This chapter gives a brief overview of Bayesian hypothesis testing. We first describe a standard Bayesian analysis of a single binomial response, going through the prior distribution choice and explaining how the posterior is calculated. We then discuss Bayesian hypothesis testing using the Bayes factor, a measure of how much the posterior odds of believing in one hypothesis changes from the prior odds. We show, using a binomial example, how the Bayes factor may be highly dependent on the prior distribution, even with extremely large sample sizes. We next discuss Bayes hypothesis testing using decision theory, reviewing the intrinsic discrepancy of Bernardo, as well as the loss functions proposed by Freedman. Freedman’s loss functions allow the posterior belief in the null hypothesis to equal the p-value. We next discuss well-calibrated null preferences priors, which applied to parameters from the natural exponential family (binomial, negative binomial, Poisson, normal), also give the posterior belief in the null hypothesis equal to valid one-sided p-values, and give credible intervals equal to valid confidence intervals.

In this chapter, we study risks associated with movements of interest rates in financial markets. We begin with a brief discussion of the term structure of interest rates. We then discuss commonly used interest rate sensitive securities. This is followed by the study of different measures of sensitivity to interest rates, including duration and convexity. We consider mitigating interest rate risk through hedging and immunization. Finally, we take a more in-depth look at the drivers of interest rate term structure dynamics.

The objective of this chapter is to extend the ad hoc least squares method of somewhat arbitrarily selected base functions to a more generic method applicable to a broad range of functions – the Fourier series, which is an expansion of a relatively arbitrary function (with certain smoothness requirement and finite jumps at worst) with a series of sinusoidal functions. An important mathematical reason for using Fourier series is its “completeness” and almost guaranteed convergence. Here “completeness” means that the error goes to zero when the whole Fourier series with infinite base function is used. In other words, the Fourier series formed by the selected sinusoidal functions is sufficient to linearly combine into a function that converges to an arbitrary continuous function. This chapter on Fourier series will lay out a foundation that will lead to Fourier Transform and spectrum analysis. In this sense, this chapter is important as it provides background information and theoretical preparation.

The chapter addresses testing when using models. We review linear models, generalized linear models, and proportional odds models, including issues such as checking model assumptions and separation (e.g., when one covariate completely predicts a binary response). We discuss the Neyman–Scott problem, that is, when bias for a fixed parameter estimate can result when the number of nuisance parameters grows with the sample size. With clustered data, we compare mixed effects models and marginal models, pointing out that for logistic regression and other models the fixed effect estimands are different in the two type of models. We present simulations showing that many models may be interpreted as a multiple testing situation, and adjustments should often be made if testing for many effects in a model. We discuss model selection using methods such as Akaike’s information criterion, the lasso, and cross-validation. We compare different model selection processes and their effect on the Type I error rate for a parameter from the final chosen model.

The objective of this chapter is to present some important relations between the Fourier Transform and correlation functions. It turns out that the cross-correlation function and autocorrelation function have some useful relationships to Fourier Transform and power spectrum of the individual functions. As a result, cospectrum and coherence (normalized statistical correlation in frequency domain) can be defined.

In this chapter, we present the frequency-severity model, which is implicit in common risk calculations used in practice. In this model, the total loss from a risk, or set of risks, is treated as a random sum of random, identically distributed individual losses. If the frequency and severity random variables are independent, then the mean and variance of the aggregate loss can easily be calculated from the moments of the frequency and severity distributions. However, numerical methods are usually required for other metrics, such as quantiles or expected shortfall. We show how to implement these methods and discuss the limitations of this type of model, arising from the independence assumptions.

The analyses we have discussed in previous chapters include the use of base functions, such as sinusoidal functions with specified frequencies, i.e. harmonic analysis; sinusoidal base functions with a frequency range from 0 to the Nyquist frequency with an interval inversely proportional to the total length of time of the data, i.e. Fourier analysis; and wavelet base functions for wavelet analysis. These base functions, however, are chosen regardless of the nature of the variability of the data themselves. In this chapter, we will discuss a different method, in which the base functions are determined empirically, that is dependent on the nature of the data. In other words, this method will find the base functions from the data and these base functions describe the nature of the data. The method is applicable to many types of data, especially to time series data at multiple locations, e.g. a sequence of weather maps or satellite images. There are several variants of the method, but here we will only provide an introduction for the basics.

This chapter introduces the fast Fourier Transform (FFT) for discrete Fourier Transform, beginning with the discretization of the Fourier Transform to its digital expression with constant time intervals. When the integral in Fourier Transform is replaced by a summation, the continuous Fourier Transform is changed to discrete. The discrete Fourier Transform and its inverse are exact relations. An example of the discrete Fourier Transform is discussed for a simple rectangular window function which results in the sinc function, useful for the interpretation of finite sampling effect. A technique of zero-padding is introduced with the discrete Fourier Transform for better visualization of the spectrum. But the computation of discrete Fourier Transform of a long time series can be quite “labor intensive” or costly in computer time with a direct computation. However, since the base functions are periodic, a direct computation can have many duplications in multiplications of terms. Algorithms can be designed to reduce the duplications so that the speed of computation is increased. The reduction of duplicated computations can be repeatedly done through an FFT algorithm. In MATLAB, this is done by a simple command fft. The efficiency of FFT is discussed.

This chapter covers paired data, such as comparing responses before and after a treatment in one group of individuals. The sign test (also called the exact McNemar’s test when responses are binary) is compared to a median test on the differences of responses within pairs, and we show that the sign test is often more appropriate. We give confidence intervals compatible with the sign test. We discuss parameters associated with the Wilcoxon signed-rank test (often more powerful than the sign test) and assumptions needed to give associated confidence intervals. When we can assume symmetric distribution on the differences within pairs, the t-test is another option, and we discuss asymptotic relative efficiency for choosing between the t-test and Wilcoxon signed-rank test. We compare parameterizing the treatment effect as differences or ratios. We discuss tests using Pearson’s and Spearman’s correlation and Kendal’s tau, and present confidence intervals assuming normality. When the paired data represent different assays or raters, then agreement coefficients are needed (e.g., Cohen’s kappa, or Lin’s concordance correlation coefficient).

In this chapter, we consider qualitative and quantitative aspects of risk related to the development, implementation, and uses of quantitative models in enterprise risk management (ERM). First, we discuss the different ways that model risk arises, including defective models, inappropriate applications, and inadequate or inappropriate interpretation of the results. We consider the lifecycle of a model – from development, through regular updating and revision, to the decommissioning stage. We review quantitative approaches to measuring model and parameter uncertainty, based on a Bayesian framework. Finally, we discuss some aspects of model governance, and some potential methods for mitigating model risk.

This chapter defines statistical hypothesis tests mathematically. Those tests assume two sets of probability models, called the null and alternative hypotheses. A decision rule is a function that depends on the data and a specified ?-level and determines whether or not to reject the null hypothesis. We define concepts related to properties of hypothesis tests such as Type I and II error rates, validity, size, power, invariance, and robustness. The definitions are general but are explained with examples such as testing a binomial parameter, or Wilcoxon–Mann–Whitney tests. P-values are defined as the smallest ?-level for observed data for which we would reject the null at that level and all larger levels. Confidence sets and confidence intervals are defined in relation to a series of hypothesis tests with changing null hypotheses. Compatibility between p-value functions and confidence intervals is defined, and an example with Fisher’s exact test shows that compatibility is not always present for some common tests.