Distinctness of the Eigenvalues of a Quadratic form in a Multivariate Sample

The joint distribution of Studentized residuals under elliptical distributions

Iwashita, T., & Klar, B. (2014) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2014.03.015 · Cited by 3

Optimal nonlinear transformations of random variables

Goia, A., & Salinelli, E. (2010) · Annales de l Institut Henri Poincaré Probabilités et Statistiques · doi:10.1214/09-aihp326 · Cited by 3

Characterization Results

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_6 · Cited by 3

Identifying the Relevant Nodes Without Learning the ModelPreprint

Peña, J. M., Nilsson, R., Björkegren, J., & Tegner, J. N. (2012) · arXiv (Cornell University) · doi:10.48550/arxiv.1206.6847 · Cited by 3

A New Approach to the Identification Problem

de Brito, C. E. F. (2002) · Lecture notes in computer science · doi:10.1007/3-540-36127-8_5 · Cited by 2

Wald tests when restrictions are locally singular

Dufour, J., Renault, É., & Zinde‐Walsh, V. (2025) · The Annals of Statistics · doi:10.1214/24-aos2398 · Cited by 2

Canonical correlation and reduction of multiple time series

Pourahmadi, M. (1994) · Annals of the Institute of Statistical Mathematics · doi:10.1007/bf00773471 · Cited by 2

BAYESIAN LOCAL INFLUENCE ASSESSMENTS IN A GROWTH CURVE MODEL WITH GENERAL COVARIANCE STRUCTURE

Bai, P., & Fei, Y. (2000) · Acta Mathematica Scientia · doi:10.1016/s0252-9602(17)30668-9 · Cited by 2

On the Bias of Directed Information Estimators

Schamberg, G., & Coleman, T. P. (2019) · IEEE International Symposium on Information Theory (ISIT) · doi:10.1109/isit.2019.8849531 · Cited by 2

The Nonlinear Problem of the 3d Datum Transformation and the Procrustes Algorithm

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_12 · Cited by 2

A multivariate approach for estimating the random effects variance component in one-way random effects model

Sutradhar, B. C. (1997) · Statistics & Probability Letters · doi:10.1016/s0167-7152(96)00144-7 · Cited by 2

Consistency of the Hotelling-, wilks- Roy- and pillai-tests for a general linear hypothesis in multiyariate linear nonnormally Distribnted models

Schulze, U. (1976) · Mathematische Operationsforschung Statistik · doi:10.1080/02331887608801274 · Cited by 2

Growth curve mixture models with unknown covariance structures

Pan, Y., Fei, Y., Mingming, N., Nummi, T., & Pan, J. (2021) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2021.104904 · Cited by 1

Decision making, symmetry and structure: Justifying causal interventions

Johnston, D., Ong, C. S., & Williamson, R. C. (2025) · Journal of Causal Inference · doi:10.1515/jci-2023-0001 · Cited by 1

Linear Model Estimation and Prediction for p > n

Christensen, R. (2025) · The American Statistician · doi:10.1080/00031305.2025.2566251 · Cited by 1

Skew Elliptically Contoured Distributions

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_11 · Cited by 1

Mixtures of Normal Distributions

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_4 · Cited by 1

A test of the hypothesis of partial common principal components

Rublík, F. (2009) · Mathematica Slovaca · doi:10.2478/s12175-009-0149-9 · Cited by 1

Noncentral matrix quadratic forms of the skew elliptical variables

Fang, B. Q. (2007) · Journal of Multivariate Analysis · doi:10.1016/j.jmva.2007.07.003 · Cited by 1

A necessary test for elliptical symmetry based on the uniform distribution over the Stiefel manifold

Iwashita, T., & Klar, B. (2020) · SUT Journal of Mathematics · doi:10.55937/sut/1609967403 · Cited by 1

Linear Phenotypic Eigen Selection Index Methods

Cerón‐Rojas, J. J., & Crossa, J. L. (2018) · Springer eBooks · doi:10.1007/978-3-319-91223-3_7 · Cited by 1

On the nonsingularity of principal submatrices of a random orthogonal matrix

Kariya, T., & Wu, C. (1985) · Journal of Statistical Planning and Inference · doi:10.1016/0378-3758(85)90083-7 · Cited by 1

Effect of tapering on accuracy of forecasts made with stable estimators of vector autoregressive processes

Zhou, Y., & Roy, A. (2005) · International Journal of Forecasting · doi:10.1016/j.ijforecast.2005.06.005 · Cited by 1

Asymptotics in multivariate linear models with optimal experimental designs

Schmidt, W. H. (1977) · Series Statistics · doi:10.1080/02331887708801390 · Cited by 1

SUPPLEMENT TO \NONPARAMETRIC ESTIMATION OF MULTIVARIATE CONVEX-TRANSFORMED DENSITIES."

Seregin, A., & Wellner, J. A. (2010) · Cited by 1

A class of nonparametric estimators of regression coefficients in an autoregressive scheme

Kulp, R. W. (1976) · ThinkTech (Texas Tech University) · Cited by 1

Determinantal Generalizations of Instrumental VariablesPreprint

Weihs, L., Robinson, B., Dufresne, E., Kenkel, J., Kubjas, K., McGee, R. L., Nguyen, N., Robeva, E., & Drton, M. (2017) · arXiv (Cornell University) · doi:10.48550/arxiv.1702.03884 · Cited by 1

Maximum Likelihood Estimation for Brownian Motion Tree Models Based on One SamplePreprint

Truell, M., Hütter, J.-C., Squires, C., Zwiernik, P., & Uhler, C. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2112.00816 · Cited by 1

Randomized Policy Optimization for Optimal StoppingPreprint

Guan, X., & Mišić, V. V. (2022) · SSRN Electronic Journal · doi:10.2139/ssrn.4066178 · Cited by 1

Linear algebra and random vectors

Lauritzen, S. L. (1996) · Oxford University Press eBooks · doi:10.1093/9780191775284.005.0002 · Cited by 1

High-dimensional inference of graphical models using regularized score matchingPreprint

Lin, L., Drton, M., & Shojaie, A. (2015) · arXiv (Cornell University) · Cited by 1

Application in Portfolio Theory

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_10

The Sixth Problem of Generalized Algebraic Regression

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_13

The Second Problem of Algebraic Regression

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_3

Quadratic Forms and Other Functions of Elliptically Contoured Matrices

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_5

The Fifth Problem of Algebraic Regression: The System of Conditional Equations: Homogeneous and Inhomogeneous Equations: $$\{\mathbf{By} = \mathbf{Bi}\ \mathbf{versus} -\mathbf{c} + \mathbf{By} = \mathbf{Bi}\}$$

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_9

Probability Density Function and Expected Values

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_3

Special Problems of Algebraic Regression and Stochastic Estimation

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_14

Testing for equality of ordered eigenvectors of two multivariate normal populations

Najarzadeh, D., Khazaei, M., & Ganjali, M. (2014) · METRON · doi:10.1007/s40300-014-0051-2

Estimation in Linear Models Based on Observations With Unknown and Possibly Unequal Scaling

Jensen, S., & Madsen, J. C. (2006) · Journal of the American Statistical Association · doi:10.1198/016214506000000393

Tilt estimation in moderate-to-strong scintillation

Burl, J. B., Roggemann, M. C., & Welsh, B. M. (2001) · Applied Optics · doi:10.1364/ao.40.002966

A Matrix-Valued Weibull Distribution

Hasofer, A. M. (1985) · Springer eBooks · doi:10.1007/978-3-642-82419-7_2

The Fourth Problem of Probabilistic Regression

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_8

The First Problem of Algebraic Regression

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_1

The First Problem of Probabilistic Regression: The Bias Problem

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_2

The maximum likelihood threshold of a path diagram

Drton, M., Fox, C., Käufl, A., & Pouliot, G. (2019) · The Annals of Statistics · doi:10.1214/18-aos1724

Multivariate Linear Models: General

Christensen, R. (2019) · Springer texts in statistics · doi:10.1007/978-3-030-29164-8_9

Partial data problems in scalar and vector field tomographyPreprint

Ilmavirta, J., & Mönkkönen, K. (2021) · arXiv (Cornell University) · doi:10.48550/arxiv.2103.14385

References

Seber, G. A. F. (1984) · Wiley series in probability and statistics · doi:10.1002/9780470316641.refs

The Third Problem of Algebraic Regression

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_5

References

(1994) · Wiley series in probability and statistics · doi:10.1002/9780470316894.refs

Bibliography

Fuller, W. A. (1987) · Wiley series in probability and statistics · doi:10.1002/9780470316665.biblio

Preliminaries

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_1

The Fifth Problem of Probabilistic Regression

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_10

Linear Models

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_9

Estimation

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_7

Hypothesis Testing

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_8

Reference

Seber, G. A. F. (2007) · doi:10.1002/9780470226797.refs

Performance Analysis

Pillai, S. U., & Burrus, C. S. (1989) · Signal processing and digital filtering · doi:10.1007/978-1-4612-3632-0_3

Basic Properties

Gupta, A. K., Varga, T. R., & Bodnar, T. (2013) · Springer eBooks · doi:10.1007/978-1-4614-8154-6_2

Multivariate Models

Seber, G. A. F. (2015) · Springer series in statistics · doi:10.1007/978-3-319-21930-1_9

Various prerequisites

Lauritzen, S. L. (1996) · Oxford University Press eBooks · doi:10.1093/9780191775284.005.0001

Multivariate normal models

Lauritzen, S. L. (1996) · Oxford University Press eBooks · doi:10.1093/9780191775284.003.0006

Models for mixed data

Lauritzen, S. L. (1996) · Oxford University Press eBooks · doi:10.1093/9780191775284.003.0007

Global strong convexity and characterization of critical points of time-of-arrival-based source localization

Pun, Y.-M., & So, A. M. (2023) · Computational Geometry · doi:10.1016/j.comgeo.2023.102077

Improved Error Bounds for the DEIM and Related Masked Projection Techniques

Tripathi, B. N., & Shekhawat, H. S. (2025) · IEEE Control Systems Letters · doi:10.1109/lcsys.2025.3623245

The Third Problem of Probabilistic Regression

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_6

Parameter identification in linear non-Gaussian causal models under general confounding

Tramontano, D., Drton, M., & Etesami, J. (2026) · The Annals of Statistics · doi:10.1214/25-aos2597

Conditional independence

Lauritzen, S. L. (2026) · Oxford University Press eBooks · doi:10.1093/9780191775284.003.0003

Preface

Lauritzen, S. L. (2026) · Graphical Models · doi:10.1093/9780191775284.002.0005

Introduction

Lauritzen, S. L. (2026) · Oxford University Press eBooks · doi:10.1093/9780191775284.003.0001

Contingency tables

Lauritzen, S. L. (2026) · Oxford University Press eBooks · doi:10.1093/9780191775284.003.0005

Markov properties

Lauritzen, S. L. (2026) · Oxford University Press eBooks · doi:10.1093/9780191775284.003.0004

Preface to first edition

Lauritzen, S. L. (2026) · Graphical Models · doi:10.1093/9780191775284.002.0006

Exponential models

Lauritzen, S. L. (2026) · Oxford University Press eBooks · doi:10.1093/9780191775284.005.0004

Structure estimation

Lauritzen, S. L. (2026) · Oxford University Press eBooks · doi:10.1093/9780191775284.003.0008

Graphical Models

Lauritzen, S. L. (2026) · doi:10.1093/9780191775284.001.0001 · Cited by 1

Maximum likelihood estimation for Brownian motion tree models based on one sample

Squires, C., Truell, M., Hütter, J.-C., Zwiernik, P., & Uhler, C. (2026) · Bernoulli · doi:10.3150/26-bej1970 · Cited by 6

Note on Estimating the Eigen System of Σ1-1Σ2

Kim, M.-G. (2003) · Communications for Statistical Applications and Methods · doi:10.5351/ckss.2003.10.2.603

The Second Problem of Probabilistic Regression

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_4

Algebraic Solutions of Systems of Equations

Grafarend, E. W., & Awange, J. L. (2011) · Springer eBooks · doi:10.1007/978-3-642-22241-2_15 · Cited by 1

Rank of a quadratic form in an elliptically contoured matrix random variable

Gupta, A. K., & Varga, T. R. (1991) · Statistics & Probability Letters · doi:10.1016/0167-7152(91)90056-w · Cited by 14

On the distribution of the multivariate quadratic form in multivariate normal samples

Hayakawa, T. (1972) · Annals of the Institute of Statistical Mathematics · doi:10.1007/bf02479752 · Cited by 19

Joint density of eigenvalues in spiked multivariate modelsPreprint

Dharmawansa, P., & Johnstone, I. M. (2014) · arXiv (Cornell University) · doi:10.48550/arxiv.1406.0267

An asymptotic expansion for the distribution of the determinant of a multivariate quadratic form in a normal sample

Hayakawa, T. (1973) · Annals of the Institute of Statistical Mathematics · doi:10.1007/bf02479385 · Cited by 5

On Distributions of Matrix Quadratic Forms

Ohlson, M., & Koski, T. (2009) · Cited by 1