On the independence of quadratic forms in a noncentral. Osaka Mathematical Journal. 2(2) P.151-P.159

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1 Title Author(s) On the independence of quadratic forms in a noncentral normal system Ogawa, Junjiro Citation Osaka Mathematical Journal. 2(2) P.5-P.59 Issue Date 95- Text Version publisher URL DOI.89/8824 rights

2 Osaka Mathematical Journal, Vol. 2, No. 2, November, 95 On the Independence of Quadratic Forms in a Non-Central Normal System* By Junjiro OGAWA The problem on the independence pf quadratic forms in a normal system originates from the famous treatise of W. G. Cochran }. Cochran proved the following theorem: "Let x l9 x 2,...,x n be independently distributed according to the identical nόrttial law ΛΓfO, ) 2), #ι, Q2, >Qm being m quadratic forms of x l9 x 2,... 9 x n, their ranks m r l9 *2, > r m respectively. If Σ QJ Σ X L then the necessary sufficient condition for q l9 q 2f..., q m to be independent statistically is that ^γ 3 =n. When these conditions are satisfied, then q l9 q 2,..., q m are distributed independently according to the chi-square distributions of degrees of freedom r lf r 2f... f r m respectively". Then, in 94, W. G. Madow 3) proved the generalization of this theorem for the non-central case, he obtained the same condition. On the other h, A. T. Craig 4), H. Hotelling 5) H. Sakamoto 6) have extended the theorem in the other direction, their point being as follows: Let x lf...,x n be independently distributed according to the normal law N(,) written as a vector χ=(a?ι,..., a? n ), furthermore let A B be two real ymmetric matrices, then the necessary sufficient condition for two quadratic forms q =ιa^c f q 2 =ϊb f to ba independent statistically is AB=. But their proofs were insufficient, K. Matsushita 73 we 8) gave the complete proofs. In this note we shall generalize the last theorem for the non-central case, prove the following two theorems show one example of their applications.. THEOREMS. Theorem I. Let #, a? 2 *»#n be normally independently distributed with means %, a z,...,a n respectively with the common variance unity. If we denote n rom variables Xι,x 2,..., x n by a vector notation =(#ι, # 2,...,#J, make two quadratic forms g x = M^

3 52 Independence of Quadratic Forms f? 2 = #ΐ'> where A B are real symmetric matrices, then the necessary sufficient condition of their statistical independence is AB = o: (i) PROOF 9) : In the sequel we shall write α for the rc -dimensional vector (a l9 a 2 >...,αj, which is neither proportional to unit vector nor to zero vector. The moment-generating function φ(t l9 2 ) of the joint distribution of q l q z is given as follows: >oo f r» * 2 )=(2τr) J J... J exp ^(h + ί a 9 2 g- Σ ( In vector notations, r /J Er -2f A--*2S -o\/r' exp L --J-(sl)( ^ \ α ααv\ι/j )(: ) r where J57 is the unit matrix of degree n primes mean the transposed vectors. We denote the matrix E 2t±A 2t 2 B by G for a system of fixed values of ^. 2 for which the matrix is positive-definite operate the variates transformation represented by E (K where then /C K- =(l)3 (θ -oc the Jacobian of this transformation is unity. Therefore, we have

4 J. OGAWA 53 =(2*Γ" ϊ Γ... Γ expγ-^ J-oo J-«L 2 i.e. φ(:t lf t 2 )=\E-2t l A-2t 2 B\ 2 The necessary sufficient condition of the statistical independence of q l q 2, as is well known, is given by which becomes after some calculations, \E-2t A\\E-2t 2 B \E-2t l A-2t 2 B\ ~^α'. (2) So we have only to show the equivalence of () (2). First, () implies (2) : If AB=Q, then A B will be ^brought into diagonal forms simultaneously by means of the same orthogonal transformation, say P, so, we shall have no loss of generality by assuming that ' - «r J M ) r OJ Ό. 5 -w r P5P'= "!. \ Ai. }n r s o o) (4) where r 5 are the respective ranks of A B. If we operate the orthogonal transformation P on both sides of the equation (2), then the left-h side of the equation is clearly seen to be unity, therefore, to obtain the equality of both sides, it suffices to show that the matrix in the curled bracket on the right -h side is zero matrix. It is certainly true, because

5 54 Independence"of Quadratic Forms =P' =p> \ i 2t l a r i ) '. > n r lj J/./ + { \ o ] i j 2(ί (^ l-2ί,a i l-2t,a i ). >7ί r s n i \ \ ''* ί ^ -2 <2^, β i --2t 2 β s N ' / *. 5τz r s i) - ';-..) Ί ' P=. Second, (2j implies (): (2) is an equation of the form where G(t lt tύ=a (E-2t l \E-2t z B\ \E-2t ί A-2t z B\ a', (3) so they are rational functions of ίj ί 2 which are analytic at the origin. Let 'F(t lt t^) G(ti, f 2 ) be analytic in the neighbourhood of the origin, I ίj I ' < 6, I ί 2 <C f If we fix a value of ί 2 such that ί 2 i <C s > then G^ίj, ί 2 ) is a rational function of t ίt which has no poles in the

6 J. OGAWA 55 whole complex t λ -plane including αo. Otherwise, a pole of G(t lf * 2 ) would be an essential singularity of e ^'*^, which contradicts (3), where Ffo, ί 2 ) was a rational function of t lβ So G(t lf f 2 ) is a constant in ίj, provided a value of t z is fixed such that f 2 <. I n the same way we can prove that G(t lf * 2 ) is a constant in t 2, provided a value of t λ is fixed such that \t λ \ < 6. Hence GC^, * 2 ) is a constant in the domain tj. <, ί 2 ] <[ <9. From the equation (3), the same is true for F(t l9 ί 2 ), so that i. e. F(t l9 * 2 )=F(, )=. Jξ?-2.ί A-2* 2 β = l^^2< A #-2 2 J3. (4) holds identically in the domain t τ \ < 6, \ t 2 \ <. (4) is equivalent to () >. Theorem 2. Let ϊ v =(»ivί χ zv> > «?*v) y==: l^ 2,..., τι ' be a rom sample of size n drawn from a fc-dimensional normal population _Λ _! 2 r Λ (2τr) I Λ I 2 exp[- A an( where A=(λ w ) is the moment-matrix of Xι,x 2 > >^fc i (λ u ) i s its inverse. If q λ g 2 are the quadratic forms in # ίv, i=l, 2,...,fc, * =,2,...,7*, then the criterion of the statistical independence of q λ ^2 is given by where A B are the respective matrices of q λ # 2 Λx 7 % denotes the Kronecker's product of Λ the unit matrix of degree n. PROOF: As in the previous paper ιυ, if we make use of the non-singular variate transformation, which brings A into unit matrix, then the Theorem 2 reduces to Theorem. 2. Application. D. S. Villars 2) has recently treated the significance test estimation of exponential regression. His point being as follows: Consider a variate z, whose distribution for a given value of a fixed variate, ί, is: where α, 6 k are real-valued parameters. The regression of z on

7 56 Independence of Quadratic Forms t is exponential, for it follows from (5) that the expected value of z, given ί, is: E(z\f) a-be- kt (6) On the basis of a rom sample O N (z l9 t λ z z, t* %*> t N \ it is required to test whether k= or oo. When N is an even integer (>6), times t lt t 2,...,t N at which measurements of z are made are arranged such that ί2«-*2-ι=δ, a constant. α=l, 2,..., n=~, ( 7 ) the odd time intervals t 3 t z,t 5 tt,... do not have to be equal. Let x a =z 2a -i y^z^ for α=l, 2,...,w. From (5) (7), it follows that n pairs x^ y a are normally independently distributed with common variance cr 2, that where μ Λ =E(x a \ v==efy3 9 'h=<sq e-**\ And then, Villars proposed 3) the criterion π, * ~~ for testing the null -hypo thesis, where Λ, a I, 2,...,n, ( 8 m=e-^. In this case, n».«=^-σ««z/= Σ /«= /6 α=ι q 2 = are both non-negative quadratic forms of a non-central normal system, so their statistical independence can be Judged by the Theorem of the previous section. The non-central normal system to be considered here is: ^)-V- 2W exp [- I Σ (*.-/O 2 + Σ ί»α-o 2 }] ΠdxJIdVa () Here we make use of the so-called 'Ήelmert's orthogonal transformation", represented by the matrix

8 r i ^n n-l \. n Πu V/» \/n(n Γ) iw -2 4n-l \/n ^»Cn-l)..j/(»-ϊχ»-2) ^ ^ \/n(n ϊ) ι/»(«) 57 V^-lXtt-2) V/^- ^- 2 ). T \ 9 /2 write On, (μι, b2> The Jacόbian of this transformation being unity, so () is transformed into (2τr)- n σ~^ exp Γ- ^2! Σ (fα- /^Λ 2 + Σ (^7α~ O 2 }] ΠdξJIdη a9 L ^σ (a = l α = l )J Further q, g 2 are transformed into respectively. Σ ^2 So the matrices of <h q 2 are transformed into - - Γ O m Ό O i O m - - Om Om 2 - m ό-' m - - w 2 ~m -m 2 '-m - - -m l- O -mo ί respectively, hence, for any real value of m, it is readily seen that

9 58 Indenpendence of Quadratic Forms AB =. Therefore, QΊ q 2 are independent statistically. In addition, q^/σ 2 ( + m 2 ) is distributed according to the % 2 -distributϊon of degrees of freedom n, ίι/σ 2 (l f m 2 ) is distributed according to the noncentral % 2 -distribution 4). Therefore,. from (9), F f is distributed according to the so-called "non-central F' distribution". (Received June 24, 95) References Notes ) W. G. Cochran, The Distribution of Quadratic Forms in a Normal System, with Applications to the Analysis of Covariance, Proc. Camb. Phil. Soc., Vol. 3, ) N (,) denotes the normal distribution law with mean unit variance, i. e., the stard normal distribution. 3) W. G. Madow, The Distribution of Quadratic Forms in Non-central Normal Rom Variable, Ann. Math. Statist. Vol. II, 94. W. G. Madow, On a Source of Downwards Bias in the Analysis of Variance Covariance, Ann. Math. Statist., Vol. 9 (948) pp ) A. T. Craig, On the Independence of Certain Estimates of Variance, Ann. Math. Statist., Vol. 9, 938. A. T. Craig, Note on the Independence of Certain Quadratic Forms, Ann. Math. Statist. Vol. 4, ) H. Retelling, Note on a Matric Theorem of A. T. Craig, Ann. Math. Statist. Vol. 5, ) H. Sakamoto, On the Independence of Statistics, Kokyuroku of the Institute of Statistical Mathematics, Vol., ) K. Matsushita, Note on the Independence of Certain Statistics, Ann. of Instit. Statist. Math. Vol., No., ) J. Ogawa, On the Ίndepedence of Billinear Quadratic Forms of a Rom Sample from a Normal Population, Ann. Instit. Statist. Math. Vol., No., ) This elegant proof, I owed to Mr. Seiji Nabeya of the Institute of Statistical Mathematics. ) This was the very point of our earlier papers (7) (8). ) See (8). 2) D. S. Villars, A Significance Test Estimation in the Case of Exponential Regression, Ann. Math. Statist., Vol. 8, 947. D. S. Villars & T. W. Anderson, Some Significance Test for Normal Bivariate Distributions, Ann. Math. Statist., Vol. 4 (943), pp ) Villars' original criterion was 4- Sy

10 J. OGAWA 59 where J / 2S x y, but, ϊ believe, it is more natural for testing the regression to be exponential, that is #z- = or oo, to adopt the criterion P _. Sxx -\~2Sxy + Syy Sχχ 2Sχy+Syy The distribution of F is the so-called.f -distribution, when m=q or oo. 4) P. C. Tang, The Power Function of the Analysis of Variance Test with Illustrations of their Use, Statist. Res. Mem., Vol.. ' *) The outline of this note was published in Japanese in Kokyuroku (Research Memoir) of the Institute of Statistical Mathematics, Vol. 5, NO. 2, May, 949.

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