Supplement to the paper Accurate distributions of Mallows C p and its unbiased modifications with applications to shrinkage estimation

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1 To aear in Economic Review (Otaru University of Commerce Vol. No Sulement to the aer Accurate distributions of Mallows C its unbiased modifications with alications to shrinkage estimation Haruhiko Ogasawara This article sulements Ogasawara (017. Proof of Theorem 1 Since 1 C ( tr( U U n n ( n tr( I + U U n 1 ( E( Σ U U Σ Λ O I (A.1 1/ 1 1/ 0 0 ( n 1 (see e.g. Siotani Hayakawa & Fujikoshi 1985 Euation (.4.11 we have When E(C Λ O ( n 1 n n 1 ( n ( ( 1(. n 1 n 1 Λ O E(GD which gives from the above result ( 1( E(C E(GD E(C 0 n 1. Proof of Theorem The exectations in (3.4 are given by (. (.4 (.5 (.6 1 for Λ O. For the variances of (3.4 noting that under normality U U are indeendent the following result will be used when X i is indeendent of Y j ( i j 1 : 1

2 When cov( X Y X Y E( X Y X Y E( X Y E( X Y =E( X X E( Y Y E( X E( Y E( X E( Y ={cov( X X +E( X E( X }{cov( Y Y +E( Y E( Y } E( X E( X E( Y E( Y 1 1 =cov( X X cov( Y Y +E( X E( X cov( Y Y cov( X X E( Y E( Y. 1 Λ O since 1 1/ 1/ 0 0 covariance matrix I( n by W( I ( n we have (A. U Σ U Σ is Wishart-distributed with the ij kl ik jl il jk degrees of freedom which is denoted cov{( U ( U } ( ( ( i j k l 1... where ( ij indicates the (i jth element of a matrix ik delta. On the other h 1 W ( I( n n 1 1 cov{( U ij( U kl} 1 1/ 1 1/ 0 0 ( i j k l 1... (see e.g. Siotani et al Euation (.4.1. From (A. 1 1 var{tr( U U } var{tr( U U } where i j k l1 is the Kronecker U Σ U Σ is inverse-wishart distributed as ( n 1( ij kl ik jl il jk ( n ( n 1 ( n 3 1 ( var ( U ii ( n 1 var ( U ii i1 i1 cov{( U ( U }cov{( U ( U } 1 1 ij kl ij kl (A ij cov{( U ii ( U jj} ( n ( n 1 ( n 3 ii jj ij ( n 1 cov{( U ( U } ( ( i j 1....

3 Conseuently 1 var{tr( U U } ( ij i j1 ( n ( n 1 ( n 3 ( n 1 ( i j1 ( n 1 ij ( n 1( ( ( ij kl ik jl il jk ik jl il jk ( ( 1 ( i j k l1 n n n 3 ( { ( n 1} ( ( n ( n 1 ( n 3 ( n 1 ( {4 ( n 1( } ( n ( n 1 ( n 3 (A.4 which gives the variances in (3.4. Euation (A.4 is artially justified in that when = 1 (A.4 with (3.3 gives the well-known variance ( n ( n var( F ( ( n ( n 4 of the central F distribution with n degrees of freedom. Proof of Corollary From (3. when =1 C ( n 1 F n ( F n ( ( C C ( F n n n F MC ( n 1 F C ( n n (A.5 (A.6 3

4 which yield the results of Corollary. Proof of Corollary 3 The roerties of the noncentral F distribution are well documented (e.g. Johnson Kotz & Balakrishnan 1994 Chater 30. The exectation when n variance when n 4 for the noncentral F distribution denoted by F in Corollary are ( ( n E( F ( ( n n ( ( ( n var( F ( n ( n 4 resectively. Then when O( n E(C ( E( F ( ( n O(1 n ( F n E(C ( E( ( ( n ( O(1 n n n F E(MC ( E( n which give (3.7. Using (A.7 ( ( O(1 ( n var( F O(1 O( n ( n (A.7. (A.8 follows. Euation (A.8 gives (3.8. From the unbiased roerty of MC the definitions of C C we have the results of (3.9 excet its last ineuality MSE(C MSE(C which is given by 4

5 ( ( n {E(C ( } n 4( ( n 1 {E(C ( } { ( ( } ( n 4 {E(C ( } ( n (recall the assumtion in Section 1 var(c var (C. Proof of Lemma 1 when Since ˆ MSE( d ( d 1 0 d n ˆ min 0 0 n V MSE( dˆ is minimized d d / ( 1/{1 c ( }. The minimized MSE is 4 MSE( ˆ n 1 ( ˆ 1 ( ˆ. n c c V V Proof of Corollary 4 First we obtain MSE(MC MSE( d C minc n 1 1 n 1 var(c ( ( n 1 {( var(c } ( n ( var(c ( n {( var(c } var(c (A.9 which can be ositive or negative as shown in the following examles. When = 1 the numerator of the first factor on the right-h side of the last euation of (A.9 is 5

6 ( n { var(c } ( n 4( n 4 O( n O(1 O( n ( ( n ( n n 4 (A.10 where for var(c (3. (A.5 are used. When n is sufficiently large (A.10 is ositive demonstrating that in this case MSE(MC MSE( d C. However when n is relatively small min C we define n a 4 (see a condition for (A.3 b 0 (recall the assumtion in Section 1. Then (A.10 becomes 4a 4 ba ( a b / ( a 4 which is negative when ba ( a b /{( a 1( a 4}. For instance when a = 5 b = 1 the last ineuality holds when 4. From this result we have the central ineuality min{ } max{ } in (4.. The remaining ineualities are given by the unbiased roerty of MC the definitions of C C. Proof of Theorem 4 From (A.6 (A.7 we have n var(mc ( var( F n ( ( ( n. n 4 (A.11 Substituting (A.11 for the first euation of (4.3 given by Lemma 1 the second euation of (4.3 follows. Results associated with Theorem 4 when O(1 0 When O(1 from (A.11 we have 1 var(mc ( O( n d ( 1 min MC O n ( ( (. (A.1 6

7 Note that when 0 (3. (A.7 yield var(c var(c ( var( F n ( ( n ( n n ( n ( n 4 n ( ( n var(mc n 4 1 var(c var(c ( O( n 1 var(mc ( O( n d min MC 1 ( var(mc (A.13 ( n 4 var(mc ( n 4 ( ( n O n ( (see (4.1. From (A.1 (A.13 when O(1 it is seen that (A.1 is given from the last two sets of results of (A.13 by relacing with ( resectively. However as described earlier generally O( n giving (A.8. References Johnson N. L. Kotz S. & Balakrishnan N. (1994. Continuous univariate distributions Vol. (nd ed.. New York: Wiley. Ogasawara H. (017. Accurate distributions of Mallows C its unbiased modifications with alications to shrinkage estimation. To aear in Journal of Statistical Planning Inference. Siotani M. Hayakawa T. & Fujikoshi Y. (1985. Modern multivariate statistical analysis: A graduate course hbook. Columbus OH: American Sciences Press. 7

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