Review of Statistical Intervals: A Guide for Practitioners and Researchers, 2d Edition, by William Q. Meeker, Gerald J. Hahn, and Luis A. Escobar

Two topics that constitute the core of statistical hydrology are regression and correlation analysis and frequency analysis. Other topics, such as analysis of variance, hypothesis testing, sampling theory, parameter estimation, and network design gravitate around these two main analysis types, which are carried out for a broad range of problems. For example, regression and correlation analysis is carried out for developing an expression for monthly watershed yield as a function of precipitation, evapotranspiration, and other measurable variables; a flood quantile corresponding to a given recurrence interval is expressed in terms of physically measurable watershed characteristics; downscaling of global circulation or climate model (GCM) future projections to a point is done by regression analysis of observation data for variables of interest (predictand), reanalysis data of predictor variables, and GCM output for predictor variables; nonlinear convolution of discharge hydrograph ordinates are expressed as functions of effective precipitation and instantaneous impulse response; and so on. A confidence interval is then determined for the regression curve. Likewise, frequency analysis, which is the basis of the design of hydraulic structures, involves determining a frequency distribution that best fits the data under consideration, estimating the distribution parameters, using a goodness-of-fit test to evaluate the distribution fit, constructing a frequency curve, and constructing a confidence interval for the frequency curve. For reservoir operation, a flow duration curve (FDC) is constructed and a confidence interval is determined before use. When probable maximum precipitation (PMP) is computed, it should be associated with a confidence interval. The usual practice in hydrology for constructing confidence intervals is to assume the normal distribution even though this assumption may not be valid in many cases.

This book, in its second edition, is a timely update of the rather popular first edition. Its subject matter encompasses 18 chapters and 10 appendices. The preface to the second edition followed by that to the first edition and a mention of the companion website set the stage for what to expect from the book and is quite informative. A unique feature of the book is that at the beginning of each chapter objectives and an overview are given, which are very helpful to the reader.

Beginning with a discussion of statistical inference, Chapter 1 provides an overview of different types of intervals and discusses assumptions of sample data and the role of practical assumptions regarding representative data, enumerative versus analytic studies, basic assumptions for inferences from enumerative studies, considerations in conducting analytical studies, convenience and judgment samples, sampling people, infinite population assumptions, practical assumptions, planning a study, the role of statistical distributions, interpretation of statistical intervals, and statistical intervals and data. The chapter scopes out the book and is written in a lucid and easy-to-understand manner.

Chapter 2 provides an overview of different types of statistical intervals, including confidence intervals, prediction intervals, and tolerance intervals, and the characteristics thereof. It provides guidelines for their application. For hydrologic and water resources engineers, who mostly deal with confidence intervals, it is a very informative chapter. Chapters 3–7 deal with the construction of both one-sided and two-sided statistical intervals, including confidence and tolerance intervals as well as prediction intervals, for different distributions, and methods of their computation. Confidence intervals for mean, standard deviation, quantile, and distribution proportion less than a specified value are presented, as are tolerance intervals for distribution proportion and prediction intervals for bounds. Chapter 4 is focused on normal distribution with a discussion of distribution normality and nonnormality, transformation and inferences from transformed data, statistical intervals for linear regression analysis, and comparison of populations. Most statistical hydrologists are familiar with the contents of this chapter. Distribution-free statistical intervals are detailed in Chapter 5, Chapter 6 deals with statistical intervals for a binomial distribution, and Chapter 7 treats statistical intervals for a Poisson distribution. These chapters are well written and highly informative.

Chapter 8 discusses sample size requirements for confidence intervals on distribution parameters. Staring with the basic requirements for sample size determination, the chapter goes on to discuss sample sizes for confidence intervals for the mean, standard deviation, and quantile of a normal distribution, and the sample size for estimating a Poisson occurrence rate. Chapter 9 extends the discussion of sample size requirements to tolerance intervals, tolerance bounds, and demonstration tests. Normal distribution, binomial distribution, and distribution-free cases are presented. Sample size requirements for prediction intervals are dealt with in Chapter 10. Normal distribution and distribution-free cases are included. Chapters 8–10 are well written and contain a wealth of practical and useful information.

Chapter 11 presents basic case studies, including operating temperature of manufactured devices, systems, regulatory limits for chemical pollution, reliability of a circuit board, success of a demonstration test, minimum sample size for a demonstration test, and reliability of devices. Likelihood-based statistical intervals are discussed in Chapter 12. Beginning with an introduction to likelihood-based inference, the chapter discusses maximum likelihood estimation, likelihood confidence intervals, likelihood-based estimation methods for location scale, likelihood-based confidence intervals for parameters, Wald approximation confidence intervals, and other likelihood-based statistical intervals. It is a good chapter. Chapters 13 and 14 are on bootstrap statistical intervals. Chapter 13 treats nonparametric bootstrap statistical intervals, consisting of nonparametric bootstrap sample generation and bootstrap operational considerations; Chapter 14 treats parametric bootstrap and other simulation-based statistical intervals consisting of bootstrap samples and estimates, bootstrap confidence intervals, generalized pivotal quantiles, simulation-based tolerance and prediction intervals, and other bootstrap and simulation methods, as well as application to other distributions. Both are very useful chapters.

Chapter 15 introduces Bayesian statistical intervals entailing Bayesian inference, specification of a prior distribution, Bayesian analyses using Markov chain Monte Carlo (MCMC) simulation, and Bayesian tolerance and prediction intervals. Chapter 16 extends the discussion of the construction of Bayesian intervals for binomial, Poisson, and normal distributions. Chapter 17 deals with statistical intervals for Bayesian hierarchical models, including normal, binomial, and Poisson distributions. Discussed in Chapter 18 are case studies dealing with confidence intervals for proportion of defective integrated circuits, components of variance in a measurement process, treatment effect in a marketing campaign, and probability of detection with missing data; tolerance interval for characterizing the distribution of output; and service life distribution of a rocket motor.

The book is concluded with an epilogue and 10 appendices. Appendix A includes notation and acronyms, Appendix B contains generic definitions of statistical intervals and formulas for computing coverage probabilities, Appendix C includes useful distributions, Appendix D presents general results from statistical theory and methods to construct statistical intervals, Appendix E deals with pivotal methods for constructing parametric statistical intervals, Appendix F treats generalized pivotal quantiles, Appendix G is on distribution-free intervals based on order statistics, Appendix H includes basic results from Bayesian inference models, Appendix I discusses the probability of successful demonstration, and Appendix J contains 21 tables.

This book is indeed a state-of-the art treatise on statistical intervals and is well written. The authors, Drs. Meeker, Hahn, and Escobar, have meticulously integrated rigorous theory, real-world practical applications, and modern information technology. They deserve much applause. Their vast and rich experience with statistical intervals is reflected in the high quality of their book. The book will have a broad appeal to graduate students, university faculty, and researchers, not only from departments of statistics and mathematics but also from engineering, meteorology, agricultural, and environmental as well as health sciences departments. Practitioners from a wide range of private-sector and government organizations will find the book equally useful.