Introduction.- Doubly Stochastic Matrices.- Schur-Convex Functions.- Equivalent Conditions for Majorization.- Preservation and Generation of Majorization.- Rearrangements and Majorization.- Combinatorial Analysis.- Geometric Inequalities.- Matrix Theory.- Numerical Analysis.- Stochastic Majorizations.- Probabilistic, Statistical, and Other Applications.- Additional Statistical Applications.- Orderings Extending Majorization.- Multivariate Majorization.- Convex Functions and Some Classical Inequalities.- Stochastic Ordering.- Total Positivity.- Matrix Factorizations, Compounds, Direct Products, and M-Matrices.- Extremal Representations of Matrix Functions.

Inequalities: Theory of Majorization and Its Applications

In semidefinite programming, one minimizes a linear function subject to the constraint that an affine combination of symmetric matrices is positive semidefinite. Such a constraint is nonlinear and nonsmooth, but convex, so semidefinite programs are convex optimization problems. Semidefinite programming unifies several standard problems (e.g., linear and quadratic programming) and finds many applications in engineering and combinatorial optimization. Although semidefinite programs are much more general than linear programs, they are not much harder to solve. Most interior-point methods for linear programming have been generalized to semidefinite programs. As in linear programming, these methods have polynomial worst-case complexity and perform very well in practice. This paper gives a survey of the theory and applications of semidefinite programs and an introduction to primaldual interior-point methods for their solution.

Semidefinite programming

The Lasso estimate for linear regression parameters can be interpreted as a Bayesian posterior mode estimate when the regression parameters have independent Laplace (i.e., double-exponential) priors. Gibbs sampling from this posterior is possible using an expanded hierarchy with conjugate normal priors for the regression parameters and independent exponential priors on their variances. A connection with the inverse-Gaussian distribution provides tractable full conditional distributions. The Bayesian Lasso provides interval estimates (Bayesian credible intervals) that can guide variable selection. Moreover, the structure of the hierarchical model provides both Bayesian and likelihood methods for selecting the Lasso parameter. Slight modifications lead to Bayesian versions of other Lasso-related estimation methods, including bridge regression and a robust variant.

The Bayesian Lasso

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

Stochastic programming can effectively describe many decision-making problems in uncertain environments. Unfortunately, such programs are often computationally demanding to solve. In addition, their solution can be misleading when there is ambiguity in the choice of a distribution for the random parameters. In this paper, we propose a model that describes uncertainty in both the distribution form (discrete, Gaussian, exponential, etc.) and moments (mean and covariance matrix). We demonstrate that for a wide range of cost functions the associated distributionally robust (or min-max) stochastic program can be solved efficiently. Furthermore, by deriving a new confidence region for the mean and the covariance matrix of a random vector, we provide probabilistic arguments for using our model in problems that rely heavily on historical data. These arguments are confirmed in a practical example of portfolio selection, where our framework leads to better-performing policies on the “true” distribution underlying the daily returns of financial assets.

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Distributionally Robust Optimization Under Moment Uncertainty with Application to Data-Driven Problems

SUMMARY A new way of introducing a parameter to expand a family of distributions is introduced and applied to yield a new two-parameter extension of the exponential distribution which may serve as a competitor to such commonly-used two-parameter families of life distributions as the Weibull, gamma and lognormal distributions. In addition, the general method is applied to yield a new three-parameter Weibull distribution. Families expanded using the method introduced here have the property that the minimum of a geometric number of independent random variables with common distribution in the family has a distribution again in the family. Bivariate versions are also considered.

A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families

For many years there has been an interest in families of bivariate distributions with marginals as parameters. Genest and MacKay (1986a,b) showed that several such families that appear in the literature can be derived by a unified method. A similar conclusion is obtained in this article through the use of mixture models. These models might be regarded as multivariate proportional hazards models with random constants of proportionality. The mixture models are useful for two purposes. First, they make some properties of the derived distributions more transparent; the positive-dependency property of association is sometimes exposed, and a method for simulation of data from the distributions is suggested. But the mixture models also allow derivation of several new families of bivariate distributions with marginals as parameters, and they indicate obvious multivariate extensions. Some of the new families of bivariate distributions given in this article extend known distributions by adding a parameter ...

Families of Multivariate Distributions

: Properties of distribution functions F (or their densities f) are related to properties of the corres onding hazard rate q defined by q(x) equals f(x)/ 1 - F(x) . Interest in the hazard rate is derived from its probabilistic interpretation: if, for example, F is a life distribution, q(x)dx is the conditional probability of death in (x, x + dx) given survival to age x. Because of this interpretation f is assumed to be the density of a positive random variable, although for many of the results this is not necessary. The hazard rate is important in a number of applications, and is known by a variety of names. It is used by actuaries under the name of force of mortality to compute mortality tables, and its reciprocal is known to statisticians as Mill's ratio. In the analysis of extreme value distributions it is called the intensity function, and in reliability theory it is usually referred to as the failure rate. A number of general results are obtained, but particular attention is paid to densities with monotone hazard rate. (Autho )

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Properties of Probability Distributions with Monotone Hazard Rate

Basics.- Preliminaries.- Ordering Distributions: Descriptive Statistics.- Mixtures.- Nonparametric Families.- Nonparametric Families: Densities and Hazard Rates.- Nonparametric Families: Origins in Reliability Theory.- Nonparametric Families: Inequalities for Moments and Survival Functions.- Semiparametric Families.- Semiparametric Families.- Parametric Families.- The Exponential Distribution.- Parametric Extensions of the Exponential Distribution.- Gompertz and Gompertz-Makeham Distributions.- The Pareto and F Distributions and Their Parametric Extensions.- Logarithmic Distributions.- The Inverse Gaussian Distribution.- Distributions with Bounded Support.- Additional Parametric Families.- Models Involving Several Variables.- Covariate Models.- Several Types of Failure: Competing Risks.- More About Semi-parametric Families.- Characterizations Through Coincidences of Semiparametric Families.- More About Semiparametric Families.- Complementary Topics.- Some Topics from Probability Theory.- Convexity and Total Positivity.- Some Functional Equations.- Gamma and Beta Functions.- Some Topics from Analysis.

Albert W. Marshall

Papers

A new method for adding a parameter to a family of distributions with application to the exponential and Weibull families

Families of Multivariate Distributions

Properties of Probability Distributions with Monotone Hazard Rate

Life Distributions Structure Of Nonparametric Semiparametric And Parametric Families

The Inverse Gaussian Distribution