The Average-Value-at-Risk (AV@R) has emerged as a superior, coherent risk measure that accounts for the magnitude of potential losses beyond a given quantile and consistently favors diversification.

This chapter begins the study of random vectors in high dimensions, starting by showing their norm concentrates. We give a probabilistic proof of the Grothendieck inequality and apply it to semidefinite optimization. We explore a semidefinite relaxation for the maximum cut, presenting the Goemans–Williamson randomized approximation algorithm. We also give an alternative proof of the Grothendieck inequality with nearly the best known constant using the kernel trick, a method widely used in machine learning. The exercises explore invariant ensembles of random matrix theory, various versions of the Grothendieck inequality, semidefinite relaxations, and the notion of entropy.

This chapter presents the main ideas behind measuring the risks of random vectors. The main point is that it may be possible to transfer assets between components of a vector, and so the risk measure becomes a convex set in Euclidean space.

Chapter 2 focuses on background information that is essential to understanding SEMs. This includes providing the general structural equation model that appears throughout the book along with definitions of the notation and the assumptions of the model. The chapter introduces path diagram symbols and their relation to the equation form of the model. It also describes differences between endogenous and exogenous variables and observed and latent variables for both continuous and categorical variables. In addition, the chapter introduces the problems of missing data, outliers and influential cases, and multiple significance testing, issues that are common in all types of models. Finally, basic rules of expected values, variances, and covariances are part of the chapter.

This chapter develops a non-asymptotic theory of random matrices. It starts with a quick refresher on linear algebra, including the perturbation theory for matrices and featuring a short proof of the Davis–Kahan inequality. Three key concepts are introduced – nets, covering numbers, and packing numbers – and linked to volume and error-correcting codes. Bounds on the operator norm and singular values of random matrices are established. Three applications are given: community detection in networks, covariance estimation, and spectral clustering. Exercises explore the power method to compute the top singular value, the Schur bound on the operator norm, Hermitian dilation,Walsh matrices, the Wedin theorem on matrix perturbations, a semidefinite relaxation of the cut norm, the volume of high-dimensional balls, and Gaussian mixture models.

Models with multiple equations rather than a single equation are the subject of Chapter 4. It covers model specification, implied moments, model identification, model estimation, and model interpretation, fit, and diagnostics in the context of such models. The consequences of measurement error and the treatment of mediation effects are part of the chapter. Finally, the chapter compares simultaneous equation models and Directed Acyclic Graphs (DAGs).

This chapter introduces fundamental concepts of monetary risk measures and their associated acceptance sets in the context of financial risk assessment.

Law-determined risk measures assign the same risk to identically distributed random variables. On atomless probability spaces, they are characterized by their minimal penalty functions being law-determined.