Distributions of characteristic roots in multivariate analysis Part II. Non-Null Distribution

DOI10.2307/3315084zbMath0378.62056OpenAlexW2160100835MaRDI QIDQ4157835

Publication date: 1977

Published in: Canadian Journal of Statistics (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.2307/3315084

Mathematics Subject Classification ID

Multivariate distribution of statistics (62H10)

Related Items

An everywhere convergent series representation of the distribution of Hotelling's generalized $T^ 2_ 0$, Testing sphericity using small samples, Generalized Hypergeometric Functions and Exact Distributions of Test Statistics, Invariant Polynomials and Related Tests, Simplicial variances, potentials and Mahalanobis distances, Second-order properties of a two-stage fixed-size confidence region for the mean vector of a multivariate normal distribution, Testing a covariance matrix: exact null distribution of its likelihood criterion

Cites Work

On Elementary Symmetric Functions of the Roots of a Multivariate Matrix: Distributions
Some Results on the Non-Central Multivariate Beta Distribution and Moments of Traces of Two Matrices
An Asymptotic Expansion for the Distribution of the Latent Roots of the Estimated Covariance Matrix
Bayesian Estimation in Multivariate Analysis
On the Distribution of the Latent Vectors for Principal Component Analysis
On a Property of a Test for the Equality of Two Normal Dispersion Matrices Against One-sided Alternatives
Admissible Bayes Character of $T^2-, R^2-$, and Other Fully Invariant Tests for Classical Multivariate Normal Problems
A Note on the Sphericity Test
A generalized multivariate analysis of variance model useful especially for growth curve problems
On Certain Distribution Problems Based on Positive Definite Quadratic Functions in Normal Vectors
Exact distributions of Wilks's likelihood ratio criterion
The Distribution of the Generalized Variance
On elementary symmetric functions of the roots of two matrices in multivariate analysis
A Class of Tests with Monotone Power Functions for Two Problems in Multivariate Statistical Analysis
On Confidence Bounds Associated with Multivariate Analysis of Variance and Non-Independence Between Two Sets of Variates
Upper percentage points of the largest root of a matrix in multivariate analysis
On the Moments of the Trace of a Matrix and Approximations to Its Non- Central Distribution
On the Exact Distributions of the Likelihood Ratio Criteria for Testing Linear Hypotheses About Regression Coefficients
On the Distribution of the Largest Latent Root and the Corresponding Latent Vector for Principal Component Analysis
On Tests of the Equality of Two Covariance Matrices
On the Non-Central Distributions of Two Test Criteria in Multivariate Analysis of Variance
On the distribution of a statistic used for testing a covariance matrix
A Note on Wilks' Internal Scatter
On the Exact Distributions of the Criterion $W$ for Testing Sphericity in a $p$-Variate Normal Distribution
On the Distribution of the Largest Latent Root of the Covariance Matrix
Generalized Asymptotic Expansions of Cornish-Fisher Type
On the Non-central Distribution of the Second Elementary Symmetric Function of the Roots of a Matrix
On the Moments of Elementary Symmetric Functions of the Roots of Two Matrices and Approximations to a Distribution
Properties of Power Functions of Some Tests Concerning Dispersion Matrices of Multivariate Normal Distributions
On the Distribution of the Largest Root of a Matrix in Multivariate Analysis
On the Exact Distributions of Likelihood Ratio Criteria for Testing Independence of Sets of Variates Under the Null Hypothesis
Distribution of the Largest Latent Root and the Smallest Latent Root of the Generalized $B$ Statistic and $F$ Statistic in Multivariate Analysis
A NOTE ON CONFIDENCE BOUNDS FOR CERTAIN RATIOS OF CHARACTERISTIC ROOTS OF COVARIANCE MATRICES
Percentage points of the extreme roots of a Wishart matrix
Calculation of Zonal Polynomial Coefficients by Use of the Laplace-Beltrami Operator
Bayesian estimation of latent roots and vectors with special reference to the bivariate normal distribution
On the exact distribution of Wilks's criterion
Asymptotic Expansions of the Non-Null Distributions of the Likelihood Ratio Criteria for Multivariate Linear Hypothesis and Independence
Distribution of a Product and the Structural Set Up of Densities
Some distance properties of latent root and vector methods used in multivariate analysis
Power Comparisons of Tests of Equality of Two Covariance Matrices Based on Individual Characteristic Roots
Unbiasedness of Some Test Criteria for the Equality of One or Two Covariance Matrices
The Distribution of the Latent Roots of the Covariance Matrix
Asymptotic Theory for Principal Component Analysis
Some Results on the Distribution of Two Random Matrices Used in Classification Procedures
On the distribution of the largest characteristic root of a matrix in multivariate analysis
On Testing a Set of Correlation Coefficients for Equality
On the Noncentral Distributions of the Second Largest Roots of Three Matrices in Multivariate Analysis
Asymptotic Expansions of the Distributions of the Likelihood Ratio Criteria for Covariance Matrix
Latent vectors of random symmetric matrices
Asymptotic Expansions for Distributions of the Roots of Two Matrices from Classical and Complex Gaussian Populations
Monotonicity of the Power Functions of Some Tests of Independence between Two Sets of Variates
On the Null Distribution of the Sum of the Roots of a Multivariate Beta Distribution
Distribution Theory of a Positive Definite Quadratic Form with Matrix Argument
Joint Distribution of the Extreme Roots of a Covariance Matrix
Asymptotic Distributions of Some Multivariate Tests
Distributions Connected with a Multivariate Beta Statistic
Note on Some Formulas for Weighted Sums of Zonal Polynomials
Some Optimum Confidence Bounds for Roots of Determinantal Equations
Noncentral Distribution of Wilks' Statistic in Manova
Exact joint distributions of any few ordered roots of a class of random matrices
On the exact distributions of the extreme roots of the Wishart and MANOVA matrices
On the Moments of the Trace of a Matrix and Approximations to its Distribution
Systems of Partial Differential Equations for Hypergeometric Functions of Matrix Argument
Equality of More Than Two Variances and of More Than Two Dispersion Matrices Against Certain Alternatives
Asymptotic formulae for the distribution of a multivariate test statistic: Power comparisons of certain multivariate tests
The distribution of a statistic used for testing sphericity of normal distributions
The Distributions of Some Test Criteria in Multivariate Analysis
On the Moment Generating Function of Pillai's $V^{(s)}$ Criterion
Percentile approximations for a class of likelihood ratio criteria
Asymptotic Formulae for the Distribution of Hotelling's Generalized $T_0^2$ Statistic. II
An Asymptotic Expansion of the Non-Null Distribution of Hotelling's Generalized $T_0^2$-Statistic
Decomposition of the Multivariate Beta Distribution with Applications
Locally Best Invariant Test for Sphericity and the Limiting Distributions
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
Unnamed Item
On the exact finite series distribution of the smallest or the largest root of matrices in three situations
Asymptotic formulas for the distributions of the determinant and the trace of a noncentral beta matrix
Bessel functions of matrix argument
Note on the utilization of the generalized student ratio in the analysis of variance or dispersion
Zonal polynomials of the real positive definite symmetric matrices
On the distribution of a quadratic form in a multivariate normal sample
A note on the union-intersection character of some MANOVA procedures
Central and noncentral distributions of Wilks' statistic in MANOVA as mixtures of incomplete beta functions
Asymptotic expansions for the joint and marginal distributions of the latent roots of the covariance matrix
The distribution of the characteristic roots of $S_1S^{-1}_2$ under violations
The exact non-central distribution of the generalized variance
On the derivation of the asymptotic distribution of the generalized Hotelling's $T^2_0$
Simultaneous tests for equality of latent roots against certain alternatives. II
A few remarks about some recent articles on the exact distributions of multivariate test criteria. I
Exact robustness studies of tests of two multivariate hypotheses based on four criteria and their distribution problems under violations
Asymptotic expansions for distributions of latent roots in multivariate analysis
Maximization of an integral of a matrix function and asymptotic expansions of distributions of latent roots of two matrices
Some recent developments on complex multivariate distributions
Upper percentage points of the smallest root of the MANOVA matrix
An approximation to the C. D. F. of the largest root of a covariance matrix
On the exact distribution of the smallest root of the Wishart matrix using zonal polynomials
Moment-generating operators for determinants of product moments in samples from a normal system
An identity involving partitional generalized binomial coefficients
On the calculation of generalized binomial coefficients
Proof of conjectures about the expected values of the elementary symmetric functions of a noncentral Wishart matrix
On the distributions of the ratios of the extreme roots to the trace of the Wishart matrix
A monotonicity property of the power functions of some invariant tests for MANOVA
On the monotonicity of the power functions of tests based on traces of multivariate beta matrices
A note on the confidence bounds for the characteristic roots of dispersion matrices of normal variates
On the distribution of the second elementary symmetric function of the roots of a matrix
Latent roots and vectors of a Wishart matrix
On the distribution of the maximum latent root of a positive definite symmetric random matrix
On the moments of traces of two matrices in multivariate analysis
Power comparisons of tests of two multivariate hypotheses based on individual characteristic roots
On the distribution of the latent roots of a positive definite random symmetric matrix. I
Non-central distributions of the largest latent roots of three matrices in multivariate analysis
On the moments of elementary symmetric functions of the roots of two matrices
On the ratios of the individual latent roots to the trace of a Wishart matrix
Computation of the null distribution of the largest or smallest latent roots of a beta matrix
Asymptotic expansion of the distribution of the generalized variance for noncentral Wishart matrix, when $\Omega= O(n)$
Asymptotic formulae for the distributions of some criteria for tests of equality of covariance matrices
Partial differential equations for hypergeometric functions of two argument matrices
Simultaneous tests for equality of latent roots against certain alternatives. I
On the distribution of the likelihood ratio criterion for testing linear hypotheses on regression coefficients
On the distributions of the latent roots and traces of certain random matrices
Expectation of elementary symmetric functions of a Wishart matrix
The distribution of the sphericity test criterion
Asymptotic non-null distributions of the likelihood ratio criteria for covariance matrix under local alternatives
Simultaneous confidence bounds on the departures from a particular kind of multicollinearity
ON THE DISTRIBUTION OF THE LARGEST OR THE SMALLEST ROOT OF A MATRIX IN MULTIVARIATE ANALYSIS
TESTS OF SIGNIFICANCE FOR THE LATENT ROOTS OF COVARIANCE AND CORRELATION MATRICES
Some Results Useful in Multivariate Analysis
Some Statistical Methods for Comparison of Growth Curves
Random Orthogonal Transformations and their use in Some Classical Distribution Problems in Multivariate Analysis
FURTHER CONTRIBUTIONS TO MULTIVARIATE CONFIDENCE BOUNDS
Some Bounds on the Distribution Functions of the Largest and Smallest Roots of Normal Determinantal Equations
On the General Canonical Correlation Distribution
Charts of Some Upper Percentage Points of the Distribution of the Largest Characteristic Root
Bartlett Decomposition and Wishart Distribution
On the Mutual Independence of Certain Statistics
The Non-Central Multivariate Beta Distribution
Distribution of the Likelihood Ratio for Testing Multivariate Linear Hypotheses
A comparison of the powers of two multivariate analysis of variance tests
Estimation of dispersion parameters
On the exact distribution of the ratio of two independent generalized sample variances of the same or different orders
Exact percentage points of the dsitribtuion of the race of a multivariate beta matrix
Forms for the distribution of ss ilkptxcxtt statistic for bivariate sphericity
CERTAIN GENERALIZATIONS IN THE ANALYSIS OF VARIANCE
The Generalization of Student's Ratio
On the Independence of k Sets of Normally Distributed Statistical Variables
On the distribution of hotelling's trace and power comparisons
Tables for the extreme roots of the wishart matrix
On the exact distribution of the ratio of (p×p) Generalized sample variance to any of its principal minors
Distribution of the Determinant of the Sum of Products Matrix in the Non‐Central Linear Case
Multivariate Beta Distributions and Independence Properties of the Wishart Distribution
A comparison of tests of the Wilks-Lawley hypothesis in multivariate analysis
Power of the likelihood-ratio test of the general linear hypothesis in multivariate analysis
On the Distribution of the Largest of Seven Roots of a Matrix in Multivariate Analysis
On the Moments of Elementary Symmetric Functions of the Roots of Two Matrices
On the Marginal Distributions of the Latent Roots of the Multivariate Beta Matrix
APPROXIMATION FOR THE DISTRIBUTION OF THE LARGEST LATENT ROOT OF A WISHART MATRIX*^,¹
An Asymptotic Expansion for the Distribution of the Eigenvalues of a 3 by 3 Wishart Matrix
Derivatives of the characteristic root of a synmetric or a hermitian matrix with two applications in multivariate analysis
On the Test of Independence Between Two Sets of Variates
Tables for the power function of box's two sided test for testing hypothesis ∑= I the bivariate case
Distribution of the likelihood ratio criterion for testing a hypothesis specifying a covariance matrix
Asymptotic formulas for the distributions of two trace statistics for testing the independence under fixed alternative
On the Expected Values of the Elementary Symmetric Functions of a Noncentral Wishart Matrix
On the Monotonic Character of the Power Functions of Two Multivariate Tests
Distributions of Matrix Variates and Latent Roots Derived from Normal Samples
The Moments of Elementary Symmetric Functions of the Roots of a Matrix in Multivariate Analysis
Some Non-Central Distribution Problems in Multivariate Analysis
Test Criteria for Statistical Hypotheses Involving Several Variables
RELATIONS BETWEEN TWO SETS OF VARIATES
Properties of sufficiency and statistical tests
A GENERALIZATION OF FISHER'S z TEST
THE APPLICATION OF THE MOMENT FUNCTION IN THE STUDY OF DISTRIBUTION LAWS IN STATISTICS
ON THE DISTRIBUTION OF ROOTS OF CERTAIN DETERMINANTAL EQUATIONS
THE SAMPLING DISTRIBUTION OF SOME STATISTICS OBTAINED FROM NON‐LINEAR EQUATIONS
On the Sampling Theory of Roots of Determinantal Equations
ON GENERALIZED ANALYSIS OF VARIANCE. (I)
On the Distribution of Wilks' Statistic for Testing the Independence of Several Groups of Variates
TESTS OF SIGNIFICANCE IN MULTIVARIATE ANALYSIS
The General Canonical Correlation Distribution
A GENERAL DISTRIBUTION THEORY FOR A CLASS OF LIKELIHOOD CRITERIA
Limiting Distribution of a Root of a Determinantal Equation
Distribution of the Sum of Roots of a Determinantal Equation under a Certain Condition
On the Distribution of the Characteristic Roots of Normal Second-Moment Matrices
On the Distribution of two Random Matrices used in Classification Procedures
On a Heuristic Method of Test Construction and its use in Multivariate Analysis
Simultaneous Confidence Interval Estimation
Normal Multivariate Analysis and the Orthogonal Group
On Multivariate Distribution Theory
Some Further Results in Simultaneous Confidence Interval Estimation
Some Extensions of the Wishart Distribution
The Non-Central Wishart Distribution and Certain Problems of Multivariate Statistics
An algebraic derivation of the distribution of rectangular coordinates
ON THE LIMITING DISTRIBUTION OF THE CANONICAL CORRELATIONS
On the Limiting Distribution of Roots of a Determinantal Equation
Sample Criteria for Testing Equality of Means, Equality of Variances, and Equality of Covariances in a Normal Multivariate Distribution
The non-central Wishart distribution
A generating function for averages over the orthogonal group
Some New Test Criteria in Multivariate Analysis