To Örebro University

This paper derives the density function of the product of a Wishart matrix and a normal vector that are independently distributed. Unlike existing results, we allow the covariance matrix of the Wishart distribution and that of the normal vector to differ, and we do not assume them to be proportional. In this general setting, the density is shown to admit a two-dimensional integral representation. We further obtain the density of linear combinations of the product. For a single linear combination, it is again expressed as a two-dimensional integral whose integrand involves the modified Bessel function of the second kind, and the special case of identity covariance matrices is treated explicitly. All resulting expressions are straightforward to evaluate numerically, and the product defines a new family of matrix-variate mixtures of multivariate normal distributions.

This paper studies hypothesis testing for structured scale matrices in the multivariate t distribution. We develop likelihood ratio and Rao score tests for general structural constraints on the scale matrix and derive the Wald test for hypotheses in which the scale matrix is restricted to a quadratic subspace. The latter class includes, in particular, diagonality, sphericity, compound symmetry, circular Toeplitz structures, and their block extensions with structured subblocks. Since the construction of the test statistics requires maximum likelihood estimation, we derive the likelihood equations under quadratic subspace restrictions. Although likelihood ratio and score tests have been investigated for covariance structure in the multivariate normal model, the Wald test has not been developed in that setting. Because the multivariate normal distribution arises as a limiting case of the multivariate t model, our results also provide a new contribution to the normal framework. Finite sample performance is examined via simulation, and a real data example illustrates the proposed methodology.

A connection of the matrix-variate gamma distributions to matrix-valued Lévy processes of a vector argument is made. Namely, a Lévy process, termed a matrix gamma-Laplace motion, is obtained by the subordination of the triangular Brownian motion of a vector argument to a vector-valued gamma motion of a vector argument. In this context, we introduce a triangular matrix-valued Rayleigh process, which, through symmetrization, leads to a new matrix-variate gamma process. This process when taken at a properly defined one-dimensional argument has the matrix gamma marginal distribution with the shape parameter equal to its argument.

We derive the exact density function of the product of an inverse Wishart random matrix and an independent normal random vector. The density is expressed as a one-dimensional integral that can be evaluated quickly and accurately by standard quadrature methods. We further obtain integral representations for the density of linear combinations of the elements of this product; in particular, the joint density of any p linear combinations can be recovered from an integral of dimension at most min(p + 1, m − p + 1, 6), where m is the dimension of the random vector. All results are established in the general setting in which the scale matrix of the Wishart distribution and the covariance matrix of the normal vector are arbitrary positive definite matrices, and simplified formulas involving integrals of dimension at most three are provided for the important special case in which the two covariance matrices are proportional.

The test for the location of the tangency portfolio on the set of feasible portfolios is proposed when both the population and the sample covariance matrices of asset returns are singular. The particular case of investigation is when the number of observations, n, is smaller than the number of assets, k, in the portfolio, and the asset returns are i.i.d. normally distributed with singular covariance matrix Σ such that rank(Σ) = r < n < k + 1. The exact distribution of the test statistic is derived under both the null and alternative hypotheses. Furthermore, the high-dimensional asymptotic distribution of that test statistic is established when both the rank of the population covariance matrix and the sample size increase to infinity so that r/n → c ∈ (0, 1). Theoretical findings are completed by comparing the high-dimensional asymptotic test with an exact finite sample test in the numerical study. A good performance of the obtained results is documented. To get a better understanding of the developed theory, an empirical study with data on the returns on the stocks included in the S&P 500 index is provided.

This paper focuses on identifying useful indicators for nowcasting GDP in Sweden. We analyze 35 monthly indicators spanning the period from 1993 to 2023. Additionally, we evaluate the group-wise performance of these indicators. The analysis is conducted using mixed-data sampling (MIDAS) and mixed-frequency VAR models in both individual and pooled setups forn owcasting. While the primary focus is on nowcasting, we also assess the performance of the indicators for backcasting and forecasting. For nowcasting, we identify 16 indicators in the individual setup and 23 indicators in the pooled setup that outperform the benchmark. Group-wise, indicators belonging to the survey, interest & exchange rates, and public finance groups exhibit strong performance in the individual setup. Notably, in the pooled setup, the output, survey, price, interest & exchange rates, and public finance groups demonstrate strong performance.

This chapter gives a brief introduction to the Festschrift. A short overview of Sune Karlsson’s career and research is provided. In addition, the content of each chapter is presented.

In this paper, estimators for the unknown parameters under two types of matrix-variate t distributions are determined, and their basic statistical properties, including bias and sufficiency, are investigated. These estimators are then applied to test hypotheses concerning the covariance structure of a multivariate t distribution associated with a collection of uncorrelated, though not necessarily independent, observation vectors, using two types of matrix-variate  distributions. A likelihood ratio test is proposed, and its distributional properties under the null hypothesis are examined, assuming either a fully specified covariance matrix or one specified up to a constant. Furthermore, it is demonstrated that the asymptotic distribution for the type I matrix-variate t distribution under both hypotheses coincides with that under the normality assumption. Finally, for testing a fully specified covariance matrix, the asymptotic distribution of the likelihood ratio test statistic is determined.