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**Confidence intervals on variance components.**
*(English)*
Zbl 0755.62055

Statistics: Textbooks and Monographs. 127. New York: Marcel Dekker. iv, 211 p. (1992).

The monograph under review presents methods for constructing confidence intervals on individual variance components, linear combinations of variance components, and ratios of variance components for both random and mixed, balanced and unbalanced models based on standard normal assumptions. A feature of the intervals presented in this book is that all required computations can be easily performed with hand-held calculators.

The monograph is divided into 7 chapters. Chapter 1 is an introduction. Chapter 2 presents general results and terminology used in the book. Chapter 3 gives methods for constructing confidence intervals for several functions of variance components in balanced random models. The following two chapters (Ch. 4 and 5) provide confidence intervals for one-fold, two-fold and \((Q-1)\)-fold balanced nested designs. The models considered in the last two chapters include crossed random models and mixed models.

The monograph contains a list of many valuable recent references which will be of great help to research workers being primarily interested in the theoretical development of these methods. All main formulas for the confidence intervals are summarized in tables which will be very convenient for practitioners using these methods in their fields of applications. The book can as well be used by advanced students of statistics as a reference work.

The contents of the monograph are as follows. Chapter 1, Introduction: background, scope of the book. Chapter 2, General concepts: introduction, variance components models, point estimation, confidence intervals, tests of hypotheses, simultaneous confidence intervals, notation and use of tables, confidence intervals on the variance of a normal population, confidence intervals on the ratio of variances from two normal populations, summary.

Chapter 3, General results for balanced designs: introduction, intervals on sums of expected mean squares, confidence intervals on linear combinations of expected mean squares with different signs, confidence intervals on ratios of expected mean squares, simultaneous confidence intervals, summary. Chapter 4, The one-fold nested design: introduction, the balanced one-fold random model, the unbalanced one-fold model, summary. Chapter 5, The two-fold and \((Q-1)\)-fold nested designs: introduction, the balanced two-fold nested random model, balanced \((Q- 1)\)-fold nested random models, the unbalanced two-fold nested random model, unbalanced \((Q-1)\)-fold nested designs, summary.

Chapter 6, Crossed random designs: introduction, the balanced two-factor crossed random model with interaction, the balanced two-factor crossed random model without interaction, the balanced three-factors crossed random models with interaction, unbalanced two-factor crossed random models with interaction, unbalanced two-factor crossed random models without interaction, balanced incomplete block random models, summary.

Chapter 7, Mixed models: introduction, balanced two-factor crossed mixed models, a balanced three-factor crossed mixed model with interaction, balanced nested mixed models, balanced mixed models with both crossed and nested effects: a split-plot design, regression models with nested error structure, unbalanced mixed models, summary. Appendix A, \(F\)-tables; Appendix B, Wald’s confidence intervals on a ratio of two variance components.

The monograph is divided into 7 chapters. Chapter 1 is an introduction. Chapter 2 presents general results and terminology used in the book. Chapter 3 gives methods for constructing confidence intervals for several functions of variance components in balanced random models. The following two chapters (Ch. 4 and 5) provide confidence intervals for one-fold, two-fold and \((Q-1)\)-fold balanced nested designs. The models considered in the last two chapters include crossed random models and mixed models.

The monograph contains a list of many valuable recent references which will be of great help to research workers being primarily interested in the theoretical development of these methods. All main formulas for the confidence intervals are summarized in tables which will be very convenient for practitioners using these methods in their fields of applications. The book can as well be used by advanced students of statistics as a reference work.

The contents of the monograph are as follows. Chapter 1, Introduction: background, scope of the book. Chapter 2, General concepts: introduction, variance components models, point estimation, confidence intervals, tests of hypotheses, simultaneous confidence intervals, notation and use of tables, confidence intervals on the variance of a normal population, confidence intervals on the ratio of variances from two normal populations, summary.

Chapter 3, General results for balanced designs: introduction, intervals on sums of expected mean squares, confidence intervals on linear combinations of expected mean squares with different signs, confidence intervals on ratios of expected mean squares, simultaneous confidence intervals, summary. Chapter 4, The one-fold nested design: introduction, the balanced one-fold random model, the unbalanced one-fold model, summary. Chapter 5, The two-fold and \((Q-1)\)-fold nested designs: introduction, the balanced two-fold nested random model, balanced \((Q- 1)\)-fold nested random models, the unbalanced two-fold nested random model, unbalanced \((Q-1)\)-fold nested designs, summary.

Chapter 6, Crossed random designs: introduction, the balanced two-factor crossed random model with interaction, the balanced two-factor crossed random model without interaction, the balanced three-factors crossed random models with interaction, unbalanced two-factor crossed random models with interaction, unbalanced two-factor crossed random models without interaction, balanced incomplete block random models, summary.

Chapter 7, Mixed models: introduction, balanced two-factor crossed mixed models, a balanced three-factor crossed mixed model with interaction, balanced nested mixed models, balanced mixed models with both crossed and nested effects: a split-plot design, regression models with nested error structure, unbalanced mixed models, summary. Appendix A, \(F\)-tables; Appendix B, Wald’s confidence intervals on a ratio of two variance components.

Reviewer: Wang Songgui (Hefei)

### MSC:

62J10 | Analysis of variance and covariance (ANOVA) |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |

62F25 | Parametric tolerance and confidence regions |

62J99 | Linear inference, regression |