Testing Equality of Natural Parameters for Generalized Riesz Distributions

Transcription

1 Testing Equality of Natural Parameters for Generalized Riesz Distributions Jesse Crawford Department of Mathematics Tarleton State University faculty.tarleton.edu/crawford April 4, 2014 Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

2 Markov Properties Decomposable Undirected Graph (DUG) U = (V, E) Random vector (X v v V ) X 3 X 4 (X 1, X 2 ) If C separates A from B, then A B C Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

3 Markov Properties for Normal Random Variables = Σ 1 = 4??????????? 0?? 0? PD0 (U) Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

4 Estimating Σ Let (X 1,..., X N ) be a sample from N(0, Σ) = Σ 1 PD 0 (U) (Markov property) ˆΣ = MLE for Σ (Essentially a sample covariance matrix) S = N 2 p(ˆσ) R,λ Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

5 Representing U as an AMG V Acyclic Mixed Graph V v 1 v 2 means there is an undirected path from v 1 to v 2 Equivalence classes are called boxes, B V / Boxes are {1, 2}, {3}, and {4}. Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

6 Representing U as an AMG V Boxes are {1, 2}, {3}, and {4}. Given a box B, define B = parents of B. If B = {3}, then B = {1, 2} Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

7 Submatrices If B = {3}, then B = {1, 2} S [B] = B B submatrix = ( ) S 33 ( S11 S S B = B B submatrix = 12 S 21 S 22 ( ) S13 S B] = B B submatrix = S 23 S [B = B B submatrix = ( ) S 31 S 32 S [B] = S [B] S [B S 1 B S B] ) Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

8 Definition (Generalized Riesz Distribution) Let PD 0 (U) Let λ = (λ B B V / ), such that [B] + B 1 λ B >, for all B V /, 2 Then the generalized Riesz distribution with natural parameter and shape parameter λ is dr,λ (S) = π V dim(p(u)) 2 ( [B] λ B B V / ) ( S [B] λ B B V / ) ( ( ) ) Γ(λ B B 2 i 1 2 ) i = 1,..., [B] B V / exp{ tr( S)}dν V (S) dν V (S) := ( ) S [B] [B]+ B +1 2 S B [B] 2 B V / ds. Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

9 Equality of Covariance Matrices for Several Samples Let (X (k) 1,..., X (k) N k ) be a sample from N(0, Σ k ), for k = 1,..., K Would like to test equality of covariance matrices H 0 : Σ 1 = = Σ K S k = N k 2 p(ˆσ k ) R k,λ k, where k = Σ 1 k Reformulated testing problem: H 0 : 1 = = K Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

10 Likelihood Ratio Statistic General Testing Problem Likelihood Ratio Statistic H 0 : θ Θ 0 vs. H A : θ Θ 0 q = L(ˆθ 0 ) L(ˆθ) Reject H 0 if q c, for some critical value c. Distribution of q can be approximated by finding its moments, E[q α ], for every α > 0. Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

11 Likelihood Ratio Statistic Random matrices S 1,..., S K, where S k R k,λ k H 0 : 1 = = K Theorem The likelihood ratio statistic for testing H 0 is q(s 1,..., S K ) = ( λ λ B[B] B S [B] λ B K S k[b] λ kb k=1 λ λ kb[b] kb B V / ), where S = S S K and λ = λ λ K. Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

12 Definition Suppose I and N are nonnegative integers, and let l > I+N 1 2. Define the constant c(α, l, I, N) = l αli [ Γ(αl + l N+i 1 2 ) Γ(l N+i 1 2 ) i = 1,..., I ]. Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

13 Moments of the Likelihood Ratio Statistic Theorem Given α > 0, the αth moment of q under H 0 is E[q α ] = ( K k=1 c(α, λ kb, [B], B ) c(α, λ B, [B], B ) where λ = λ λ K. Proof. Induct on the number of boxes. Let M be a maximal box. B V / q = q M q M Find conditional expectation of qm α given the other boxes. Find E(q M α ) by the induction hypothesis. ), Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

14 References I S.A. Andersson, H.K. Brøns, and S.T. Jensen. Distribution of eigenvalues in multivariate statistical analysis. The Annals of Statistics, 11(2): , S.A. Andersson and T. Klein. On Riesz and Wishart distributions associated with decomposable undirected graphs. Journal of Multivariate Analysis, 101(3): , S.A. Andersson, D. Madigan, and M.D. Perlman. Alternative Markov properties for chain graphs. Scandinavian Journal of Statistics, 28(1):33 85, T.W. Anderson. An Introduction to Multivariate Statistical Analysis, 3rd ed. John Wiley and Sons, New Jersey, Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

15 References II M.S. Bartlett. Properties of sufficiency and statistical tests. Proceedings of the Royal Society of London, 160(A): , M.S. Bartlett. Further aspects of the theory of multiple regression. Proceedings of the Cambridge Philosophical Society, 34:33 40, G.E.P. Box. A general distribution theory for a class of likelihood criteria. Biometrika, 36: , G.W. Brown. On the power of the L 1 test for equality of several variances. Ann. Math. Statist., 10: , Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

16 References III J. Faraut and A. Korányi. Analysis on symmetric cones, in: Oxford Mathematical Monographs. Clarendon Press, Oxford, A. Hassairi and S. Lajmi. Riesz exponential families on symmetric cones. Journal of Theoretical Probability, 14(4): , S.L. Lauritzen. Graphical Models. Clarendon Press, Oxford, J. Neyman and E.S. Pearson. On the problem of k samples. Bull. Acad. Polonaise Sci. Let., Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

17 References IV M. Perlman. Unbiasedness of the likelihood ratio tests for equality of several covariance matrices and equality of several multivariate normal populations. Ann. Statist., 8(2): , E.J.G. Pitman. Tests of hypotheses concerning location and scale parameters. Biometrika, 31: , N. Sugiura and N. Nagao. Unbiasedness of some test criteria for the equality of one or two covariance matrices. Ann. Math. Statist., 39: , Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

18 References V N. Sugiura and N. Nagao. On Bartlett s test and Lehmann s test for homogeneity of variances. Ann. Math. Statist., 40: , S.S. Wilks. Certain generalizations in the analysis of variance. Biometrika, 24(3/4): , J. Wishart. The generalised product moment distribution in samples from a normal multivariate population. Biometrika, 20A(1/2):32 52, Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19

19 Thank You! Jesse Crawford (Tarleton State University) Generalized Riesz Distributions April 4, / 19