Testing Statistical Hypotheses of Equivalence

The next topic covered (Chap. 7) is discrete-state, continuous-time Markov processes. The presentation of this topic begins with an introduction to the generator matrix of the general discrete-state, continuous-time Markov process and derivation of the forward and backward Kolmogorov difference differential equations. Following this, speci c discrete-state, continuous-time Markov processes (the Poisson process and its natural extensions—the pure birth, the pure death, and the birth and death processes) are discussed. One attractive feature of this book’s presentation of discrete-state, continuous-time Markov processes is that it systematically derives the transition probabilities of a Markov process from its generator matrix, which in turn is derived from the de nition of that process. This leads to a uni ed approach to analyzing and  nding the properties of these processes. Returning to the general discrete-state, continuous-time Markov process, conditions for the existence of limits of the transition probabilities as time t ! 1 as well as the conditions that these limits must satisfy are given without proof. Coverage of discrete-state, continuoustime Markov processes ends with an introduction to Markovian networks (i.e., networks whose state distributions exhibit the Markov property, a network being a collection of nodes connected by a set of paths). As in the case of Markov chains, the discussion of discrete-state, continuoustime Markov processes next turns to statistical inference and applications (Chaps. 8 and 9). The material is predominantly on Poisson processes and birth and death processes. A point of interest is the inclusion of a clever test of the hypothesis that a process is Poisson (based on the uniform distribution). Some of the  elds covered in applications include queueing theory, queueing networks, inventory systems, reliability models, Markovian combat models, and stochastic models for social networks. As can be seen, the range of  elds covered is wide. With its coverage of Markov processes completed, the book turns to discreteand continuous-time renewal processes in Chapter 10. In discussing discretetime renewal processes, the concepts of recurrence and transience, encountered in Markov chains, are extended to the event of interest (the renewal event) in a discrete-time renewal process. The discrete renewal function is also introduced, and its limiting behavior is stated without proof. The treatment of continuous renewal processes is similar; that is, the renewal density function is introduced, and its limiting behavior studied among other things. Generalizations of the renewal process (the alternating renewal process, with two renewal periods and the process alternating between them in an independent manner, and the Markov renewal process, in which the process passes through different states as determined by the Markovian transitions of a transition probability matrix) are given. The discussion of renewal processes closes by touching on the estimation of the interevent distribution itself or the parameters that determine it. Chapter 11 covers stationary processes and time series analysis. After  rst de ning strictand wide-sense stationary processes and providing examples of these, the authors state ergodic results that give conditions under which the sequence sample means converge without proof. The authors then introduce, with examples to illustrate, the following: representation of a wide-sense stationary process as a sum of periodic functions with random coef cients, spectral representation of the covariance function, and transformations of wide-sense stationary processes and their spectral densities by linear  lters. Next, they introduce the standard framework for time series analysis (linear  lters of sequences of independent normal random variables with mean 0 and variance 3⁄4 2) and standard time series models (i.e., autoregressive, moving average, autoregressive moving average, and autoregressive integrated moving average). The treatment of time series analysis closes with a brief description of the Box– Jenkins approach (time series analysis in the time domain) and an introduction to the problem of estimating the spectral density (spectral analysis of time series data). The book ends with a brief introduction to simulation and Monte Carlo techniques in Chapter 12. Standard methods for simulating random variables (e.g., using the inverse cumulative distribution function) and the simulation of stochastic processes (viewed as being accomplished by simulating a sequence of random variables) are described. The introduction to Monte Carlo methods includes Markov chain Monte Carlo (MCMC). In the MCMC method, rather than require the observations to be independent, the observations are allowed to come from a Markov chain. One of the authors’ stated goals is to give high priority to the concepts of stochastic processes. This book achieves this goal. Further, besides conveying the concepts of stochastic processes, it succeeds in providing insight into the reasons why for a particular topic certain lines of investigation are pursued and why certain variables/functions are introduced. Examples showing how reallife problems translate into mathematical questions about stochastic processes help accomplish this. Another attractive feature is the inclusion of topics usually discussed in a more advanced setting. Putting these topics in a setting that does not have extensive requirements makes the concepts and motivations for these topics available to a wider audience. The pluses mentioned earlier, however, are balanced by some minuses. First, there are a number of typos/errors and omissions of important qualifying words. For example, there is the statement “the sum of Poisson processes is a Poisson process. ” This statement as it stands is not true. The phrase “Poisson processes” needs to be quali ed by the word “independent. ” Although the corrections usually can be  gured out, removing these typos/omissions will make the book even easier to read. As a closing comment, here is a suggestion. The inclusion of statistical inference for stochastic processes ties this book closer to solving real-life problems. However, to complete the treatment of model building using stochastic models, material on assessing the models built should be included.