Random matrices: Universality of ESDs and the circular law

Given an n x n complex matrix A, let mu(A)(x, y) := 1/n vertical bar{1 <= i <= n, Re lambda(i) <= x, Im lambda(i) <= y}vertical bar be the empirical spectral distribution (ESD) of its eigenvalues lambda(i) is an element of C, i = l, ... , n. We consider the limiting distribution (both in probability and in the almost sure convergence sense) of the normalized ESD mu(1/root n An) of a random matrix A(n) = (a(ij))(1 <= i, j <= n), where the random variables a(ij) - E(a(ij)) are i.i.d. copies of a fixed random variable x with unit variance. We prove a universality principle for such ensembles, namely, that the limit distribution in question is independent of the actual choice of x. In particular, in order to compute this distribution, one can assume that x is real or complex Gaussian. As a related result, we show how laws for this ESD follow from laws for the singular value distribution of 1/root n A(n) - zI for complex z. As a corollary, we establish the circular law conjecture (both almost surely and in probability), which asserts that mu(1/root n An) converges to the uniform measure on the unit disc when the a(ij) have zero mean.

The role of Minitab in teaching and learning statistics

This paper describes how the statistical package Minitab is used in teaching statistics in our undergraduate programmes in Mathematics and Statistics to enhance student learning. How the sophisticated recent versions of Minitab can be used to help students understand statistical concepts, develop their statistical thinking and gain valuable skills in performing statistical analysis are discussed.

Bounded Rationality and Repeated Network Formation

We define a finite-horizon repeated network formation game with consent and study the differences induced by two levels of individual rationality. Perfectly rational players will remain unconnected at the equilibrium, while nonempty equilibrium networks may form when players are assumed to behave as finite automata of limited complexity. We provide structural properties of the sequences of networks which are likely to form in Nash and subgame perfect Nash equilibria of the repeated game. For instance, players can form totally different connected networks at each period or the sequence of networks can exhibit a total order relationship.

Network formation under mutual consent and costly communication

We consider two different approaches to describe the formation of social networks under mutual consent and costly communication. First, we consider a network-based approach; in particular Jackson–Wolinsky’s concept of pairwise stability. Next, we discuss a non-cooperative game-theoretic approach, through a refinement of the Nash equilibria of Myerson’s consent game. This refinement, denoted as monadic stability, describes myopically forward looking behavior of the players. We show through an equivalence that the class of monadically stable networks is a strict subset of the class of pairwise stable networks that can be characterized fully by modifications of the properties defining pairwise stability.

Labor unions and Nash bargaining using coalition formation games

In this paper we examine the properties of stable coalitions under sequential and simultaneous bargaining by competing labor unions. We do this using the Nash bargaining solution and various notions of stability, namely, Nash, coalitional, contractual and core stability. (C) 2011 Elsevier B.V. All rights reserved,

Psychology Statistics for Dummies

The quick, easy way to master all the statistics you'll ever need The bad news first: if you want a psychology degree you'll need to know statistics. Now for the good news: Psychology Statistics For Dummies. Featuring jargon-free explanations, step-by-step instructions and dozens of real-life examples, Psychology Statistics For Dummies makes the knotty world of statistics a lot less baffling. Rather than padding the text with concepts and procedures irrelevant to the task, the authors focus only on the statistics psychology students need to know. As an alternative to typical, lead-heavy statistics texts or supplements to assigned course reading, this is one book psychology students won't want to be without. Ease into statistics – start out with an introduction to how statistics are used by psychologists, including the types of variables they use and how they measure them Get your feet wet – quickly learn the basics of descriptive statistics, such as central tendency and measures of dispersion, along with common ways of graphically depicting information Meet your new best friend – learn the ins and outs of SPSS, the most popular statistics software package among psychology students, including how to input, manipulate and analyse data Analyse this – get up to speed on statistical analysis core concepts, such as probability and inference, hypothesis testing, distributions, Z-scores and effect sizes Correlate that – get the lowdown on common procedures for defining relationships between variables, including linear regressions, associations between categorical data and more Analyse by inference – master key methods in inferential statistics, including techniques for analysing independent groups designs and repeated-measures research designs Open the book and find: Ways to describe statistical data How to use SPSS statistical software Probability theory and statistical inference Descriptive statistics basics How to test hypotheses Correlations and other relationships between variables Core concepts in statistical analysis for psychology Analysing research designs Learn to: Use SPSS to analyse data Master statistical methods and procedures using psychology-based explanations and examples Create better reports Identify key concepts and pass your course

Dyslexia in higher education: Implications for maths anxiety, statistics anxiety and psychological well-being

This study examined levels of mathematics and statistics anxiety, as well as general mental health amongst undergraduate students with dyslexia (n = 28) and those without dyslexia (n = 71). Students with dyslexia had higher levels of mathematics anxiety relative to those without dyslexia, while statistics anxiety and general mental health were comparable for both reading ability groups. In terms of coping strategies, undergraduates with dyslexia tended to use planning-based strategies and seek instrumental support more frequently than those without dyslexia. Higher mathematics anxiety was associated with having a dyslexia diagnosis, as well as greater levels of worrying, denial, seeking instrumental support and less use of the positive reinterpretation coping strategy. By contrast, statistics anxiety was not predicted by dyslexia diagnosis, but was instead predicted by overall worrying and the use of denial and emotion focused coping strategies. The results suggest that disability practitioners should be aware that university students with dyslexia are at risk of high mathematics anxiety. Additionally, effective anxiety reduction strategies such as positive reframing and thought challenging would form a useful addition to the support package delivered to many students with dyslexia.

Measuring and testing for the systemically important financial institutions

This paper analyzes the measure of systemic importance ∆CoV aR proposed by Adrian and Brunnermeier (2009, 2010) within the context of a similar class of risk measures used in the risk management literature. In addition, we develop a series of testing procedures, based on ∆CoV aR, to identify and rank the systemically important institutions. We stress the importance of statistical testing in interpreting the measure of systemic importance. An empirical application illustrates the testing procedures, using equity data for three European banks.

Risk, ambiguity, and diversification

Attitudes toward risk influence the decision to diversify among uncertain options. Yet, because in most situations the options are ambiguous, attitudes toward ambiguity may also play an important role. I conduct a laboratory experiment to investigate the effect of ambiguity on the decision to diversify. I find that diversification is more prevalent and more persistent under ambiguity than under risk. Moreover, excess diversification under ambiguity is driven by participants who stick with a status quo gamble when diversification among gambles is not feasible. This behavioral pattern cannot be accommodated by major theories of choice under ambiguity.