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

0 Summary In this paper we obtain some multivariate generalizations of Chebyshev's inequality, two of which are extended to continuous parameter stochastic processes The extensions are obtained in a natural way by taking into account separability and letting the number of variables approach infinity Particular attention is paid to the question of sharpness To show that the bound of the inequality cannot be improved, examples are given in a number of cases that attain equality 1 Introduction We begin by discussing a model for the various generalizations of Chebyshev's inequality, and for a standard proof that we shall use Examination of this proof will enable us to make some general comments concerning the problems of deriving inequalities and of proving sharpness Let (Q, B, P) be a probability space, and let ($C, (1) be a measurable space For each i c I, an arbitrary index set, let Bi C (t and let 5i be a class of random

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Some multivariate chebyshev inequalities with extensions to continuous parameter processes

From the inception of the proportional representation movement it has been an issue whether larger parties are favored at the expense of smaller parties in one apportionment of seats as compared to another apportionment. A number of methods have been proposed and are used in countries with a proportional representation system. These apportionment methods exhibit a regularity of order, as discussed in the present paper, that captures the preferential treatment of larger versus smaller parties. This order, namely majorization, permits the comparison of seat allocations in two apportionments. For divisor methods, we show that one method is majorized by another method if and only if their signpost ratios are increasing. This criterion is satisfied for the divisor methods with power-mean rounding, and for the divisor methods with stationary rounding. Majorization places the five traditional apportionment methods in the order as they are known to favor larger parties over smaller parties: Adams, Dean, Hill, Webster, and Jefferson.

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A majorization comparison of apportionment methods in proportional representation

The multivariate distribution of a set of random variables has exponential minimums if the minimum over each subset of the variables has an exponential distribution. Such distributions are shown equivalent to the more strongly structured multivariate exponential distributions described by Marshall and Olkin in 1967 in the sense that a multivariate exponential distribution can be found that gives the same marginal distribution for each minimum. The basic application of the result is that in computing the reliability of a coherent system a joint distribution for the component life lengths with exponential minimums can be replaced by a multivariate exponential distribution. It follows that the life length of the system has an increasing hazard rate average distribution. Other applications include characterizations of multivariate exponential distributions and the derivation of a positive dependence condition for multivariate distributions with exponential minimums.

Multivariate Distributions with Exponential Minimums

Suppose that $n$ devices are subjected to shocks occurring randomly in time as events in a Poisson process. Upon occurrence of the $i$th shock the devices suffer nonnegative random damages with joint distribution $F_i$. Damages from successive shocks are independent and accumulate additively. Failure of the $j$th device occurs at the time $T_j$ when its accumulated damage first exceeds its breaking threshold $x_j$. If $\tau$ is the life function of a coherent system, then the system life length $\tau(T_1, \cdots, T_n)$ has a distribution with increasing hazard rate average providing that $F_1, F_2, \cdots$ satisfy a multivariate stochastic ordering condition that depends upon $\tau$. If $F_1 = F_2 = \cdots$ and $\bar{H}$ is the joint survival function of $T_1, \cdots, T_n$, then $\lbrack\bar{H}(\alpha\mathbf{t})\rbrack^{1/\alpha}$ is decreasing in $\alpha$ for all $\mathbf{t} \geqslant 0. \bar{H}$ also satisfies a multivariate "new better than used" property. Moreover $T_1, \cdots, T_n$ are associated when $F_1 = F_2 = \cdots$. Examples of specific distributions are given which arise from the shock model, including a new bivariate gamma distribution which reduces to the bivariate exponential distribution of Marshall and Olkin as a special case.

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Multivariate Shock Models for Distributions with Increasing Hazard Rate Average

This paper presents in a form convenient for applications a number of bounds for distributions with monotone hazard rate. These bounds, together with their proofs and various generalizations have been obtained by Barlow and Marshall (1964). However, many of them can be characterized only through solutions of transcendental equations, and machine calculation has been necessary to make them accessible. Section 2 gives a listing of the results, and those without explicit forms are tabulated. Section 3 lists various related bounds which are of interest for purposes of comparison, and for use when the hypothesis of monotone hazard rate is not satisfied. Applications of the bounds are discussed in Section 4, together with some numerical examples.

Albert W. Marshall

Papers

Some multivariate chebyshev inequalities with extensions to continuous parameter processes

A majorization comparison of apportionment methods in proportional representation

Multivariate Distributions with Exponential Minimums

Multivariate Shock Models for Distributions with Increasing Hazard Rate Average

Tables of Bounds for Distributions with Monotone Hazard Rate