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

This paper provides a brief introduction to the theory of majorization and its use in deriving inequalities. Majorization is a preordering of vectors, and inequalities are obtained from the fact that ϕ(x) ≤ ϕ(y) whenever x is majorized by y and ϕ is an order-preserving function. For majorization, the order-preserving functions are called Schur-convex functions. Examples of vectors ordered by majorization, and of Schur-convex functions, are given. To illustrate their usefulness, some inequalities are derived.

Inequalities via Majorization — An Introduction

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Inequalities for the trace function

It is well known (see e.g. [4] p. 198) that for every positive $\epsilon$ and every square integrable random variable $X$ with zero expectation, $P{X \geqq \epsilon} \leqq E (X^2)/\lbrack\epsilon^2 + E(X^2)\rbrack$. In this paper an inequality is obtained that generalizes this in the same way that Kolmogorov's inequality generalizes Chebyshev's inequality. The inequality is proved in Section 2 and an example is given to show that equality can be achieved. In Section 3 an extension to continuous parameter martingales is obtained, and a condition under which equality can be achieved is given.

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A One-Sided Analog of Kolmogorov's Inequality

Various multivariate NBU (new better than used) notions are surveyed in this paper. The relationships among them are studied. Some preservation properties of these notions under different kinds of operations are listed and open questions are indicated. Multivariate NBU distributions with exponential marginals are discussed. Finally some examples are given.

Multivariate new better than used distributions: a survey

The univariate distribution uniform on the unit interval [0,1] is important primarily because of the following characterization: Let X be a random variable taking values in [0,1]. Then the distribution of X + U (mod 1) is the same as the distribution of X for all nonnegative random variables U independent of X if and only if X has a distribution uniform on [0,1].

Albert W. Marshall

Papers

Inequalities via Majorization — An Introduction

Inequalities for the trace function

A One-Sided Analog of Kolmogorov's Inequality

Multivariate new better than used distributions: a survey

A Bivariate Uniform Distribution