# Developments in Statistics, Volume 2

## 1st Edition

**Editors:**Paruchuri R. Krishnaiah

**eBook ISBN:**9781483264196

**Imprint:**Academic Press

**Published Date:**28th March 1979

**Page Count:**336

**View all volumes in this series:**Developments in Statistics

## Table of Contents

List of Contributors

Preface

Contents of Volume 1

Chapter 1 Random Vibration of One- and Two-Dimensional Structures

I. Introduction

A. Description of the Problem

B. Statistical Terminology

II. One-Dimensional Structures

A. Formal Analysis for Stationary Response

B. Approximate Solutions Based on Modal Sums

C. Approximate Solutions Based on Image Sums

III. Two-Dimensional Structures

A. Formal Analysis for Stationary Response

B. Approximate Solutions Based on Modal Sums

C. Approximate Solutions Based on Image Sums

D. Experimental Techniques

References

Chapter 2 The Statistical Theory of Linear Systems

1. Introduction

2. The Structure of Linear Systems with Rational Transfer Functions

3. The Asymptotic Properties of ML Estimates

4. The Construction of Estimates of the Parameters

5. Recursive Methods

6. Some Further Problems

References

Chapter 3 Bispectra and Energy Transfer in Grid-Generated Turbulence

1. Introduction

2. Bispectra and Energy Transfer

3. Symmetry Conditions and Isotropy

4. Experimental Arrangements

5. Analysis of Hot-Wire Calibration

6. Computational Method

7. Computational Symmetries

8. Bispectra and One-Dimensional Energy Transfer

9. Experimental Results

9.1. Moments and Second-Order Spectra

9.2. Bispectra

9.3. One-Dimensional Energy Transfer Spectra

10. Concluding Remarks

References

Chapter 4 Some Developments on Simultaneous Test Procedures

1. Introduction

2. Multivariate t, Multivariate F, and Multivariate Gamma Distributions

3. Distributions of Correlated Quadratic Forms and Related Distributions

4. Finite Intersection Tests for Multiple Comparisons of Means

5. Finite Intersection Tests and Step-Down Procedure for Multiple Comparisons of Mean Vectors

6. Largest Root Test and T2max Test

7. Simultaneous Tests under Regression Models

8. Simultaneous Tests under Polynomial Growth Curve Models When Errors Are Autocorrelated

9. Multiple Comparisons of Means of Correlated Normal Populations

10. Multiple Comparisons of Variances of Correlated Normal Populations

11. Simultaneous Tests under Growth Curve Models

12. Simultaneous Tests for the Equality of the Covariance Matrices of Multivariate Normal Populations

13. Simultaneous Tests When Covariance Matrices Have Special Structures

14. Tests for the Structure of Interaction under Two-Way Classification Model

15. Selection of Populations

References

Chapter 5 Stochastic Markovian Fields

1. Splitting Subspaces

2. Markovian Sets

3. Dual Stochastic Fields and Markov Property

4. Linear Stochastic Equations

5. Stationary Fields

References

Chapter 6 Stopping Time of Invariant Sequential Probability Ratio Tests

1. Introduction

1.1. Purpose of This Chapter

1.2. Simple Hypotheses

1.3. Composite Hypotheses

1.4. Historical Outline

1.5. Outline of the Contents of This Chapter

2. Exponential Boundedness and Exponential Convergence

2.1. Definitions and Properties

2.2. Stein’s Lemma and Its Generalization

3. Probability Ratios of Maximal Invariants

3.1. The Integral Ratio Method

3.2. Approximations: Laplace’s Method

3.3. Examples of the Laplace Method

4. Derivation of the log Probability Ratio of Several Invariant SPRTs

4.1. Sequential Test of the Standard Deviation in a Normal Population

4.2. Sequential Test of the Covariance Matrix in a Multivariate Normal Population

4.3. Sequential t-Test

4.4. Two-Sided Sequential t-Test

4.5. Sequential F-Test for the General Linear Hypothesis

4.6. Sequential T2-Test for the Mean of a Multivariate Normal Population

4.7. Tests for the Characteristic Roots of a 2 x 2 Covariance Matrix, and for the Mean Vector When the Covariance Matrix Is Known (Sequential x2-Test)

4.8. Sequential Tests for Ordinary and Multiple Correlation Coefficients

4.9. Sequential Two-Sample Wilcoxon Test

4.10. Sequential Rank-Order Test Based on Lehmann Alternatives

5. General Theorems on Exponentially Bounded Stopping Time

6. Exponential Boundedness of N in the Tests of Section 4

6.1. Standard Deviation in a Normal Population (Test of Section 4.1)

6.2. Covariance Matrix in a Multivariate Normal Population (Test of Section 4.2)

6.3. Sequential t-Test

6.4. Two-Sided Sequential t-Test

6.5. Sequential F-Test

6.6. Sequential T2-Test

6.7. Tests of Section 4.7 (Characteristic Roots and Mean Vector)

6.8. Sequential Correlation Coefficient Tests

6.9. Sequential Two-Sample Wilcoxon Test

6.10. Sequential Rank-Order Test Based on Lehmann Alternatives

7. Proof of Theorem 5.3

8. Obstructive Distributions and Termination with Probability One

9. Summary of Results

9.1. Complete Results

9.2. Incomplete Results

10. Suggestions for Further Study

References

Author Index

Subject Index

## Description

Development in Statistics, Volume 2 is a collection of papers that deals with one- and two- dimensional structures, the statistical theory of linear systems, bispectra, and energy transfer in grid-generated turbulence. Several papers discuss simultaneous test procedures, stochastic Markovian fields, as well as the stopping of invariant sequential probability ratio tests. One paper examines the relationships between excitation and response statistics for one-dimensional structures, and then as extended to two-dimensional structures. The special features issuing from these extensions are related to simple supported rectangular and square plates excited by a stationary random force applied at a single point. Another paper discuses the relationship between the measurable bispectra and the one-dimensional energy transfer terms, and which bispectra will vanish in an isotropic turbulent flow field. One paper reviews simultaneous test procedures, including the evaluation of the probability integrals of multivariates, multivariate gamma distributions, distributions of correlated quadratic forms. Another paper analyzes two concerns regarding the random sample size N, also known as stopping time. These are if N is finite with a probability of one, or the rate that the tail probabilities in the distribution of N go to zero. Mathematicians, statisticians, students, and professors of calculus or advanced mathematics will surely appreciate the collection.

## Details

- No. of pages:
- 336

- Language:
- English

- Copyright:
- © Academic Press 1979

- Published:
- 28th March 1979

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483264196