View of Distribution of the likelihood ratio statistic for testing homogeneity of covariance matrices of completely symmetric Gaussian models

DISTRIBUTION OF THE LIKELIHOOD RATIO STATISTIC FOR TESTING HOMOGENEITY OF COVARIANCE MATRICES OF COMPLETELY SYMMETRIC GAUSSIAN MODELS

L. Cardeño, A. Gómez, V. Tayal

1.INTRODUCTION

Let X_i be a _{p u}1 random vector, which has a multivariate normal distribution with mean vector P_i eP_i and covariance matrix 6i Vi2[(1Ui)Ip UiJ] where V_i2 !0,P_i and U_i( 1/( p   are unknown scalars, 1) U 1) i }1, , ,m

1 (1,1, ,1)c_pu

e ! ,

p

I is the identity matrix of order p and J is a p pu matrix with each entry unity. Under this model consider the problem of testing equality of covariance matrices, i. e.,

2

0 : 1 m [(1 ) p ]

H 6 } 6 V U I UJ

against general alternatives.

Han (1975) derived the likelihood ratio criterion (LRC) for testing homogene-ity of covariance matrices under compound symmetric Gaussian models and has shown that the test based on the modified LRC is better than the test derived by using Roy’s union intersection procedure (see Srivastava, 1965; Krishnaiah and Pathak, 1967). Han (1975) and Gupta and Nagar (1986, 1987) have derived dis-tributional results in the null as well as non-null case. Compound symmetry is of-ten found in repeated measures designs which are not time dependent. If the or-der of the responses does not matter than the covariances are exchangeable. The term exchangeable is synonymous with compound symmetry. The completely symmetric Gaussian models arise in symmetries in animals and plants. These models have also been shown to be useful in areas like medical research and psy-chometrics.

The problem of testing H₀ was considered by Tayal, Gupta and Nagar (1988). They derived likelihood ratio criterion ƌ, its null moments and several distribu-tional results. Suppose that independent random samples ₁, ,

i

i } iN

from the i -th population, i }1, , .m Let Q be an orthogonal matrix with first

row e'/ p . Then diag( , , , ) *

i i i i i

Q6 Qc a b ! b 6 and Q_i P_iQe

1

i

pPe , say, where a_i V_i2[1 ( p1) ],U_i b_i V_i2(1U_i), and e₁ (1,0, ,0)'! _pu1. The null hypothesis for testing equality of covariance matrices is equivalent to

1 1

: _m; _m

H a " a b " b . Let Y_iu QX_iu and y

iju be the j -th component

( j } of 1, , )p Y_iu,u 1, ,!N_i, _i }1, , ._m Further, let 2 1 ₁( 1 1) , i N i _u i u i W

¦

y  y _˜ 2 1 i₁ 1 , 2 ₂ i₁ 1 N p N i i _u i u i _{j u} i u N y ˜

¦

y W

¦ ¦

y , and 0 ₁ m i i

N

¦

N . The likelihood ratio

sta-tistic / for testing H can be expressed as (Tayal, Gupta and Nagar, 1988), 0 0 0 /2 ( 1) /2 /2 1 2 0 1 1 /2 ( 1) /2 /2 1 ₁ 1 ₁ 2 . i i i m _N m _{p N} N p i i i i m _{N p m} N _m p N i i _i i _i i W W N N _{W W}   / ˜ ˜ ª º ª º ¬ ¼ ¬ ¼

–

–

–

_¦

_¦

(1)

The h -th null moment of ƌ is available as 0 /2 0 0 0 /2 0 0 1 1 [( )/ 2] [( 1) /2] E( ) [{ (1 ) }/2] [( 1) (1 )/ 2] [{ (1 ) 1}/2] [( 1) (1 )/ 2] [( 1)/2] [( 1) /2] i N ph h m _{N ph} i i m i i i i i N N m p N N h m p N h N N h p N h N p N * * / * * * * * *           u  

–

–

(2)

where Re(N h_i )! (N_i  ,1) i } and Re( )˜ denotes the real part of( )˜ .1, ,m Tayal, Gupta and Nagar (1988) used the above moment expression to derive exact and asymptotic null distributions of ƌ. They obtained, for arbitrary p, as-ymptotic expansion of the distribution of a constant multiple of 2 lnƌ in series involving chi-square distributions. They also derived exact distribution of ƌ2/N

when p =2 and N1 " Nm N using the residue theorem (see Nagar, Jain,

and Gupta, 1988; Gupta and Nagar, 1999). As observed by them, the exact distri-bution of ƌ for arbitrary p , using the residue theorem, does not seem possible to derive. We therefore, in this article, derive the distribution of _U (/2/N0) ,1/s where (s s !0) is an adjustable constant, in terms of beta series for arbitrary p.

The distribution of _U (/2/N0) ,1/s in a series of beta distributions, is derived in Section 2. In Section 3, a comparison is made between the significance points obtained by using results derived in Section 2 and exct results obtained by Tayal, Gupta and Nagar (1988). It is found that the expansion in terms of beta is fairly good and can be used to compute percentage points for p t3.

2.NULL DISTRIBUTION

Let _U (/2/N0)1/s. Substituting

0

2 /h N s for ,h ƣ_i =N_i /N₀, _i=1, ,… _m and then N₀ =n+2Ƥin (2), we obtain

/ 1 1 [( )/2 ] [( 1)( /2 )] 1 E( ) [( )/2 / ] [( 1)( /2 / )] [ ( /2 / ) 1/2] [ ( 1)( /2 / )] . [ ( /2 ) 1/2] [ ( 1)( /2 )] i h m _{ph s} i i m i i i i i n m p n U n m h s p n h s n h s p n h s n p n J G G G G J J G J G J G J G * * * * * * * *     ˜             u    

–

–

Now taking the inverse Mellin transform of the above expression, we get the density of U as 1 1 / 1 1 1 [ ( /2 / ) 1/ 2] ( ) (2 ) [( )/2 / ] [ ( 1)( / 2 / )] , [( 1)( / 2 / )] i m i i m _{ph s} i i m i h i n h s f u K n m h s p n h s u dh p n h s L J L J G SL J G J G G  f   f   * * * *          u   

–

³ –

–

₍₃₎ where 0 u 1,L  and 1 1 1 [( )/ 2 ] [( 1)( / 2 )] . [ ( /2 ) 1/2] [ ( 1)( / 2 )] m m i i i i n m p n K n p n G G J G J G * * * *        

–

–

(4)

Now substituting /2n h s/ t s/ , we have /2 1 /2 1 1 ( ) (2 ) i ( ) ,0 1, c m pn ns t i i _c f u K u t u dt u L J L SL J M  f     f  

–

_³

(5) where c ns/ 2 and 1 1 / 1 ( ) [ ( / ) 1/2] [ ( 1)( / )] ( ) . / /2 [( 1)( / )] i m m i i i i m _{pt s} i i t s p t s t t s m p t s J J G J G M G G J * * *        

–

–

–

(6)

Expandig the logarithm of ( )M t using Barnes’ expansion and converting back, one obtains 3/2 1 ( 1)/2 1 1 ( ) (2 ) ( 1) i 1 m p m m i i Q t t p s t Q GJ D D D I S J  _f   _   § · ª _ º ¨ ¸ « » © ¹ ¬

¦

¼

–

, (7)

where Q  ,m 1 0 1 1 , 1, 2,3, , 1 r r r Q rA Q Q D D D D D

¦

 } , (8) 1 1 1 1 1 1 1 ( 1/2) ( ( 1) ) (-1) ( 1) ( 1) (( 1) ) 2 ( 1) r r m r i r i r r r r r i i i r r r B B p s A r r p B p m B p D _{J G J G} J J G G      ª  _  «  _¬  º  _{§ ·}   _¨  _¸»  © _¹¼

¦

¦

(9) and ( )B ˜_r is the Bernoulli polynomial of degree r and order one. Substituting (7) in (5) we get /2 1 ( /2 ) 3/2 1 ( 1)/2 1 1 1 ( ) (2 ) ( 1) (2 ) 1 , 0 1 i m _{p n n s} m m i i c t c f u K p s u Q t u dt u t G Q Q D D D J L L S J SL        f _f     f  ª º u _«  _»   ¬ ¼

–

¦

³

(10) Also, since - [1 ₁ r] ( ) r r

t Q 

¦

f Q t O tQ , using Nair (1940), one can expand -1 [1 r] r r t Q 

¦

f Q t in factorial series as -1 0 ( ) 1 ( ) r r r t a t Q t R t a Q D D Q D f f  * * ª _ º  « » _{  } ¬

¦

¼

¦

(11)

where a is a constant to be determined later and the coefficients R 's can _ơ be determined explicitly as done below. Expanding logarithm of

(t a)/ (t a Q D)

*  *    using Barnes’ expansion and converting back, one

ob-tains 1 ( ) 1 ( ) j j j t a t C t t a Q D D Q D f    * * ª º  _ « »    _«¬

¦

_»¼ (12) where , 0 1 1 , 1, 2, , , 1 j j j C A C j C j D

¦

DA D A } } D A A , (13) and

1 1 ( 1) [ ( ) ( )] ( 1) AD _ B  a  Q D B  a A A _{A A} A A . (14)

Sustituting (12) in (11) and comparing the coefficients of like powers of t on both sides, one gets

, 0 0 , 1, 2, , , R 1. j j j j Q R C D D

¦

D D D } } (15)

Now using (11) in (10) and integrating term by term which is valid, the density of U is obtained as ( /2 ) 3/2 1 ( 1)/2 1 /2 1 1 0 ( ) (2 ) ( 1) (1 ) ,0 1. ( ) i m _{p n} m m i i n s a f u K p s u u R u J G Q Q D D D S J Q D          f   u   * 

–

¦

(16)

Since the series in (16) is uniformly convergent for 0  , the distribution u 1 function can be derived by integrating term by term, and this procedure would then lead to a series in incomplete beta function given by

( /2 ) 3/2 1 ( 1)/2 1 0 ( ) (2 ) ( 1) ( /2 ) ( /2 , ) , ( /2 ) i m p n m m i i u F u K p s ns a R I ns a ns a J G Q D D S J Q D Q D      f _* *   u     

–

¦

(17)

where I ( , )_u ˜ ˜ is the incomplete beta function. Also expanding K and ( /2ns a)/ ( /2ns a Q D)

*  *    and using Barnes’ expansion, one has

* ( /2 ) 3/2 ( 1) ( 1)/2 1 1 (2 ) ( 1) 1 2 ( /2) i m _{p n} m m i i Q n K p n J G D D D Q S   _  J    § · ª _ f º « » ¨ ¸ © ¹ ¬

¦

¼

–

, (18) where * * * * 0 1 1 , 1, r r r QD rA QD Q D D

¦

 (19) * r r r A A s  , (20)

1 ( /2 ) 1 1 1 ( /2 ) 2 2 j j j ns a ns C ns ns a Q D D Q D   _f  * * ª º  § · _{« } § · _» ¨ ¸ ¨ ¸    © ¹ _«¬

¦

© ¹ _»¼ (21)

and CD_j's are given by (13). Substituting (18) and (21) in (17) and simplifying we get the following result:

1 1 1 ( ) , 2 ( /2) u F u I ns a G n D D D Q f § _ ·_ ¨ ¸ © ¹

¦

(22) where * , -0 0 1 , - . 2 j j j j u j Q C G R I ns a j s D D D D Q D D    §¨   ·¸ © ¹

¦

¦

A A A A (23) Substituting D 1, 2in (2.21) we get * * * 0 10 0 01 1 00 1 1 , 1 0 , ₀ 2 2 u u ns Q C ns Q C Q C G R I a R I a s s s Q Q ­ ½ § _{ } · _ § _ · _ ® ¾ ¨ ¸ ¨ ¸ © ¹ © _{¹¯ ¿} (24) and * * * 0 20 0 11 1 10 2 2 2 1 2 1 * * * 0 02 1 01 1 00 0 2 1 0 , 2 , 1 2 2 , . 2 u u u ns Q C ns Q C Q C G R I a R I a s s s ns Q C Q C Q C R I a s s s Q Q Q ­ ½ § _{ } · _ § _{ } · _ ® ¾ ¨ ¸ ¨ ¸ © ¹ © _{¹¯ ¿} ­ ½ § ·  _¨  _¸®   _¾ © _{¹¯ ¿} (25)

From (15), (8), (13), (10) and (20) we have R ₀ 1, Q₁ R C_{1 10}R C_{0 01},

2 2 20 1 11 0 02 Q R C R C R C , Q₀ 1, Q₁ A₁, Q₂ A₁2/ 2 A₂, C₀₀ C₁₀ C₂₀ 1, 11 11, 01 01, C A C A C₀₂ A₀₁2 /2 A₀₂, Q₀* 1, Q₁* A₁*, Q₂* A₁*2/2A₂*, * * 2 1 1/ , 2 2/ A A s A A s and G₁ simplifies to 1 1 01 1 1 1 ( ) , 1 , 2 2 u u G A A I ns a I ns a s Q Q ­ § · § ·½  _{® ¨}   _{¸ ¨}  _¸¾ © ¹ © ¹ ¯ ¿

where from (9) and (14) we get

1 1 11 9 1 1 24( 1) 3( 1)( 3) 24 1 m i i p s A m m m p J G ª  § _{ }· _{   } º « _ ¨ ¸ » « © ¹ » ¬

¦

¼

and A₀₁ Q(2a  Q 1 /2) . Now if we choose a  (Q 1)/2 and 1 (11 9) (1/ ) 1 3( 1)( 1)( 3) 24( 1)( 1) m i i p p m m p m J G ª º  _¬  _¼     

¦

then G ₁ 0 and G₂ in (25) reduces to

* 02 2 2 2 1 1 , 2 , 2 2 u u A G A I ns a I ns a s Q Q ­ ½ § · § · § · _¨  _{¸® ¨}   _¸ _¨  _¸¾ © ¹ © ¹ © ¹¯ ¿

where from (9), (20) and (14), we have

* 2 2 2 1 1 1 1 ( 1)( 2) 4( 1) 8 6 m i i A m m m m G J ª º  _«      _» ¬

¦

¼

and A₀₂ Q(1-Q2)/ 24. In order to make G ₂ 0 we choose s2 such that

* 2 2 02/ 0 A A s   . Thus we obtain 2 2 * 2 (1 )/24 s Q Q A .

It may be noted here that the choice of s is possible only if Q ! . For 1 m 2, we have Q and in this case we choose 1 s 1. Finally, simplifying G₃ in (14), from (22) we get the following asymptotic expansion of the distribution of U :

4 3 3 8 ( ) , , 3 , ( ). 2 2 2 u u u ns R ns ns F u I a I a I a O n n Q ­ Q Q ½  § _ ·_ § _{ } ·_ § _ · _ ® ¾ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ ¯ © ¹ © ¹¿ (26)

3.COMPARISON OF PERCENTAGE POINTS

For N₁ " N_m N and p 2, m 3, m , Tayal, Gupta and Nagar 4 (1988) have given the following exact results of the distribution of ƌ1/N

V in

series involving Psi function (Abramowitz and Stegun, 1965; Luke, 1969):

2, 3 p m 2 2 0 2 2 2 ( 1/3) ( 1/3) 1 ( ) ln 2 ( 1) 3 ( 1) ( 1) ( 1/3) ( 1/3) 1 , 0 1. 3 ( !) (1/3) (2/3) N i N N g v v v i i N i i i v i \ \ \ f  * * * * * * *   ­_{   } § _ · ® ¨ ¸  ¯ © ¹    ½ § ·  _¨  _¸¾   © ¹¿

¦

2, 4 p m

^

2 2 2 2 2 -1 0 2 ( -1) ( 1/2) ( 1/2) ( ) ( ln ) 4(ln 4 1)( ln ) ( 1) 2 1 4(ln 4 1) 4 ( ln ) ( ) 2 ( 1) 3 ( 1/2) ( 1/2) 1 1 , 0 1. 2 2 ( !) N i i N N N g v v v v N v v i i i i i i v i i S S \ S \ \  f * * * * *   _{ª    } ¬           º   ½ § · § ·  _¨  _¸ _¨  _{¸¾ »}   © ¹ © ¹¿ _¼

¦

The computation of the exact percentaje points has been carried out by using 0

( ) x ( )

F x

³

g t dt where ( )g t is the density of V . The computation is carried out by using series repreentation given above. First ( )F x is computed for vari-ous values of x . It is checked for monotonicity and for conditions ( )F x o0 as

0

x o and ( )F x o1 as x o1. Then x is computed for various values of ( )

F x . The approximate percentage points have been obtained by using first term in (26). A comparison of the exact and approximate percentage points is given in Table 1 and table 2. It is observed that beta approximation performs fairly good.

TABLE 1

Comparison of the Exact significance points of V with Beta approximation, m=3, p=2

0.05

D D 0.01

N Exact Beta Exact Beta

5 0.193203 0.192961 0.309137 0.308957 10 0.480094 0.48007 0.591986 0.591972 15 0.623450 0.623445 0.713466 0.713464 20 0.705784 0.705782 0.779584 0.779583 25 0.758814 0.758813 0.821004 0.821004 30 0.795732 0.795731 0.849352 0.849352 35 0.822883 0.822883 0.869962 0.869962 40 0.843681 0.84368 0.885618 0.885618 TABLE 2

Comparison of the Exact significance points of V with Beta approximation, m=4, p=2

0.05

D D 0.01

N Exact Beta Exact Beta

5 0.12604 0.125765 0.212278 0.212038 10 0.395871 0.395836 0.499626 0.499603 15 0.550343 0.550335 0.639382 0.639377 20 0.643634 0.643631 0.718926 0.718924 25 0.705319 0.705318 0.769926 0.769926 30 0.74896 0.74896 0.805333 0.805333 35 0.781411 0.781411 0.831326 0.831326 40 0.806466 0.806465 0.85121 0.85121

ACKNOWLEDGMENT

The researche work of Liliam Cardeño and Armando Gómez was supported by the Comité para el Desarrollo de la Investigación, Universidad de Antioquia research grant no.

Departamento de Matemáticas LILIAM CARDEÑO

Universidad de Antioquia ARMANDO GÓMEZ

Medellín, Colombia

Department of Statistics VPIN TAYAL

University of Rajasthan Jaipur, India

REFERENCES

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York.

A. K. GUPTA, D. K. NAGAR, (1986), Testing equality of covariance matrices under intraclass correlation structure, “Tamkang Journal of Mathematics”, 17 , pp. 7-16.

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Statis-tics”, Series A , 16(11), pp. 3323-3341.

A. K. GUPTA, D. K. NAGAR, (1988), Asymptotic expansion of the nonnull distribution of likelihood ratio statistic for testing multisample sphericity, “Communications in Statistics”, Series A, 17(9) pp.

3145-3155.

A. K. GUPTA, D. K. NAGAR, (1999), Distribution of Votaw’s ƫ (vc)₁ criterion for testing compound sym-metry of a covariance matrix, “Acta Mathematica Scientia”, 19(1), pp. 62-80.

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RIASSUNTO

Distribuzione del rapporto di verosimiglianza per verificare l’ipotesi di omogeneità delle matrici di cova-rianza di modelli gaussiani simmetrici completi

In questo lavoro viene derivata la distribuzione del rapporto di verosimiglianza per saggiare l’ipotesi di uguaglianza delle matrici di covarianza di modelli gaussiani simmetrici completi nel caso di serie beta. La soluzione trovata è confrontata con i risultati esatti.

SUMMARY

Distribution of the likelihood ratio statistic for testing homogeneity of covariance matrices of completely symmetric Gaussian models

Iin this article the distribution of the likelihood ratio test statistic for testing equality of covariance matrices of completely symmetric Gaussian models has been derived in beta series. Comparison of results with the exact results has also been done.