Traces of Wishart matrices appear in many applications, for example in finance, discriminant analysis, Mahalanobis distances and angles, loss functions and many more. These applications typically involve mixtures of traces of Wishart and inverse Wishart matrices that are concerned in this paper. Of particular interest are the sampling moments and their limiting joint distribution. The covariance matrix of the marginal positive and negative spectral moments is derived in closed form (covariance matrix of where ). The results are obtained through convenient recursive formulas for and Moreover, we derive an explicit central limit theorem for the scaled vector Y, when and present a simulation study on the convergence to normality and on a skewness measure.

Moments of functions of Wishart distributed matrices appear frequently in multivariate analysis. Although a considerable number of such moments have long been available in the literature, they appear in rather dispersed sources and may sometimes be difficult to locate. This paper presents a collection of moments of the Wishart and inverse Wishart distribution, involving functions such as traces, determinants, Kronecker, and Hadamard products, etc. Moments of factors resulting from decompositions of Wishart matrices are also included.

The problem of how to determine portfolio weights so that the variance of portfolio returns is minimized has been given considerable attention in the literature, and several methods have been proposed. Some properties of these estimators, however, remain unknown, and many of their relative strengths and weaknesses are therefore difficult to assess for users. This paper contributes to the field by comparing and contrasting the risk functions used to derive efficient portfolio weight estimators. It is argued that risk functions commonly used to derive and evaluate estimators may be inadequate and that alternative quality criteria should be considered instead. The theoretical discussions are supported by a Monte Carlo simulation and two empirical applications where particular focus is set on cases where the number of assets (p) is close to the number of observations (n).

This volume is a tribute to Professor Dietrich von Rosen on the occasion of his 65^th birthday. It contains a collection of twenty original papers. The contents of the papers evolve around multivariate analysis and random matrices with topics such as high-dimensional analysis, goodness-of-fit measures, variable selection and information criteria, inference of covariance structures, the Wishart distribution and growth curve models.

In this paper we derive weighted squared risk measures for a commonly used Stein-type estimator of the global minimum variance portfolio. The risk functions are conveniently split in terms of variance and squared bias over different weight matrices. It is argued that the common out-of-sample variance criteria should be used with care and that a simple unweighted risk function may be more appropriate.

Statistical analysis frequently involves methods for reducing high-dimensional data to new variates of lower dimension for the purpose of assessing distributional properties, identification of hidden patterns, for discriminant analysis, etc. In classical multivariate analysis such matters are usually analysed by either using principal components (PC) or the Mahalanobis distance (MD). While the distributional properties of PC’s are fairly well established in high-dimensional cases, no explicit results appear to be available for the MD under such cases. The purpose of this chapter is to bridge that gap by deriving weak limits for the MD in cases where the dimension of the random vector of interest is proportional to the sample size (n, p-asymptotics). The limiting distributions allow for normality-based inference in cases when the traditional low-dimensional approximations do not apply.

This paper argues that traditional measures of innovation as a univariate phenomenon may not be dynamic enough to adequately describe the complex nature of innovation. Consequently, the purpose is to develop a multidimensional index of innovation that is able to reflect innovation enablers and outputs. The index may then be used (i) to assess and quantify temporal changes of innovation, (ii) to describe regional differences and similarities of innovation, and (iii) serve as exogenous variables to analyze the importance of innovation for other economic phenomena. Our index is defined in a four-dimensional space of orthogonal axes. An empirical case study is used for demonstration of the index, where 44 variables are collected for all municipalities in Sweden. The index spanning the four-dimensional innovation comprises size, accessibility, firm performance, and agglomeration. The proposed index offers a new way of defining and analyzing innovation and should have a wide range of important applications in a world where innovation is receiving a great deal of recognition.

This paper derives first-order sampling moments of individual Mahalanobis distances (MD) in cases when the dimension p of the variable is proportional to the sample size n. Asymptotic expected values when n, p → ∞ are derived under the assumption p/n → c, 0 ⩽ c < 1. It is shown that some types of standard estimators remain unbiased in this case, while others are asymptotically biased, a property that appears to be unnoticed in the literature. Second order moments are also supplied to give some additional insight to the matter.

The problem of how to maximize the return on a given portfolio of assets within the theory of Markowitz has been given considerable attention in the literature and improvements of standard methods continues to progress. Recent developments, often based on Stein estimators or other regularized estimators, usually focus on settings when the numbers of assets (say p) is close to the number of observations (n) since this is the scenario met in most real applications. Before any specific method is applied investors would want to know the basic properties and the relative performance of them. The performance of any estimation method, however, depends on which quality criterea of judgement is being used. Proposed methods may be optimal with respect to precision of the parameters involved in the portfolio procedure, on the proximity between estimated vs true global minimum variance portfolio (GMVP) weights, on the out-of-sample performance etc. Moreover, regularized estimators are often associated with very complicated or even unknown sampling distributions, which in turn complicate statistical inference drastically. The extent to which a method allows for statistical inference therefore also becomes an important matter when judging the properties of a data driven GMVP estimator. In this paper we give an in-depth discussion of risk critereas and their impact on GMVP optimization. A Monte Carlo simulation investigating the properties of some common estimators, including a new one proposed by the authors, with respect to several quality critereas is included to compare and contrast recent proposals. An empirical study is also included using Stockholm stock exchange data. 

Course syllabuses, outlines or similar academic documents specifying the content of a course will often be a helpful tool both for teachers and students to grasp the content and purpose of a course. In many cases, however, the compilation of such documents is a painstaking process for the educator designing it, and is a task that many teachers will shun. In this paper we propose a fairly simple pedagogical model for designing specific learning outcomes that the students are expected to attain after completion of a course.