Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

This book provides the most comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management. Whether you are a financial risk analyst, actuary, regulator or student of quantitative finance, Quantitative Risk Management gives you the practical tools you need to solve real-world problems. Describing the latest advances in the field, Quantitative Risk Management covers the methods for market, credit and operational risk modelling. It places standard industry approaches on a more formal footing and explores key concepts such as loss distributions, risk measures and risk aggregation and allocation principles. The book's methodology draws on diverse quantitative disciplines, from mathematical finance and statistics to econometrics and actuarial mathematics. A primary theme throughout is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. Proven in the classroom, the book also covers advanced topics like credit derivatives. Fully revised and expanded to reflect developments in the field since the financial crisis Features shorter chapters to facilitate teaching and learning Provides enhanced coverage of Solvency II and insurance risk management and extended treatment of credit risk, including counterparty credit risk and CDO pricing Includes a new chapter on market risk and new material on risk measures and risk aggregation

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Quantitative Risk Management: Concepts, Techniques, and Tools

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Chapter 8 – Modelling Dependence with Copulas and Applications to Risk Management

The conventional approach to social programme evaluation focuses on estimating mean impacts of programmes. Yet many interesting questions regarding the political economy of programmes, the distribution of programme benefits and the option values conferred on programme participants require knowledge of the distribution of impacts, or features of it. This paper presents evidence that heterogeneity in response to programmes is empirically important and that classical probability inequalities are not very informative in producing estimates or bounds on the distribution of programme impacts. We explore two methods for supplementing the information in these inequalities based on assumptions about participant decision-making processes and about the strength in dependence between outcomes in the participation and non-participation states. Dependence is produced as a consequence of rational choice by participants. We test for stochastic rationality among programme participants and present and implement methods for estimating the option values of social programmes.

Making the Most out of Programme Evaluations and Social Experiments: Accounting for Heterogeneity in Programme Impacts

An Introduction to Probability Theory and its Applications, Volume I

On the theory of elliptically contoured distributions

When k(x, y) is a quasi-monotone function and the random variables X and Y have fixed distributions, it is shown under some further mild conditions that ℰ k(X, Y) is a monotone functional of the joint distribution function of X and Y. Its infimum and supremum are both attained and correspond to explicitly described joint distribution functions.

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Inequalities for Ek(X, Y) when the marginals are fixed

We are concerned here with establishing the consistency and asymptotic normality for the maximum likelihood estimator of a “merit vector” $(u_0,\dots,u_t)$, representing the merits of $t +1$ teams (players, treatments, objects), under the Bradley–Terry model, as $t \to \infty$. This situation contrasts with the well-known Neyman–Scott problem under which the number of parameters grows with $t$ (the amount of sampling), and for which the maximum likelihood estimator fails even to attain consistency. A key feature of our proof is the use of an effective approximation to the inverse of the Fisher information matrix. Specifically, under the Bradley–Terry model, when teams $i$ and $j$ with respective merits $u_i$ and $u_j$ play each other, the probability that team $i$ prevails is assumed to be $u_i/(u_i + u_j)$. Suppose each pair of teams play each other exactly $n$ times for some fixed $n$. The objective is to estimate the merits, $u_i$’s, based on the outcomes of the $nt(t +1)/2$ games. Clearly, the model depends on the $u_i$’s only through their ratios. Under some condition on the growth rate of the largest ratio $u_i/u_j (0 \leq i, j \leq t)$ as $t \to \infty$, the maximum likelihood estimator of $(u_1/u_0,\dots,u_t/u_0)$ is shown to be consistent and asymptotically normal. Some simulation results are provided.

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Asymptotics when the number of parameters tends to infinity in the Bradley-Terry model for paired comparisons

On α-symmetric multivariate distributions☆

It is shown that the mean of a normal distribution with unknown variance $\sigma^2$ may be estimated to lie within an interval of given fixed width at a prescribed confidence level using a procedure which overcomes ignorance about $\sigma^2$ with no more than a finite number of observations. That is, the expected sample size exceeds the (fixed) sample size one would use if $\sigma^2$ were known by a finite amount, the difference depending on the confidence level $\alpha$ but not depending on the values of the mean $\mu$, the variance $\sigma^2$ and the interval width $2d$. A number of unpublished results on the moments of the sample size are presented. Some do not depend on an assumption of normality.

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Gordon Simons

Papers

On the theory of elliptically contoured distributions

Inequalities for Ek(X, Y) when the marginals are fixed

Asymptotics when the number of parameters tends to infinity in the Bradley-Terry model for paired comparisons

On α-symmetric multivariate distributions☆

On the Cost of not Knowing the Variance when Making a Fixed-Width Confidence Interval for the Mean