Chapter 13: In this chapter, we discuss several problems in which a similarity transformation to Jordan canonical form facilitates a solution. For example, we find that A is similar to AT; limp→∞Ap = 0 if and only if every eigenvalue of A has modulus less than 1; and the invertible Jordan blocks of AB and BA are the same. We begin by considering coupled systems of ordinary differential equations.

Claim severity refers to the monetary loss of an insurance claim. Unlike claim frequency, which is a nonnegative integer-valued random variable, claim severity is usually modeled as a nonnegative continuous random variable. Depending on the definition of loss, however, it may also be modeled as a mixed distribution, i.e., a random variable consisting of probability masses at some points and continuous otherwise.

Chapter 3: A matrix is not just an array of scalars. It can be thought of as an array of submatrices in many different ways. We begin by regarding a matrix as an array of columns and we explore some implications of this viewpoint for matrix products and Cramer's rule. We turn to arrays of rows, which lead to additional insights for matrix products. We discuss determinants of block matrices, block versions of elementary matrices, and Cauchy's formula for the determinant of a bordered matrix. Finally, we introduce the Kronecker product, which provides a way to construct block matrices with a special structure.

Social inquiry needs representations of social reality, and such representations involve ontological choices of how much to freeze history and geography. The freezing metaphor relates to how much social inquiry relaxes conditional independence and unit homogeneity, the two key assumptions of VBA. Unfreezing history and geography brings to light historical and geographic particularities, makes possible new inductive insights into updating theories, and generates new research questions. Comparisons help turn such explorations into discoveries by identifying patterns. CHA employs four comparisons: cross-sectional, serial, contextual, and historical. It thus differs from VBA, which uses comparison for the strictly methodological purpose of controlling for confounders. CHA goes back to the late nineteenth century and has explored four key themes: the evolution of capitalism, regime changes, the administrative transformation of states, and the changing nature of war. CHA does not have a monopoly on historical thinking. VBA employs historical thinking when exploring confounders and historical boundary conditions that might bias their test results.

Physical time refers to five context-independent elements of time: tempo, duration, timing, sequencing, and stages. It makes two contributions. It refines the analysis of historical time by focusing on the rhytms in which it unfolds. It thus supplements the analysis of what changes in content by paying attention to how it changes in rhythm. It also refines causal analysis by treating the elements of physical time that produce distinct causal effects; these are frequently overlooked by linear notions of causality. CHA employs both physical time and historical time, and this distinguishes it from other methodologies. It configures these elements of time in distinct ways, which define the three strands of CHA: eventful, longue durée, and macro-causal analysis.

In this chapter we consider the Bayesian approach in updating the prediction for future losses. We consider the derivation of the posterior distribution of the risk parameters based on the prior distribution of the risk parameters and the likelihood function of the data. The Bayesian estimate of the risk parameter under the squared-error loss function is the mean of the posterior distribution. Likewise, the Bayesian estimate of the mean of the random loss is the posterior mean of the loss conditional on the data.

This book is about modeling the claim losses of insurance policies. Our main interest is nonlife insurance policies covering a fixed period of time, such as vehicle insurance, workers compensation insurance and health insurance. An important measure of claim losses is the claim frequency, which is the number of claims in a block of insurance policies over a period of time. Though claim frequency does not directly show the monetary losses of insurance claims, it is an important variable in modeling the losses.

We consider models for analyzing the surplus of an insurance portfolio. Suppose an insurance business begins with a start-up capital, called the initial surplus. The insurance company receives premium payments and pays claim losses. The premium payments are assumed to be coming in at a constant rate. When there are claims, losses are paid out to policyholders. Unlike the constant premium payments, losses are random and uncertain, in both timing and amount. The net surplus through time is the excess of the initial capital and aggregate premiums received over the losses paid out. The insurance business is in ruin if the surplus falls to or below zero. The main purpose of this chapter is to consider the probability of ruin as a function of time, the initial surplus and the claim distribution. Ultimate ruin refers to the situation where in ruin occurs at finite time, irrespective of the time of occurrence.

Chapter 12: In the preceding chapter, we found that each square complex matrix A is similar to a direct sum of upper triangular unispectral matrices. We now show that A is similar to a direct sum of Jordan blocks (unispectral upper bidiagonal matrices with 1s in the superdiagonal) that is unique up to permutation of its direct summands.

This chapter offers a primer of the literature on the origins of proportional representation. Subsequent chapters use this literature to illustrate the elements of CHA. It therefore provides the background information to make those illustration easier tofollow.

Short-term insurance policies often take multiple years before the final settlement of all losses incurred is completed. This is especially true for liability insurance policies, which may drag on for a long time due to legal proceedings. To set the premiums of insurance policies appropriately so that the premiums are competitive and yet sufficient to cover the losses and expenses with a reasonable profit margin, accurate projection of losses cannot be overemphasized. Loss reserving refers to the techniques to project future payments of insurance losses based on policies in the past.

We have discussed the limited-fluctuation credibility method, the Bühlmann and Bühlmann–Straub credibility methods, as well as the Bayesian method for future loss prediction. The implementation of these methods requires the knowledge or assumptions of some unknown parameters of the model. For the limited-fluctuation credibility method, Poisson distribution is usually assumed for claim frequency.