Estimation of security betas—that is, coefficients that measure the security's systematic risk—is crucial for application of the Capital Asset Pricing Model. The standard estimation procedure is to use the least-squares regression applied to historical data. This technique consists of fitting a linear relationship between the rates of return on the security and those on the market portfolio.

The regression coefficient estimate, however, does not capture all available information. Suppose the estimated beta of a stock is b = .2. In the absence of any additional information, this estimate is taken by the sampling theory as being the best estimate, because the true beta is equally likely to be overestimated as underestimated by the sample b. This, however, does not imply that given the sample estimate, the true parameter is equally likely to be below or above the value of .2. It is known from previous measurements that betas of all stocks are concentrated around unity, most of them ranging in value between .5 and 1.5. An observed beta of .2 is more likely to be a result of an underestimation than overestimation.

Bayesian decision theory provides a framework for incorporating prior information in estimation of unknown parameters. The paper “A Note on Using Cross-Sectional Information in Bayesian Estimation of Security Betas” (Chapter 32) from 1973 presents a method for Bayesian estimation of the regression coefficients that is optimal with respect to the minimization of the expected squared estimation error.

The bivariate normal distribution function appears often in mathematical finance. It is required in pricing of options whose payout depends on two assets, such as rainbow options, of calls on the maximum of three assets, of extendible options, cross-country swaps, and so on. It is also required for calculation of covariances of derivatives and corporate liabilities.

In some cases, the bivariate normal distribution involves correlations that are close to unity. Suppose an investor wants to calculate the variance of the portfolio value change over a horizon of length H, and suppose the portfolio contains options and derivatives. The variance over an interval of length H of the price of a call option with time to expiration T > H is given by the bivariate normal function with correlation H/T. If the option expires shortly after the end of the horizon period, the correlation can be very high. The same correlation figures in the formula for variance in the change over the horizon H in the market value of a loan maturing at time T.

The standard method of evaluating the bivariate normal distribution function is the tetrachoric series. This series converges only slightly faster than a geometric series with quotient equal to the correlation coefficient. If the correlation is close to unity, the tetrachoric series is not practical. The paper “A Series Expansion for the Bivariate Normal Integral” (Chapter 33) from 1998 gives an alternative series that converges approximately as a geometric series with quotient equal to one minus the correlation squared, which makes it a convenient means of calculation when the correlation is close to one in absolute value.

The article “A Conditional Law of Large Numbers” (Chapter 34), originally written in 1980, is a purely mathematical work. The law of large numbers in probability theory justifies interpreting limiting frequencies as probabilities; the conditional law of large numbers provides a similar foundation for the principle of maximum entropy, an extremely useful proposition in many areas of physics, which had never been formally proven. Informally stated, the theorem asserts that in the equiprobable case, the frequencies conditional on given constraints converge in probability to the distribution that has the maximum entropy subject to these constraints.

A generalization of that result is also given, which relaxes the assumption of all states being equally likely. In the general case, the frequencies conditional on a set of constraints converge in probability to the distribution that maximizes the entropy relative to the underlying distribution.

The paper “A Test for Normality Based on Sample Entropy” (Chapter 35) written in 1976 also deals with the subject of entropy, although in a completely different setting and for a completely different purpose. This paper proposes a statistical goodness-of-fit test to determine whether a given sample came from the normal (Gaussian) distribution. It is based on the fact that the normal distribution has the maximum entropy among all distributions with the same variance. The test statistic is the exponential of a sample estimate of the population entropy, based on higher-order spacings, divided by the sample standard deviation. The test is shown to be a consistent test of the composite hypothesis of normality. The power of the test is estimated against a number of different alternative distributions. It is observed that the power of the test compares favorably to that of several standard tests of goodness-of-fit.

Another paper in probability theory is the joint 1998 work with Julian Keilson, “Monotone Measures of Ergodicity for Markov Chains” (Chapter 36). Finite irreducible Markov chains in continuous time approach ergodicity—that is, they possess a limiting state probability distribution. The speed of approaching the asymptotic distribution is provided by measures of ergodicity. The paper provides a systematic discussion of a certain set of norms, each a measure of ergodicity. Monotonicity of these norms is proven, whether or not the chain is time-reversible. That is a novel and useful result, because up to then monotonicity of these measures had been proven only for time-reversible chains, a small subset of Markov chains. Similar results are noted for Markov chains in discrete time.

The paper “An Inequality for the Variance of Waiting Time under a General Queueing Discipline” (Chapter 37), originally written in 1977, belongs to the field of operations research and, specifically, to the area of queueing systems. Queueing systems are mathematical models of structures into which “customers” arrive at random times to receive some kind of service that takes an uncertain amount of time. Typical examples are telephone exchanges connecting phone calls, or airports accommodating arriving planes. When all servers are busy, arriving customers must wait in line (“queue”) until a server is available. The waiting customers are selected for service by some rule, called the queueing discipline. Of interest are various characteristics of the operation, such as the average waiting time, the number of customers in the queue, the idle periods of the servers, etc.

The chapter proves an interesting inequality about the variance of the waiting time. The queueing discipline can take a variety of forms, such as serving next the customer who came the earliest (“First-come-first-served”), or the most recent arrival (“Last-come-first-served”) as is often the case in warehousing, or selecting a customer from the queue at random, or various priority rules. The expected, or average, waiting time is the same under any queuing discipline, as long as the selection rule does not depend on the serving time of the customers in the queue. The variance, however, is the lowest under the first-come-first-served discipline and the largest under the last-come-first-served discipline. For any other rule of selecting the next customer, the variance is in between these two extremes. It means that the waiting times are the most and the least equitably distributed, respectively, under the two extreme rules.