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

/pdf/quantitative-risk-management-concepts-techniques-and-tools-afazrwn4wm.pdf

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.

/pdf/distributionally-robust-optimization-under-moment-4d1au1dpky.pdf

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

: Two (narrow and wide) multivariate geometric analogues of the Marshall-Olkin multivariate exponetial distribution are derived from the following cumulative damage model. A set of devices is exposed to a common damage process. Damage occurs in discrete cycles. On each cycle the amount of damage is an independent observation on a nonnegative random variable. Damages accumulate additively. Each device has its own random breaking threshold. A device fails when the accumulated damage exceeds its threshold. Thresholds are independent of damages, and have a Marshall-Olkin multivariate exponential distribution. The joint distribution of the random numbers of cycles up to and including failure of the devices has the wide multivariate geometric distribution. It has the narrow multivariate geometric distribution if the damage variable is infinitely divisible. (Author)

https://calhoun.nps.edu/bitstream/10945/30003/1/multivariategeom00esar.pdf

Multivariate geometric distributions generated by a cumulative damage process

Jensen's Inequality states that if p is an extended real valued convex function defined on a convex set A of n X n matrices, A,,... ,Ak E A and a.1... ,ak are nonnegative numbers satisfying Eai = 1, then 4(4aAl + +aikAk) < ao1.(Aj) + * +OakO(Ak). (2) Finally, a Convexity Result of Minkowski states that if H is an n X n positive semidefinite Hermitian matrix, then (with the convention log 0 = oo) log det H is concave, (3) (det H)1/n is concave. (4) (See e.g., Bellman (1970), p. 128, 132, Marcus and Minc (1964), p. 115, or Marshall and Olkin (1979), p. 475, 476.) Note that (4) implies (3) by virtue of the fact that concavity implies logconcavity. To prove Hadamard's Inequality, let

http://www.jstor.org/stable/pdfplus/2975653.pdf

A Convexity Proof of Hadamard's Inequality

A general theory of multivariate new better than used NBU distributions is developed. Using the general theory some multivariate NBU conditions are introduced and studied. Some of the results apply as well to multivariate increasing hazard rate average IHRA and exponential distributions. Proofs of some related well-known implications are simplified.

Multivariate New Better than Used Distributions

If $x$ is a random variable with mean zero and variance $\sigma^2$, then, according to Chebyshev's inequality, $P{|x| \geqq 1} \leqq \sigma^2$. The corresponding one-sided inequality $P{x \geqq 1} \leqq \sigma^2/(\sigma^2 + 1)$ is also known (see e.g. [2, p. 198]). Both inequalities are sharp. A generalization of Chebyshev's inequality was obtained by Olkin and Pratt [1] for $P{|x_1| \geqq 1 or \cdots or |x_k| \geqq 1}$, where $Ex_i = 0, Ex_i^2 = \sigma^2$, $Ex_ix_j = \sigma^2 \rho (i \neq j), i, j = 1, \cdots, k;$ we give here the corresponding generalization of the one-sided inequality, and we consider also the case where only means and variances are known. To obtain an upper bound for $P\{x \varepsilon T\} \equiv P\{x_1 \geqq 1 \text{or} \cdots \text{or} x_k \geqq 1\}$, we consider a nonnegative function, $f(x) \equiv f(x_1, \cdots, x_k)$, such that $f(x) \geqq 1$ for $x \varepsilon T$. Then $Ef(x) \geqq \int_{\{x \varepsilon T\}} f(x) dP \geqq P\{x \varepsilon T\}$. Since the bound is to be a function of the covariance matrix, $\Sigma, f(x)$ must be of the form $(x - a)A(x - a)'$, where $a = (a_1, \cdots, a_k), A = (a_{ij}): k \times k$. A "best" bound is one which minimizes $Ef(x) = \operatorname{tr} A(\Sigma + a'a)$, subject to $f(x) \geqq 0, f(x) \geqq 1$ on $T$.

https://projecteuclid.org/download/pdf_1/euclid.aoms/1177705913

A One-Sided Inequality of the Chebyshev Type

Waiting times for many kinds of events can be represented as first passage times for appropriate stochastic processes. In some applications the most natural stochastic process is multidimensional. In other applications it may be more natural to consider a stochastic process which takes on values in a more general space. In this paper we consider new better than used NBU stochastic processes on partially ordered separable Banach spaces, that is processes which have NBU first passage times to closed upper sets. Various equivalent conditions for a process to be NBU are derived. Some of the results can be used also to characterize increasing hazard rate average IHRA processes. Examples of such processes include processes with shocks and recovery and processes with random repair times. Some applications in reliability theory and in other areas of probability are given.

Albert W. Marshall

Papers

Multivariate geometric distributions generated by a cumulative damage process

A Convexity Proof of Hadamard's Inequality

Multivariate New Better than Used Distributions

A One-Sided Inequality of the Chebyshev Type

NBU Processes with General State Space