t) @t h for these arameters, wh 3. t0 (h ; : : : ; 6) (5) Proof: If i the characteristic fuctio of W s ( ) s we obtai i (3) we relace (t) by W (t) s ( ) s Y! + (?) b c! + (?) +b c (t) ; that is the characteristic fuctio of the sum of ideedet Gamma radom variables, of which wh eger shae arameters r j ad rate arameters j give by (0), ad a further Gamma radom variable wh rate arameter s > 0 ad shae arameter. This characteristic fuctio is thus the c.f. of the GNIG distributio of deth wh distributio fuctio give i (9). The arameters s ad are determied i such a way that (6) holds. This comels s ad to be give by () ad maes the two rst momets of this ear-exact distributio for W to be the same as the two rst exact momets of W. If i the characteristic fuctio of W i (3) we relace (t) by P r ( ) r we obtai W (t) X r ( Y ) r! + (?) b c! + (?) +b c (t) that is the characteristic fuctio of the mixture of two GNIG distributios of deth wh desy fuctio give i (). The arameters,, r ad r are deed i such a way that (7) holds, givig rise to the evaluatio of these arameters as the umerical solutio of the system of equatios i (3) ; 0
ad to a ear-exact distributio that matches the rst four exact momets of W. If i the characteristic fuctio of W i (3) we relace (t) by 3P r ( ) r we obtai W (t) 3X r ( Y ) r! + (?) b c! + (?) +b c (t) ; that is the characteristic fuctio of the mixture of three GNIG distributios of deth wh desy fuctio give i (). The arameters,,, r, r ad r 3 are deed i such a way that (8) holds, what gives rise to the evaluatio of these arameters as the umerical solutio of the system of equatios i 5, givig rise to a ear-exact distributio that matches the rst six exact momets of W. Corollary 3 Distributios wh cdf 's give by i) F ( log zjr ; : : : ; r ; s; ; : : : ; ; ), ii) iii) X 3X F ( log zjr ; : : : ; r ; s ; ; : : : ; ; ), F ( log zjr ; : : : ; r ; s ; ; : : : ; ; ), or where the arameters are the same as i Theorem, ad 0 < z < reresets the ruig value of the statistic e W, may be used as ear-exact distributios for this statistic. Proof: Sice the ear-exact distributios develoed i Theorem were for the radom variable W log we oly eed to mid the relatio F (z) F W ( log z) where F ( ) is the cumulative distributio fuctio of ad F W ( ) is the cumulative distributio fuctio of W, i order to obtai the corresodig ear-exact distributios for.
Ideed i order to obtai ear-exact -quatiles for we do ot eve eed the ear-exact distributios for, sice if we cosider the relatio () e W ( ) ; where () is the -quatile of ad W ( ) is the ( )-quatile of W we may easily obtai the ear-exact -quatiles of from the corresodig ( )-quatiles of W. 3 Near-exact distributios for the lielihood ratio test statistic of shericy Lemma 4 The c.f. of W log, where is the test statistic i (5) may be wre as W (t) Y + j + Y + j + j + + j j + j (t) Y! (?) b c! (?) +b c (t) ; (6) where bc. I (6) above, (t) is the c.f. of the sum of ideedet Logbeta r.v.'s, of which wh arameters ( + ) ad (j ) (j + ; : : : ; ) ad the remaiig wh arameters ad (j ) (j ; : : : ; ) ad (t) is the c.f. of a GIG distributio j of deth, wh rate arameters ( ) ad shae arameters (?) ( ; : : : ; ). Proof: From (7) we have E h h j + h Y + h so that taig W log ad usig the multilicatio formula for the Gamma fuctio Y (z) () ( ) z i z + i ;
we have (t) E e W W E e log E Y j + j Y j + j : (7) Thus, give the deio of i the revious sectio, which may ideed be wrte as 8 >< eve >: odd ; ad give that the for + j, j + sice for + j we have j + + 8 > + < + eve >: ( ) + + odd ; ad give that uder H 0 i (), ad are ideeet, we may wre, from (3) ad (7), the c.f. of W log, where is the statistic i (5), as W (t) W (t) W (t) (t) (t) 0 @ Y j + j + j A Y j Y j + + j + j + j + j (t) Y j + j + j Y j + + + j + + j 3
Y j + Y j + + j + j + (t)! (t) (t) + j + Y Y + + j + + + + j + j + j + + j j + j (t) : The, by relacig (t) i (6) by the c.f. of a Gamma distributio or the c.f. of a mixture of Gamma distributios, we will get ear-exact distributios for W log uder the form of a GNIG distributio or mixtures of GNIG distributios. Surrisigly eough, as we will see i the ext sectio, these ear-exact distributios have a eve better erformace tha the already well-t oes i [9]. Theorem 5 Usig for (t) i (6), the aroximatios: - s ( ) s wh s; > 0, such that @ h @t h s ( ) s t0 @h @t h (t) for h ; ; (8) t0 - - X 3X s ( ) s, where wh ; s ; > 0, such that @ h @t h X s ( ) s @h t0 @t h (t) for h ; : : : ; 4 ; (9) t0 s ( ) s, where 3 wh ; s ; > 0, such that @ h @t h 3X s ( ) s @h t0 @t h (t) for h ; : : : ; 6 ; (30) t0 4
we obtai as ear-exact distributios for W, resectively, i) a GNIG distributio of deth wh cdf (usig the otatio i (4) i Aedix A) F (wjr ; : : : ; r ; s; ; : : : ; ; ) ; (3) where r j $ % j (?) ; j j ; (j ; ::: ; ); (3) ad m m m ad s m m m (33) wh m h i h @h @t h (t) ; h ; ; t0 ii) a mixture of two GNIG distributios of deth, wh cdf (usig the otatio i (4) i Aedix A) X F (wjr ; : : : ; r ; s ; ; : : : ; ; ) ; (34) where r j ad j (j ; : : : ; ) are give by (3) above ad,, r ad r are obtaied from the umerical solutio of the system of four equatios X (r + h) (r ) h i h @h @t h for these arameters, wh ; (t) (h ; : : : ; 4) (35) t0 iii) or a mixture of three GNIG distributios of deth, wh cdf (usig the otatio i (4) i Aedix A) 3X F (wjr ; : : : ; r ; s ; ; : : : ; ; ) ; (36) wh r j ad j (j ; : : : ; ) give by (3) above ad,,, s, s ad s 3 3X j obtaied from the umerical solutio of the system of six equatios (r + h) (r ) h i h @h @t h for these arameters, wh 3. (t) (h ; : : : ; 6) (37) t0 5
Proof: The roof of this Theorem is i all similar to the roof of Theorem, more recisely, if i the characteristic fuctio of W i (6) we relace (t) by s ( ) s we obtai Y W (t) s ( ) s! (?) b c! (?) +b c (t) ; that is the characteristic fuctio of the sum of ideedet Gamma radom variables, of which wh eger shae arameters r j ad rate arameters j give by (3), ad a further Gamma radom variable wh rate arameter s > 0 ad shae arameter. This characteristic fuctio is thus the c.f. of the GNIG distributio of deth wh distributio fuctio give i (3). The arameters s ad are determied i such a way that (8) holds. This comels s ad to be give by (33) ad maes the two rst momets of this ear-exact distributio for W to be the same as the two rst exact momets of W. If i the characteristic fuctio of W i (6) we relace (t) by P r ( ) r we obtai W (t) X r ( ) r Y! (?) b c! (?) +b c (t) ; that is the characteristic fuctio of the mixture of two GNIG distributios of deth wh desy fuctio give i (34). The arameters,, r ad r are deed i such a way that (9) holds, givig rise to the evaluatio of these arameters as the umerical solutio of the system of equatios i (35) ad to a ear-exact distributio that matches the rst four exact momets of W. If i the characteristic fuctio of W i (6) we relace (t) by 3P r ( ) r we obtai W (t) 3X r ( ) r Y! (?) b c! (?) +b c (t) ; 6
that is the characteristic fuctio of the mixture of three GNIG distributios of deth wh desy fuctio give i (34). The arameters,,, r, r ad r 3 are deed i such a way that (30) holds, what gives rise to the evaluatio of these arameters as the umerical solutio of the system of equatios i (37), givig rise to a ear-exact distributio that matches the rst six exact momets of W. Corollary 6 Distributios wh cdf 's give by i) F ( log zjr ; : : : ; r ; s; ; : : : ; ; ), ii) iii) X 3X F ( log zjr ; : : : ; r ; s ; ; : : : ; ; ), F ( log zjr ; : : : ; r ; s ; ; : : : ; ; ), or where the arameters are the same as i Theorem 5, ad 0 < z < reresets the ruig value of the statistic e W, may be used as ear-exact distributios for this statistic. The roof of this Corollary is i all similar to the roof of Corollary 3 ad also similar cosideratios to the oes right after Corollary 3, cocerig the comutatio of ear-exact quatiles of the statistics W ad, aly here to the comutatio of ear-exact quatiles of the statistics W ad. 4 Numerical ad comarative studies I order to evaluate the qualy of the ear-exact aroximatios develoed for the lielihood ratio test statistics for testig ideedece i a set of variables ad for the shericy test we use, wheever the c.f.'s are available, two measures of roximy, Z j Y (t) (t)j dt ad Z Y (t) (t) t dt ; (38) wh max ys jf Y (y) f (y)j ad max ys jf Y (y) F (y)j ; (39) where Y reresets a cotiuous radom variable deed o S wh distributio fuctio F Y (y), desy fuctio f Y (y) ad characteristic fuctio Y (t), 7
ad (t), F (y) ad f (y) rereset resectively the characteristic, distributio ad desy fuctio of a radom variable X. These two measures may be derived directly from iversio formulas, ad may be see as based o the Berry-Essee uer boud o jf Y (y) F (y)j (see [5], [3], [6], Cha. VI, sec. i [8]). We should ote that for cotiuous radom variables, lim 0 () lim!! 0 (40) ad eher oe of the equalies above imly that X d! Y : (4) For further details o these measures see [9], where they are used to study the qualy of ear-exact distributios for the shericy test statistic. 4. Studies for the ideedece test statistic Mudholar et al. i [0] develoed a Normal aroximatio to the distributio of the lielihood ratio test statistic used for testig H 0 i (). These authors reseted umerical studies comarig their Normal aroximatio wh the aroximatios due to Box ad Bartlett ([6], [4]). Sice the asymtotic Normal aroximatio from Mudholar et al. i [0] yields ideed for log a o-cetral geeralized Gamma distributio, whose c.f. is ot maageable, i order to comare the erformace of the ear-exact distributio develoed wh this Normal asymtotic aroximatio, istead of usig measures ad, we decided to use a similar method to the oe used i [0] to assess the erformace of their Normal asymtotic aroximatio. We used the exact quatiles for comuted directly from the umerical iversio of the c.f. of log by usig the Gil-Pelaez iversio formulas (see [4]) what gives us a recisio at least equal to the oe used i [0] i terms of exact quatiles, which i tur give for the Normal asymtotic aroximatio of [0] exactly the same results obtaied by these authors. 8
Table { Values of the tail robabily error (arox: rob distributios ) 0 5 for the ear-exact 0:005 0:0 0:05 0:0 0:0 0:50 3 6 GNIG.9 0 0.7 0 0 :3 0 0 7:9 0 0 :65 0 5:75 0 0 MGNIG 9:9 0 9:77 0 4:0 0 - :8 0 5.4 0 6:78 0 8 GNIG.37 0 0.0 0 0 5:45 0 5:36 0 0 9:40 0 0 : 0 0 MGNIG 5:5 0 3:37 0 3:6 0 6:0 0 4.6 0 3:7 0 3 GNIG.6 0 0 3.47 0 :6 0 :76 0 0 3:30 0 0 5:33 0 MGNIG 7:79 0 :85 0 :75 0 4:88 0 4.69 0 :3 0 4 7 GNIG.93 0 0.55 0 0 :7 0 0 :57 0 0 6:80 0 0 7:90 0 0 MGNIG 7:73 0 6:58 0 3:44 0 3 :4 0 8.95 0 :7 0 9 GNIG 8.44 0.0 0 0 5:08 0 :38 0 0 4:60 0 0 4:48 0 0 MGNIG :58 0 :6 0 5:76 0 3 :80 0 6.38 0 3:87 0 4 GNIG 3.94 0 4.63 0 :80 0 7:6 0 :99 0 0 :67 0 0 MGNIG :48 0 4:79 0 3 6:4 0 :90 0 3 4.56 0 3:7 0 5 8 GNIG.08 0 0 5.6 0 6:09 0 8:5 0 :69 0 0 3:05 0 0 MGNIG 7:6 0 :33 0 3:49 0 3 6:88 0 3 7.96 0 3 :40 0 0 GNIG 3.4 0 3.8 0 3:73 0 :88 0 :35 0 0 :0 0 0 MGNIG :9 0 3:04 0 3:85 0 3 :07 0 3 8.56 0 3 5:78 0 3 5 GNIG.55 0.54 0 :79 0 :35 0 6:74 0 8:84 0 MGNIG :47 0 3 4:37 0 3:36 0 :7 0 3.85 0 4:0 0 4 6 9 GNIG.05 0.97 0 4:5 0 9:65 0 7:8 0 :83 0 0 MGNIG :58 0 3 8:79 0 3 :85 0 3 3:4 0 3 3:6 0 3 7:9 0 3 GNIG.33 0 3.0 0 3:0 0 :0 0 3 6:97 0 :35 0 0 MGNIG 5:09 0 5:59 0 :03 0 3 :0 0 3 6:30 0 3 4:94 0 3 6 GNIG.5 0.59 0 :43 0 4:3 0 4:7 0 6:8 0 MGNIG :80 0 3 4:49 0 3 :08 0 3 4:80 0 4 : 0 3 :56 0 3 7 0 GNIG 9.08 0.30 0 :0 0 8: 0 :8 0 8:07 0 MGNIG :3 0 3 4:00 0 3 9:5 0 4 7:33 0 4 4:77 0 4 :44 0 3 GNIG 8.85 0.5 0 :6 0 3:58 0 :8 0 6:47 0 MGNIG :55 0 3 :68 0 4 :05 0 3 :6 0 3 :3 0 3 :9 0 3 7 GNIG 7.50 0 8. 0 8:80 0 6:53 0 :93 0 3:65 0 MGNIG :68 0 :78 0 4 4:0 0 4 6:63 0 5:78 0 4 8:09 0 4 0 3 GNIG. 0 3.5 0 5:45 0 3:6 0 4:5 0 :87 0 MGNIG 7:0 0 6 :87 0 7 7:4 0 5 9:9 0 5 5:8 0 6 :0 0 4 5 GNIG.35 0 3.49 0 5:9 0 :53 0 5:39 0 :75 0 MGNIG :09 0 5 4:74 0 6 8:76 0 5 8:87 0 5 :4 0 5 :09 0 4 0 GNIG.97 0.8 0 3:70 0 :3 0 4:80 0 :5 0 MGNIG :50 0 5 :3 0 5 5:39 0 5 6:3 0 5 :7 0 5 :48 0 4 However, give that the exact quatiles comuted i this way have a recisio that does ot go beyod digs ad give that this recisio is ot eough for maig comarisos wh the ear-exact distributio M3GNIG, which requires a higher recisio, we have used i Table oly the ear-exact distributios GNIG ad MGNIG. I Table the errors dislayed are evaluated usig the exact same method used by Mudholar et al. i [0], the dierece betwee the aroximate ad the exact tail robabilies multilied by 0 5. The values cosidered for ad corresod to the same cases cosidered by Mudholar et al. i [0]. We ca observe that the errors obtaied whe usig the ear-exact distributios are always much smaller tha the oes give i Table of Mudholar et al. i [0] for their Normal aroximatio, maily for larger values of. 9
Table { Values of the measures ad for the ear-exact distributios GNIG MGNIG M3GNIG GNIG MGNIG M3GNIG 3 6 5.8 0 5.7 0 3 :4 0 3 5: 0 4 : 0 5 4:6 0 6 8 4.4 0 4. 0 3 5:3 0 4 :7 0 4 : 0 5 8:9 0 7 3.7 0.9 0 3 :7 0 5 9:4 0 5 3: 0 6 3:3 0 8 4 7 4.7 0 3.8 0 4 :4 0 5 :7 0 4 3:7 0 6 :0 0 7 9 4.0 0 3.4 0 4 5:4 0 6 : 0 4 :3 0 6 6: 0 8 4.7 0 3 7.8 0 5 3:7 0 7 4:4 0 5 7:4 0 7 3:0 0 9 5 8 8. 0 4.4 0 5 4:4 0 7 4:7 0 5 5: 0 7 : 0 8 0 7.5 0 4.3 0 5 : 0 7 3: 0 5 3:6 0 7 4:4 0 9 5 5.8 0 4 8. 0 6 :8 0 8 :5 0 5 :4 0 7 :6 0 0 6 9 3.3 0 4 3.5 0 6 3:6 0 8 :4 0 5 :7 0 7 :3 0 9 3.3 0 4 3.4 0 6 :9 0 8 :8 0 5 :3 0 7 5:5 0 0 6.8 0 4.4 0 6 : 0 9 9:5 0 6 5:6 0 8 3:5 0 7 0. 0 4 7.3 0 7 3:7 0 9 9:7 0 6 4: 0 8 :7 0 0. 0 4 7.7 0 7 : 0 9 7:9 0 6 3:4 0 8 7:4 0 7. 0 4 6. 0 7 4:0 0 0 4:6 0 6 :8 0 8 8:3 0 0 3.9 0 5 4. 0 8 6: 0 :0 0 6 3: 0 9 3:5 0 3 5. 0 5 5. 0 8 : 0 :9 0 6 3: 0 9 : 0 0.4 0 5 5. 0 8 4:6 0 :4 0 6 : 0 9 :5 0 0 3 5. 0 7.4 0 0 3:0 0 4 7:8 0 8 :5 0 :7 0 5 50. 0 6.8 0 0 7:4 0 4 4:9 0 8 9:7 0 : 0 5 00 7.4 0 7. 0 0 :5 0 4 :5 0 8 :8 0 3: 0 6 50 53 5.8 0 9.0 0 3 :5 0 9 : 0 9 :5 0 4 :9 0 0 00. 0 6.8 0 0 :9 0 7 4:9 0 8 9:7 0 7:5 0 9 50.3 0 8 4.6 0 3 4:3 0 9 8:5 0 0 :3 0 4 :0 0 0 00.0 0 8 3. 0 3 3:3 0 9 5:3 0 0 6:5 0 5 5:7 0 500 9.8 0 9 7. 0 4 5:0 0 0 :0 0 5:4 0 6 3: 0 I Table we use measures ad to better assess the relative erformace of the three ear-exact distributios GNIG, MGNIG ad M3GNIG as aroximatig distributios for the ideedece test statistic. From Table we may easily see that the ear-exact distributio M3GNIG has always a better erformace tha the other two ear-exact distributios ad we ca also see that the ear-exact distributio MGNIG always outerforms the GNIG ear-exact distributio. The values exhibed for the M3GNIG distributio for both measures, maily for the measure, which reresets a uer boud for the absolute value of the dierece betwee s c.d.f. ad the exact c.d.f., would lead us to recommed s use as a relacemet for the exact distributio, maily for larger values of. The three ear-exact distributios dislay a mared asymtotic behavior both for icreasig samle sizes ad icreasig umber of variables, although for larger values of we eed large eough samle sizes i order to be able to observe their asymtotic behavior i terms of icreasig values of. 0
4. Studies for the shericy test statistic The tables i this subsectio reset the values of the measures ad give i (38) for the ew ear-exact distributios develoed i this aer for the lielihood ratio test statistic used for testig shericy. I this subsectio our urose is to assess the qualy of the ew ear-exact distributios comarig them wh the oes already develoed, usig a dieret method, i [9]. I order to achieve our urose we have cosidered the exact same values for ad already cosidered i the umerical studies reseted i that referece. We will deote the ew ear-exact distributios corresodig to the GNIG distributio, the mixture of two GNIG distributio ad the mixture of three GNIG distributios resectively by GNIGew, MGNIGew ad M3GNIGew (leavig the ames GNIG, MGNIG ad M3GNIG, used i Table 7, for the corresodig ear-exact distributios develoed i [9]). Table 3 { Values of ad for the ear-exact distributios for W log, for 4, 6 ad 5, 7 4; 6 5; 7 GNIGew 3:85 0 5 :408 0 6 :80 0 5 :300 0 6 MGNIGew 5:67 0 7 :98 0 8 :580 0 7 :378 0 8 M3GNIGew 3:875 0 9 :80 0 0 4:936 0 9 :99 0 0 Table 4 { Values of ad for the ear-exact distributios for W log, for 7, 9 ad 0, 7; 9 0; GNIGew 3:953 0 6 4:34 0 7 3:366 0 7 4:40 0 8 MGNIGew :363 0 8 :007 0 9 :863 0 0 :667 0 M3GNIGew 5:67 0 3:4 0 8:847 0 4 6:58 0 5 Comarig Tables 3 ad 4 wh Tables ad i [9] we ca observe that the values for the ew aroximatios are always better wh the excetio of M3GNIGew for 5, 7 ad 7, 9. Table 5 { Values of ad for the ear-exact distributios for W log, for 4; 5; 7 ad 50 4; 50 5; 50 7; 50 GNIGew 9:70 0 6 6:005 0 8 9:490 0 6 8:493 0 8 :376 0 6 3:33 0 8 MGNIGew :68 0 8 :05 0 0 :936 0 8 :835 0 0 3:574 0 9 3:68 0 M3GNIGew :33 0 0 3:88 0 3 4:93 0 :389 0 3 :760 0 :43 0 4
Table 6 { Values of ad for the ear-exact distributios for W log, for 0; 0; 30 ad 50 0; 50 0; 50 30; 50 GNIGew 3:83 0 7 8:53 0 9 3:86 0 8 :53 0 9 5:78 0 9 4:646 0 0 MGNIGew :73 0 0 :686 0 3:6 0 :83 0 3 :609 0 3 :567 0 4 M3GNIGew 6:36 0 4 8:006 0 6 3:60 0 6 :057 0 7 : 0 7 5:588 0 9 Comarig Tables 5 ad 6 wh Tables 3 ad 4 i [9] we may verify that i almost all cases we have for the ew ear-exact aroximatios smaller values for the measures ad. The oly case where this fact does ot hae is whe 4 ad 50 for the measures of the MGNIGew ad M3GNIGew distributios. These ew ear-exact aroximatios also exhib the good asymtotic roerties revealed by the ear-exact aroximatios i [9]. We may say as a geeral coclusio that the ew ear-exact distributios have a better erformace tha the oes develoed i [9] for large values of wh large eough ( 4). Moreover i the ext Table we may see that for large values of ad values of close to we also have better values of both measures for the ear-exact distributios develoed i this aer. Table 7 { Values of ad for the ear-exact distributios for W log, for 0; 0; 30 ad ; ; 3 0; 0; 30; 3 GNIGew :058 0 6 :383 0 7 3:56 0 8 6:034 0 9 5:043 0 9 9:685 0 0 GNIG 8:940 0 6 :7 0 6 3:634 0 7 6: 0 8 5:78 0 8 9:945 0 9 MGNIGew 8:994 0 0 8:380 0 3:664 0 4:587 0 3 :543 0 3 :85 0 4 MGNIG 3:394 0 9 3:89 0 0 :73 0 :470 0 4:55 0 3 6:408 0 4 M3GNIGew :779 0 3 :048 0 4 :956 0 6 3:04 0 7 4:43 0 8 5: 0 9 M3GNIG 3:60 0 :706 0 3 :04 0 5 :8 0 6 :766 0 7 :05 0 8 5 Coclusios The rocess used to factorize the characteristic fuctios ivolved allowed us to obtai ear-exact distributios almost simultaeously for the ideedece ad the shericy test statistics ad also to obtai simle exressios for the shae arameters of (t) i (3) ad of (t) i (6), wh the shae arameters for the ear-exact distributios for the shericy test statistic havig much simler exressios tha the oes for the ear-exact distributios i [9]. The ear-exact distributios develoed for the ideedece test statistic show a much better recisio tha oe that is obtaied wh the Normal
aroximatio of [0] while the ew ear-exact distributios develoed for the shericy test statistic are more accurate tha the oes develoed i [9] for larger values of ( 0) or eve for smaller values of as log as the samle size is large eough. Acowledgmet This research was acially suorted by the Portuguese Foudatio for Sciece ad Techology, through the Ceter for Mathematics ad s Alicatios (CMA) from the New Uiversy of Lisbo. Aedix A Cumulative distributio fuctio for the GNIG distributio The desy ad distributio fuctios for the GNIG distributio are give i [0]. Let Z Z + Z where Z (r; ), wh > 0 ad r a osive o-eger ad Z gx i X i ; wh X i (r i ; i ) ; ideedet; where r ; : : : ; r g are osive egers ad ; : : : ; g > 0 are all dieret. The distributio of Z is a GIG distributio of deth g ([8]), while the distributio of Z, if Z ad Z are assumed ideedet, is a GNIG distributio of deth g +. We will deote this by Z GNIG(r ; : : : ; r g ; r; ; : : : ; g ; ) : The cumulative distributio fuctio of Z is give by F Z (zjr ; : : : ; r g ; r; ; : : : ; g ; ) r K r gx r X j e jz c X j; j i0 z r (r+) F (r; r+; z) z r+i i j (r++i) F (r; r++i; ( (4) j )z) (z > 0) where K gy j r j j ad c j; c j; j () 3
wh c j; give by () through (3) i [8]. I the above exressio F (a; b; z) is the Kummer couet hyergeometric fuctio (see []). This fuctio has usually very good covergece roerties ad is owadays easily hadled by a umber of software acages. Refereces [] M. Abramowz, I. A. Stegu, I.A., Hadboo of Mathematical Fuctios, 9th r, Dover, New Yor, 974. [] R. P. Alberto, C. A. Coelho, C. A., Study of the qualy of several asymtotic ad ear-exact aroximatios based o momets for the distributio of the Wils Lambda statistic, Joural of Statistical Plaig ad Iferece 37 (007) 6-66. [3] T. W. Aderso, A Itroductio to Multivariate Statistical Aalysis, 3rd ed., J. Wiley & Sos, New Yor, 003. [4] M. S. Bartlett, A ote o multilyig factors for various aroximatios, Joural of the Royal Statistical Society, Ser. B, 6 (954) 96-98. [5] A. Berry, The accuracy of the Gaussia aroximatio to the sum of ideedet variates, Trasactios of the America Mathematical Society, 49 (94) -36. [6] G. E. P. Box, A geeral distributio theory for a class of lielihood creria, Biometria, 36 (949) 37-346. [7] C. A. Coelho, Geeralized Caoical Aalysis, Ph.D. Thesis, The Uiversy of Michiga, A Arbor, MI, 99. [8] C. A. Coelho, The Geeralized Iteger Gamma distributio - a basis for distributios i Multivariate Statistics, J. Multivariate aalysis 64 (998) 86-0. [9] C. A. Coelho, The Geeralized Iteger Gamma distributio as a asymtotic relacemet for the logbeta radom variable - Alicatios, America Joural of Mathematical ad Maagemet Scieces, 3 (003) 383-399. [0] C. A. Coelho, The Geeralized Near-Iteger Gamma distributio: a basis for 'ear-exact' aroximatios to the distributios of statistics which are the roduct of a odd umber of ideedet Beta radom variables, J. Multivariate Aalysis, 89 (004) 9-8. [] C. A. Coelho, The wraed Gamma distributio ad wraed sums ad liear combiatios of ideedet Gamma ad Lalace distributios, Joural of Statistical Theory ad Practice, (007) -9. 4
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