Shuangzhe Liu

Matrix results on the Khatri-Rao and Tracy-Singh products

Abstract We establish a connection between the Khatri-Rao and Tracy-Singh products introduced by Khatri and Rao (C.G. Khatri, C.R. Rao, Sankhya 30 (1968) 167–180) and Tracy and Singh (D.S. Tracy, R.P. Singh, Statistica Neerlandica 26 (1972) 143–157, respectively, and present further results including matrix equalities and inequalities involving the two products. Also, we give two statistical applications.

Experiments with Mixtures: Optimal Allocations for Becker's Models

Extending Scheffé’s simplex-centroid design for experiments with mixtures, we introduce aweighted simplex-centroid design for a class of mixture models. Becker’s homogeneous functions of degree one belong to this class. By applying optimal design theory, we obtainA-, D- andI-optimal allocations of observations for Becker’s models.

A V-optimal design for Scheffé's polynomial model

In experiments with mixtures, the so-called weighted simplex-centroid design is applied to obtain V-optimal allocation of observations which is shown to be an optimal design over the entire simplex by using the equivalence theorem.

The Hadamard product and some of its applications in statistics

Algebraic properties of the Hadamard product are used to establish some statistical properties.

The heteroskedastic linear regression model and the Hadamard product a note

A unified matrix approach to the heteroskedastic linear regression model and its estimation is presented. The Hadamard product plays an essential role. Our approach creates the possibility of treating not only the standard linear but also nonlinear specifications. Special attention is being paid to maximum-likelihood estimation.

On Matrix Trace Kantorovich-type Inequalities

In this chapter, a variety of matrix trace versions of the Kantorovich-type inequalities are collected. Some relevant matrix and determinant inequalities are also included. Their mathematical and statistical applications are studied. In the mathematical applications, some known results are used to derive new inequalities involving the Hadamard product, and to propose conjectures. In the statistical applications, new criteria for efficiency comparisons in the linear model are introduced. Examples, illustrated by figures, are used to examine upper bounds for known and new criteria.

The density of the Moore-Penrose inverse of a random matrix

Abstract The density of the Moore-Penrose inverse of a random matrix is derived by standard matrix calculus.

Matrix-trace Cauchy-Schwarz inequalities and applications in canonical correlation analysis

Various matrix-trace Cauchy-Schwarz and related inequalities involving positive semidefinite matrices are obtained. Applications of some of these results to canonical correlation analysis are presented.

Maximum Likelihood Estimation for the VAR-VARCH Model: A New Approach

We consider a general multivariate conditional heteroskedastic time series model and derive the information matrix of the maximum likelihood estimator by using the matrix differential calculus techniques of Magnus and Neudecker (1991). We discuss the VAR VARCH model as a special case, and demonstrate the maximum likelihood estimation of the information matrix in an example with simulated data.

Testing constraints and misspecification in VAR-ARCH models

Vector autoregressive models with conditional heteroskedastic errors (abbreviated as VAR-ARCH models) have become increasingly important for applications in financial econometrics. In this paper, we propose likelihood ratio and Wald tests for constraints and the White (1982) misspecification test for VAR-ARCH models which are estimated by the maximum likelihood (ML) method. The tests are discussed for a general class of multivariate conditional heteroskedastic time series models including the VAR-ARCH models. We derive the exact analytic expression for the gradient vector and the conditional information matrix from the log-likelihood function under the normality assumption.