ON A ZONAL POLYNOMIAL INTEGRAL
|
|
- Tamsyn Lester
- 5 years ago
- Views:
Transcription
1 ON A ZONAL POLYNOMIAL INTEGRAL A. K. GUPTA AND D. G. KABE Received September 00 and in revised form 9 July 003 A certain multiple integral occurring in the studies of Beherens-Fisher multivariate problem has been evaluated by Mathai et al. 995 in terms of invariant polynomials. However, this paper explicitly evaluates the context integral in terms of zonal polynomials, thus establishing a relationship between zonal polynomial integrals and invariant polynomial integrals.. Introduction Consider two p p independent random matrices S and S.The matrix S, with noncentrality parameter matrix Ω and covariance matrix Σ, is Wishart with n degrees of freedom and S is Wishart with covariance matrix Σ and n degrees of freedom. Then the random matrix F = S / S S / has the density gf=k F /n p S /n +n p { exp trσ S } trσ S/ FS / 0 F n ; 4 ΩΣ / S / FS / ds = k θ δ,λ φ C φλ δ,λ:φ k!l! n / λ C φi C δ,λ φ Σ S, Σ ΩΣ / S ds,. Copyright c 003 Hindawi Publishing Corporation Journal of Applied Mathematics 003: Mathematics Subject Classification: 6H0, 6H URL:
2 570 On a zonal polynomial integral where Λ is the p p diagonal matrix of the roots of F and K as a generic letter denotes the normalizing constants of density functions in this paper. Mathai et al. [4, equations 5.5., 5.5., pages 74 75] do not evaluate the integral.. However, they derive the density of the roots of F in terms of a series of invariant polynomials of two matrix arguments. This paper explicitly obtains the density of F in terms of zonal polynomials, a result which has not been available in the literature so far, not even in terms of invariant polynomials. An expression for the density of the roots of F is also derived in terms of invariant polynomials of two arguments, which is slightly different from the one presented by Mathai et al. [4, equation 5.5., page 74]. The next section states some useful results and the main results of the paper are derived in Section 3. Although it is not always possible to derive invariant polynomial results in terms of zonal polynomials, sometimes it is as done in this paper. We assume that all matrices occurring in this paper are full-rank matrices, except the noncentrality parameter Ω which may have any rank.. Some useful results If P is any p p matrix, then it is known that see Gupta and Nagar [] K exp{trpv} I VV /n p dv = 0 F n; 4 P P.. Now we can write P =P P / H for some p p orthogonal H, see G. A. Korn and T. M. Korn [3, page 4]. It follows that K exp{trqpv } I VV /n p dv = K exp { trqp P / HV } I VV /n p dv = K exp { trp P / Q V } I VV /n p dv = 0 F n; 4 P PQ Q = 0 F n; 4 Q QP P = K exp{trpqv} I VV /n p dv..
3 A. K. Gupta and D. G. Kabe 57 Again it follows from. that K exp { trqp P / HV } I VV /n p dv = K exp { tr H P P / Q } I VV /n p dv = K exp { tr H P P / Q Q / R } I VV /n p dv = 0 F n; 4 R Q QRH P PH = 0 F n;q QP P,.3 where Rp p and Hp p are any two arbitrary orthogonal matrices which may depend on Q Q and P P, respectively. Now it follows from.3 that 0F n; 4 P PQ QT T = 0 F n; 4 H P PHR Q QRU T TU,.4 where H, R, and U are any p p orthogonal matrices, and note that 0F n; 4 ABC = 0 F n; 4,.5 where is any permutation of ABC and also holds for any permutation of four symmetric matrices. Obviously,.4 in terms of zonal polynomials yields From.6, note that C θ Q QP PT T=C θ R Q QRH P PHU T TU..6 exp{trp PQ QR R} = 0 F exp{tr } = t=0 t! C θp PQ QR R,.7 where is any permutation of P PQ QR R. Further, it is easy to verify that 0F n; 4 P PQ QT TG G = 0 F n; 4 P PQ QG GT T.8 because.8 is true for any permutation of the four distinct matrices in.8. Now we proceed with the main result.
4 57 On a zonal polynomial integral 3. Main results In view of.4,.5, and.7 and using., the density of F is given by gf=k F /n p { exp trσ S } trσ / FΣ / S S /n +n p 0 F n ; 4 Σ / ΩΣ / F / SF ds / = K F /n p Σ +Σ / FΣ / /n +n F n + n, n ; F/ Σ / ΩΣ / F / Σ +Σ / FΣ / = K F /n p Σ /n +n F n + n, n ; +Σ / FΣ / Σ / ΩΣ / / F Σ / ΩΣ / / Σ +Σ / FΣ /. 3. Now if Σ / Σ Σ/ = θi for any scalar θ, then note that gf=k F /n p θi + F /n +n F n + n, n ; ΩFθI + F. 3. Thus the density of Λ=diagλ,...,λ p of the roots of F is gλ = K Λ /n p θi +Λ /n +n i<j F n + n, n ; ΩΛθI +Λ. 3.3 λi λ j The expression is not very suitable for obtaining the density of Λ for the expression 3.. ForsomeQ p p orthogonal, let QFQ be still denoted by F,letLbe the p p diagonal matrix of the roots of Σ / ΩΣ /, and let be the diagonal matrix of the roots of Σ / Σ Σ/. Then the
5 A. K. Gupta and D. G. Kabe 573 density of this F is gf=k F /n p +F /n +n F n + n, n ; LF + F. 3.4 Mathai et al. [4, equation A.3.5, page 338] showed that +F a F a,u;lf + F = δ,λ;φ a φ C δ,λ φ F, LF. 3.5 kk!l!u λ We have observed earlier that C θ Q QP P=C θ R Q QRH P PH and thus the invariant polynomial argument for matrices can also be replaced by their roots. Thus the density of Λ for 3.4 is gλ = K Λ /n p i<j λi λ j δ,λ;φ / n + n φ Cδ,λ φ Λ,LΛ k!l!. /n λ 3.6 However, 3.6 is not suitable for obtaining the density of trf. Acknowledgment The authors are thankful to the referee for a critical reading of the paper. References [] Y. Chikuse, Distributions of some matrix variates and latent roots in multivariate Behrens-Fisher discriminant analysis, Ann. Statist. 9 98, no., [] A. K. Gupta and D. K. Nagar, Matrix Variate Distributions, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 04, Chapman & Hall/CRC, Boca Raton, Fla, USA, 000. [3] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, 96. [4] A. M. Mathai, S. B. Provost, and T. Hayakawa, Bilinear Forms and Zonal Polynomials, Lecture Notes in Statistics, vol. 0, Springer-Verlag, New York, 995. A. K. Gupta: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, OH , USA address: gupta@bgnet.bgsu.edu D. G. Kabe: Department of Mathematics and Computer Science, Saint Mary s University, Halefax, Nova Scotia, Canada B3H 3C3
ON A ZONAL POLYNOMIAL INTEGRAL
ON A ZONAL POLYNOMIAL INTEGRAL A. K. GUPTA AND D. G. KABE Received September 00 and in revised form 9 July 003 A certain multiple integral occurring in the studies of Beherens-Fisher multivariate problem
More informationQUATERNIONS AND ROTATIONS
QUATERNIONS AND ROTATIONS SVANTE JANSON 1. Introduction The purpose of this note is to show some well-known relations between quaternions and the Lie groups SO(3) and SO(4) (rotations in R 3 and R 4 )
More informationJournal of Multivariate Analysis. Sphericity test in a GMANOVA MANOVA model with normal error
Journal of Multivariate Analysis 00 (009) 305 3 Contents lists available at ScienceDirect Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva Sphericity test in a GMANOVA MANOVA
More informationOn the conservative multivariate Tukey-Kramer type procedures for multiple comparisons among mean vectors
On the conservative multivariate Tukey-Kramer type procedures for multiple comparisons among mean vectors Takashi Seo a, Takahiro Nishiyama b a Department of Mathematical Information Science, Tokyo University
More informationMultivariate Statistical Analysis
Multivariate Statistical Analysis Fall 2011 C. L. Williams, Ph.D. Lecture 4 for Applied Multivariate Analysis Outline 1 Eigen values and eigen vectors Characteristic equation Some properties of eigendecompositions
More informationProperties and approximations of some matrix variate probability density functions
Technical report from Automatic Control at Linköpings universitet Properties and approximations of some matrix variate probability density functions Karl Granström, Umut Orguner Division of Automatic Control
More informationMIT Algebraic techniques and semidefinite optimization May 9, Lecture 21. Lecturer: Pablo A. Parrilo Scribe:???
MIT 6.972 Algebraic techniques and semidefinite optimization May 9, 2006 Lecture 2 Lecturer: Pablo A. Parrilo Scribe:??? In this lecture we study techniques to exploit the symmetry that can be present
More informationLinear Models for Multivariate Repeated Measures Data
THE UNIVERSITY OF TEXAS AT SAN ANTONIO, COLLEGE OF BUSINESS Working Paper SERIES Date December 3, 200 WP # 007MSS-253-200 Linear Models for Multivariate Repeated Measures Data Anuradha Roy Management Science
More informationMultivariate Gaussian Analysis
BS2 Statistical Inference, Lecture 7, Hilary Term 2009 February 13, 2009 Marginal and conditional distributions For a positive definite covariance matrix Σ, the multivariate Gaussian distribution has density
More informationarxiv:math/ v5 [math.ac] 17 Sep 2009
On the elementary symmetric functions of a sum of matrices R. S. Costas-Santos arxiv:math/0612464v5 [math.ac] 17 Sep 2009 September 17, 2009 Abstract Often in mathematics it is useful to summarize a multivariate
More informationLinear Algebra- Final Exam Review
Linear Algebra- Final Exam Review. Let A be invertible. Show that, if v, v, v 3 are linearly independent vectors, so are Av, Av, Av 3. NOTE: It should be clear from your answer that you know the definition.
More informationTesting a Normal Covariance Matrix for Small Samples with Monotone Missing Data
Applied Mathematical Sciences, Vol 3, 009, no 54, 695-70 Testing a Normal Covariance Matrix for Small Samples with Monotone Missing Data Evelina Veleva Rousse University A Kanchev Department of Numerical
More informationBare minimum on matrix algebra. Psychology 588: Covariance structure and factor models
Bare minimum on matrix algebra Psychology 588: Covariance structure and factor models Matrix multiplication 2 Consider three notations for linear combinations y11 y1 m x11 x 1p b11 b 1m y y x x b b n1
More informationHigh-dimensional asymptotic expansions for the distributions of canonical correlations
Journal of Multivariate Analysis 100 2009) 231 242 Contents lists available at ScienceDirect Journal of Multivariate Analysis journal homepage: www.elsevier.com/locate/jmva High-dimensional asymptotic
More informationLecture 3: QR-Factorization
Lecture 3: QR-Factorization This lecture introduces the Gram Schmidt orthonormalization process and the associated QR-factorization of matrices It also outlines some applications of this factorization
More informationHigh-Dimensional AICs for Selection of Redundancy Models in Discriminant Analysis. Tetsuro Sakurai, Takeshi Nakada and Yasunori Fujikoshi
High-Dimensional AICs for Selection of Redundancy Models in Discriminant Analysis Tetsuro Sakurai, Takeshi Nakada and Yasunori Fujikoshi Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku,
More informationLecture 2. LINEAR DIFFERENTIAL SYSTEMS p.1/22
LINEAR DIFFERENTIAL SYSTEMS Harry Trentelman University of Groningen, The Netherlands Minicourse ECC 2003 Cambridge, UK, September 2, 2003 LINEAR DIFFERENTIAL SYSTEMS p1/22 Part 1: Generalities LINEAR
More informationA Note on Some Wishart Expectations
Metrika, Volume 33, 1986, page 247-251 A Note on Some Wishart Expectations By R. J. Muirhead 2 Summary: In a recent paper Sharma and Krishnamoorthy (1984) used a complicated decisiontheoretic argument
More informationMatrix Theory and Differential Equations Homework 6 Solutions, 10/5/6
Matrix Theory and Differential Equations Homework 6 Solutions, 0/5/6 Question Find the general solution of the matrix system: x 3y + 5z 8t 5 x + 4y z + t Express your answer in the form of a particulaolution
More informationOn the conservative multivariate multiple comparison procedure of correlated mean vectors with a control
On the conservative multivariate multiple comparison procedure of correlated mean vectors with a control Takahiro Nishiyama a a Department of Mathematical Information Science, Tokyo University of Science,
More informationStat260: Bayesian Modeling and Inference Lecture Date: February 10th, Jeffreys priors. exp 1 ) p 2
Stat260: Bayesian Modeling and Inference Lecture Date: February 10th, 2010 Jeffreys priors Lecturer: Michael I. Jordan Scribe: Timothy Hunter 1 Priors for the multivariate Gaussian Consider a multivariate
More informationOn the Distribution of a Quadratic Form in Normal Variates
On the Distribution of a Quadratic Form in Normal Variates Jin Zhang School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan, 650091, China E-mail: jinzhang@ynu.edu.cn Abstract It is a
More informationOn the Efficiencies of Several Generalized Least Squares Estimators in a Seemingly Unrelated Regression Model and a Heteroscedastic Model
Journal of Multivariate Analysis 70, 8694 (1999) Article ID jmva.1999.1817, available online at http:www.idealibrary.com on On the Efficiencies of Several Generalized Least Squares Estimators in a Seemingly
More informationResearch Article On the Marginal Distribution of the Diagonal Blocks in a Blocked Wishart Random Matrix
Hindawi Publishing Corporation nternational Journal of Analysis Volume 6, Article D 59678, 5 pages http://dx.doi.org/.55/6/59678 Research Article On the Marginal Distribution of the Diagonal Blocks in
More informationQuadratic forms in skew normal variates
J. Math. Anal. Appl. 73 (00) 558 564 www.academicpress.com Quadratic forms in skew normal variates Arjun K. Gupta a,,1 and Wen-Jang Huang b a Department of Mathematics and Statistics, Bowling Green State
More informationJacobians of Matrix Transformations and Functions of Matrix Argument
Jacobians of Matrix Transformations and Functions of Matrix Argument A. M. Mathai Department of Mathematics & Statistics, McGill University World Scientific Singapore *New Jersey London Hong Kong Contents
More informationinv lve a journal of mathematics 2008 Vol. 1, No. 2 Invariant polynomials and minimal zero sequences mathematical sciences publishers
inv lve a journal of mathematics Invariant polynomials and minimal zero sequences Bryson W. Finklea, Terri Moore, Vadim Ponomarenko and Zachary J. Turner mathematical sciences publishers 2008 Vol. 1, No.
More informationOn Bayesian Inference with Conjugate Priors for Scale Mixtures of Normal Distributions
Journal of Applied Probability & Statistics Vol. 5, No. 1, xxx xxx c 2010 Dixie W Publishing Corporation, U. S. A. On Bayesian Inference with Conjugate Priors for Scale Mixtures of Normal Distributions
More informationOptimisation Problems for the Determinant of a Sum of 3 3 Matrices
Irish Math. Soc. Bulletin 62 (2008), 31 36 31 Optimisation Problems for the Determinant of a Sum of 3 3 Matrices FINBARR HOLLAND Abstract. Given a pair of positive definite 3 3 matrices A, B, the maximum
More informationNoncentral Student distributed LS and IV Estimators.
Noncentral Student distributed LS and IV Estimators. 0. Introduction. Leon L. Wegge The distribution of the least squares and the instrumental variable estimators of the coefficients in a linear relation
More informationLikelihood Ratio Tests in Multivariate Linear Model
Likelihood Ratio Tests in Multivariate Linear Model Yasunori Fujikoshi Department of Mathematics, Graduate School of Science, Hiroshima University, 1-3-1 Kagamiyama, Higashi Hiroshima, Hiroshima 739-8626,
More informationA high-dimensional bias-corrected AIC for selecting response variables in multivariate calibration
A high-dimensional bias-corrected AIC for selecting response variables in multivariate calibration Ryoya Oda, Yoshie Mima, Hirokazu Yanagihara and Yasunori Fujikoshi Department of Mathematics, Graduate
More informationTypical Problem: Compute.
Math 2040 Chapter 6 Orhtogonality and Least Squares 6.1 and some of 6.7: Inner Product, Length and Orthogonality. Definition: If x, y R n, then x y = x 1 y 1 +... + x n y n is the dot product of x and
More informationAnalysis of variance, multivariate (MANOVA)
Analysis of variance, multivariate (MANOVA) Abstract: A designed experiment is set up in which the system studied is under the control of an investigator. The individuals, the treatments, the variables
More informationTHE GAUSS-BONNET THEOREM FOR VECTOR BUNDLES
THE GAUSS-BONNET THEOREM FOR VECTOR BUNDLES Denis Bell 1 Department of Mathematics, University of North Florida 4567 St. Johns Bluff Road South,Jacksonville, FL 32224, U. S. A. email: dbell@unf.edu This
More informationABOUT PRINCIPAL COMPONENTS UNDER SINGULARITY
ABOUT PRINCIPAL COMPONENTS UNDER SINGULARITY José A. Díaz-García and Raúl Alberto Pérez-Agamez Comunicación Técnica No I-05-11/08-09-005 (PE/CIMAT) About principal components under singularity José A.
More informationProblem 1. CS205 Homework #2 Solutions. Solution
CS205 Homework #2 s Problem 1 [Heath 3.29, page 152] Let v be a nonzero n-vector. The hyperplane normal to v is the (n-1)-dimensional subspace of all vectors z such that v T z = 0. A reflector is a linear
More informationInvariant Polynomials and Minimal Zero Sequences
Invariant Polynomials and Minimal Zero Sequences Bryson W. Finklea St. John s College (undergraduate Terri Moore University of Washington (undergraduate Vadim Ponomarenko Department of Mathematics and
More information1. Basic Operations Consider two vectors a (1, 4, 6) and b (2, 0, 4), where the components have been expressed in a given orthonormal basis.
Questions on Vectors and Tensors 1. Basic Operations Consider two vectors a (1, 4, 6) and b (2, 0, 4), where the components have been expressed in a given orthonormal basis. Compute 1. a. 2. The angle
More informationSingular Inverse Wishart Distribution with Application to Portfolio Theory
Singular Inverse Wishart Distribution with Application to Portfolio Theory Bodnar, Taras; Mazur, Stepan; Podgórski, Krzysztof Published: 2015-01-01 Link to publication Citation for published version APA:
More informationSubspace Classifiers. Robert M. Haralick. Computer Science, Graduate Center City University of New York
Subspace Classifiers Robert M. Haralick Computer Science, Graduate Center City University of New York Outline The Gaussian Classifier When Σ 1 = Σ 2 and P(c 1 ) = P(c 2 ), then assign vector x to class
More informationOn the eigenvalues of Euclidean distance matrices
Volume 27, N. 3, pp. 237 250, 2008 Copyright 2008 SBMAC ISSN 00-8205 www.scielo.br/cam On the eigenvalues of Euclidean distance matrices A.Y. ALFAKIH Department of Mathematics and Statistics University
More informationGaussian representation of a class of Riesz probability distributions
arxiv:763v [mathpr] 8 Dec 7 Gaussian representation of a class of Riesz probability distributions A Hassairi Sfax University Tunisia Running title: Gaussian representation of a Riesz distribution Abstract
More informationSOME ASPECTS OF MULTIVARIATE BEHRENS-FISHER PROBLEM
SOME ASPECTS OF MULTIVARIATE BEHRENS-FISHER PROBLEM Junyong Park Bimal Sinha Department of Mathematics/Statistics University of Maryland, Baltimore Abstract In this paper we discuss the well known multivariate
More informationGeostatistical Modeling for Large Data Sets: Low-rank methods
Geostatistical Modeling for Large Data Sets: Low-rank methods Whitney Huang, Kelly-Ann Dixon Hamil, and Zizhuang Wu Department of Statistics Purdue University February 22, 2016 Outline Motivation Low-rank
More informationAdjoint Orbits, Principal Components, and Neural Nets
Adjoint Orbits, Principal Components, and Neural Nets Some facts about Lie groups and examples Examples of adjoint orbits and a distance measure Descent equations on adjoint orbits Properties of the double
More informationQuantum Computing Lecture 2. Review of Linear Algebra
Quantum Computing Lecture 2 Review of Linear Algebra Maris Ozols Linear algebra States of a quantum system form a vector space and their transformations are described by linear operators Vector spaces
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science : Dynamic Systems Spring 2011
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.24: Dynamic Systems Spring 20 Homework 9 Solutions Exercise 2. We can use additive perturbation model with
More informationMultivariate Regression Generalized Likelihood Ratio Tests for FMRI Activation
Multivariate Regression Generalized Likelihood Ratio Tests for FMRI Activation Daniel B Rowe Division of Biostatistics Medical College of Wisconsin Technical Report 40 November 00 Division of Biostatistics
More informationIntroduction to Matrix Algebra
Introduction to Matrix Algebra August 18, 2010 1 Vectors 1.1 Notations A p-dimensional vector is p numbers put together. Written as x 1 x =. x p. When p = 1, this represents a point in the line. When p
More informationCharacterization of Diagonalizable Matrices: An Algorithm
Applied Mathematical Sciences, Vol. 3, 2009, no. 12, 575-580 Characterization of Diagonalizable Matrices: An Algorithm Arman Shokrollahi Ramanujan Institute for Advanced Study in Mathematics University
More informationCSL361 Problem set 4: Basic linear algebra
CSL361 Problem set 4: Basic linear algebra February 21, 2017 [Note:] If the numerical matrix computations turn out to be tedious, you may use the function rref in Matlab. 1 Row-reduced echelon matrices
More informationNon-trivial solutions to certain matrix equations
Electronic Journal of Linear Algebra Volume 9 Volume 9 (00) Article 4 00 Non-trivial solutions to certain matrix equations Aihua Li ali@loyno.edu Duane Randall Follow this and additional works at: http://repository.uwyo.edu/ela
More informationPhD in Theoretical Particle Physics Academic Year 2017/2018
July 10, 017 SISSA Entrance Examination PhD in Theoretical Particle Physics Academic Year 017/018 S olve two among the four problems presented. Problem I Consider a quantum harmonic oscillator in one spatial
More informationEffective Sample Size in Spatial Modeling
Int. Statistical Inst.: Proc. 58th World Statistical Congress, 2011, Dublin (Session CPS029) p.4526 Effective Sample Size in Spatial Modeling Vallejos, Ronny Universidad Técnica Federico Santa María, Department
More informationThree Skewed Matrix Variate Distributions
Three Skewed Matrix Variate Distributions ariv:174.531v5 [stat.me] 13 Aug 18 Michael P.B. Gallaugher and Paul D. McNicholas Dept. of Mathematics & Statistics, McMaster University, Hamilton, Ontario, Canada.
More informationSpectral Theorem for Self-adjoint Linear Operators
Notes for the undergraduate lecture by David Adams. (These are the notes I would write if I was teaching a course on this topic. I have included more material than I will cover in the 45 minute lecture;
More informationA NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS. Masaru Nagisa. Received May 19, 2014 ; revised April 10, (Ax, x) 0 for all x C n.
Scientiae Mathematicae Japonicae Online, e-014, 145 15 145 A NOTE ON RATIONAL OPERATOR MONOTONE FUNCTIONS Masaru Nagisa Received May 19, 014 ; revised April 10, 014 Abstract. Let f be oeprator monotone
More informationTotal Differentiation Under Jet Composition
Journal of Nonlinear Mathematical Physics Volume 13, Supplement (2006), 102 109 AGMF Tallin 05 Total Differentiation Under Jet Composition Maido RAHULA and Vitali RETŠNOI Institute of Pure Mathematics,
More informationREPRESENTATION THEORY WEEK 5. B : V V k
REPRESENTATION THEORY WEEK 5 1. Invariant forms Recall that a bilinear form on a vector space V is a map satisfying B : V V k B (cv, dw) = cdb (v, w), B (v 1 + v, w) = B (v 1, w)+b (v, w), B (v, w 1 +
More informationHolonomic gradient method for hypergeometric functions of a matrix argument Akimichi Takemura, Shiga University
Holonomic gradient method for hypergeometric functions of a matrix argument Akimichi Takemura, Shiga University (joint with H.Hashiguchi and N.Takayama) June 20, 2016, RIMS, Kyoto Items 1. Summary of multivariate
More informationProblem structure in semidefinite programs arising in control and signal processing
Problem structure in semidefinite programs arising in control and signal processing Lieven Vandenberghe Electrical Engineering Department, UCLA Joint work with: Mehrdad Nouralishahi, Tae Roh Semidefinite
More informationT 2 Type Test Statistic and Simultaneous Confidence Intervals for Sub-mean Vectors in k-sample Problem
T Type Test Statistic and Simultaneous Confidence Intervals for Sub-mean Vectors in k-sample Problem Toshiki aito a, Tamae Kawasaki b and Takashi Seo b a Department of Applied Mathematics, Graduate School
More informationNumerische Mathematik
Numer. Math. (2003) 94: 195 202 Digital Object Identifier (DOI) 10.1007/s002110100308 Numerische Mathematik Some observations on Babuška and Brezzi theories Jinchao Xu, Ludmil Zikatanov Department of Mathematics,
More information1. Introduction. Let an n m complex Gaussian random matrix A be distributed
COMMUNICATIONS IN INFORMATION AND SYSTEMS c 23 International Press Vol 3, No 2, pp 119-138, October 23 3 COMPLEX RANDOM MATRICES AND RAYLEIGH CHANNEL CAPACITY T RATNARAJAH, R VAILLANCOURT, AND M ALVO Abstract
More informationMotivating the Covariance Matrix
Motivating the Covariance Matrix Raúl Rojas Computer Science Department Freie Universität Berlin January 2009 Abstract This note reviews some interesting properties of the covariance matrix and its role
More informationClifford Algebras and Spin Groups
Clifford Algebras and Spin Groups Math G4344, Spring 2012 We ll now turn from the general theory to examine a specific class class of groups: the orthogonal groups. Recall that O(n, R) is the group of
More informationCONSTRUCTION OF SLICED ORTHOGONAL LATIN HYPERCUBE DESIGNS
Statistica Sinica 23 (2013), 1117-1130 doi:http://dx.doi.org/10.5705/ss.2012.037 CONSTRUCTION OF SLICED ORTHOGONAL LATIN HYPERCUBE DESIGNS Jian-Feng Yang, C. Devon Lin, Peter Z. G. Qian and Dennis K. J.
More informationRECURSIVE ESTIMATION AND KALMAN FILTERING
Chapter 3 RECURSIVE ESTIMATION AND KALMAN FILTERING 3. The Discrete Time Kalman Filter Consider the following estimation problem. Given the stochastic system with x k+ = Ax k + Gw k (3.) y k = Cx k + Hv
More informationLeast-squares data fitting
EE263 Autumn 2015 S. Boyd and S. Lall Least-squares data fitting 1 Least-squares data fitting we are given: functions f 1,..., f n : S R, called regressors or basis functions data or measurements (s i,
More informationA NOTE ON ESTIMATION UNDER THE QUADRATIC LOSS IN MULTIVARIATE CALIBRATION
J. Japan Statist. Soc. Vol. 32 No. 2 2002 165 181 A NOTE ON ESTIMATION UNDER THE QUADRATIC LOSS IN MULTIVARIATE CALIBRATION Hisayuki Tsukuma* The problem of estimation in multivariate linear calibration
More informationA NOTE ON SIRIVASTAVA S PAPER SINGULAR WISHART AND MULTIVARIATE BETA DISTRIBUTIONS : JACOBIANS
A NOTE ON SIRIVASTAVA S PAPER SINGULAR WISHART AND MULTIVARIATE BETA DISTRIBUTIONS : JACOBIANS José A. Díaz-García Comunicación Técnica No I-03-19/17-10-2003 (PE/CIMAT) A note on Sirivastava s paper Singular
More informationEXPLICIT DECOMPOSITION OF A RATIONAL PRIME IN A CUBIC FIELD
EXPLICIT DECOMPOSITION OF A RATIONAL PRIME IN A CUBIC FIELD ŞABAN ALACA, BLAIR K. SPEARMAN, AND KENNETH S. WILLIAMS Received 19 June 5; Revised 19 February 6; Accepted 1 March 6 We give the explicit decomposition
More informationVerifying Regularity Conditions for Logit-Normal GLMM
Verifying Regularity Conditions for Logit-Normal GLMM Yun Ju Sung Charles J. Geyer January 10, 2006 In this note we verify the conditions of the theorems in Sung and Geyer (submitted) for the Logit-Normal
More informationTests for Covariance Matrices, particularly for High-dimensional Data
M. Rauf Ahmad Tests for Covariance Matrices, particularly for High-dimensional Data Technical Report Number 09, 200 Department of Statistics University of Munich http://www.stat.uni-muenchen.de Tests for
More informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More informationOn minors of the compound matrix of a Laplacian
On minors of the compound matrix of a Laplacian R. B. Bapat 1 Indian Statistical Institute New Delhi, 110016, India e-mail: rbb@isid.ac.in August 28, 2013 Abstract: Let L be an n n matrix with zero row
More informationTHE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION. Sunday, March 14, 2004 Time: hours No aids or calculators permitted.
THE UNIVERSITY OF TORONTO UNDERGRADUATE MATHEMATICS COMPETITION Sunday, March 1, Time: 3 1 hours No aids or calculators permitted. 1. Prove that, for any complex numbers z and w, ( z + w z z + w w z +
More informationEigenvalues and Eigenvectors
/88 Chia-Ping Chen Department of Computer Science and Engineering National Sun Yat-sen University Linear Algebra Eigenvalue Problem /88 Eigenvalue Equation By definition, the eigenvalue equation for matrix
More informationarxiv: v1 [math.co] 10 Aug 2016
POLYTOPES OF STOCHASTIC TENSORS HAIXIA CHANG 1, VEHBI E. PAKSOY 2 AND FUZHEN ZHANG 2 arxiv:1608.03203v1 [math.co] 10 Aug 2016 Abstract. Considering n n n stochastic tensors (a ijk ) (i.e., nonnegative
More informationPrincipal Component Analysis (PCA) Our starting point consists of T observations from N variables, which will be arranged in an T N matrix R,
Principal Component Analysis (PCA) PCA is a widely used statistical tool for dimension reduction. The objective of PCA is to find common factors, the so called principal components, in form of linear combinations
More informationFormal Groups. Niki Myrto Mavraki
Formal Groups Niki Myrto Mavraki Contents 1. Introduction 1 2. Some preliminaries 2 3. Formal Groups (1 dimensional) 2 4. Groups associated to formal groups 9 5. The Invariant Differential 11 6. The Formal
More informationEigenvalue and Eigenvector Homework
Eigenvalue and Eigenvector Homework Olena Bormashenko November 4, 2 For each of the matrices A below, do the following:. Find the characteristic polynomial of A, and use it to find all the eigenvalues
More informationOn Expected Gaussian Random Determinants
On Expected Gaussian Random Determinants Moo K. Chung 1 Department of Statistics University of Wisconsin-Madison 1210 West Dayton St. Madison, WI 53706 Abstract The expectation of random determinants whose
More informationLast Time. Social Network Graphs Betweenness. Graph Laplacian. Girvan-Newman Algorithm. Spectral Bisection
Eigenvalue Problems Last Time Social Network Graphs Betweenness Girvan-Newman Algorithm Graph Laplacian Spectral Bisection λ 2, w 2 Today Small deviation into eigenvalue problems Formulation Standard eigenvalue
More informationThe p-quotient Algorithm
The p-quotient Algorithm Marvin Krings November 23, 2016 Marvin Krings The p-quotient Algorithm November 23, 2016 1 / 49 Algorithms p-quotient Algorithm Given a finitely generated group Compute certain
More informationMATH 425-Spring 2010 HOMEWORK ASSIGNMENTS
MATH 425-Spring 2010 HOMEWORK ASSIGNMENTS Instructor: Shmuel Friedland Department of Mathematics, Statistics and Computer Science email: friedlan@uic.edu Last update April 18, 2010 1 HOMEWORK ASSIGNMENT
More informationEmpirical properties of large covariance matrices in finance
Empirical properties of large covariance matrices in finance Ex: RiskMetrics Group, Geneva Since 2010: Swissquote, Gland December 2009 Covariance and large random matrices Many problems in finance require
More informationZonal Polynomials and Hypergeometric Functions of Matrix Argument. Zonal Polynomials and Hypergeometric Functions of Matrix Argument p.
Zonal Polynomials and Hypergeometric Functions of Matrix Argument Zonal Polynomials and Hypergeometric Functions of Matrix Argument p. 1/2 Zonal Polynomials and Hypergeometric Functions of Matrix Argument
More informationA linear algebra proof of the fundamental theorem of algebra
A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional
More informationRepresentation Theory
Representation Theory Representations Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ : G Aut(V ). The degree (or dimension) of ρ is just dim
More informationA linear algebra proof of the fundamental theorem of algebra
A linear algebra proof of the fundamental theorem of algebra Andrés E. Caicedo May 18, 2010 Abstract We present a recent proof due to Harm Derksen, that any linear operator in a complex finite dimensional
More informationSome Notes on Least Squares, QR-factorization, SVD and Fitting
Department of Engineering Sciences and Mathematics January 3, 013 Ove Edlund C000M - Numerical Analysis Some Notes on Least Squares, QR-factorization, SVD and Fitting Contents 1 Introduction 1 The Least
More informationMAXIMUM LIKELIHOOD IN GENERALIZED FIXED SCORE FACTOR ANALYSIS 1. INTRODUCTION
MAXIMUM LIKELIHOOD IN GENERALIZED FIXED SCORE FACTOR ANALYSIS JAN DE LEEUW ABSTRACT. We study the weighted least squares fixed rank approximation problem in which the weight matrices depend on unknown
More informationIntroduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX
Introduction to Quantitative Techniques for MSc Programmes SCHOOL OF ECONOMICS, MATHEMATICS AND STATISTICS MALET STREET LONDON WC1E 7HX September 2007 MSc Sep Intro QT 1 Who are these course for? The September
More informationModified Simes Critical Values Under Positive Dependence
Modified Simes Critical Values Under Positive Dependence Gengqian Cai, Sanat K. Sarkar Clinical Pharmacology Statistics & Programming, BDS, GlaxoSmithKline Statistics Department, Temple University, Philadelphia
More informationSTATISTICAL METHODS FOR SIGNAL PROCESSING c Alfred Hero
STATISTICAL METHODS FOR SIGNAL PROCESSING c Alfred Hero 1999 32 Statistic used Meaning in plain english Reduction ratio T (X) [X 1,..., X n ] T, entire data sample RR 1 T (X) [X (1),..., X (n) ] T, rank
More information13-2 Text: 28-30; AB: 1.3.3, 3.2.3, 3.4.2, 3.5, 3.6.2; GvL Eigen2
The QR algorithm The most common method for solving small (dense) eigenvalue problems. The basic algorithm: QR without shifts 1. Until Convergence Do: 2. Compute the QR factorization A = QR 3. Set A :=
More informationPartial Sums of Powers of Prime Factors
1 3 47 6 3 11 Journal of Integer Sequences, Vol. 10 (007), Article 07.1.6 Partial Sums of Powers of Prime Factors Jean-Marie De Koninck Département de Mathématiques et de Statistique Université Laval Québec
More informationRecurrence Relations between Symmetric Polynomials of n-th Order
Applied Mathematical Sciences, Vol. 8, 214, no. 15, 5195-522 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.214.47525 Recurrence Relations between Symmetric Polynomials of n-th Order Yuriy
More information