Methodology and Applications of Statistics: A Volume in Honor of C.R. Rao on the Occasion of his 100th Birthday (Contributions to Statistics)

Preface
Dedication
Contents
Contributors
Inference
Robust Statistical Inference for One-Shot Devices Based on Density Power Divergences: An Overview
	1 Introduction
		1.1 Model Description
		1.2 The Weighted Minimum Density Power Divergence
		1.3 Wald-Type Tests
		1.4 Influence Function
	2 Robust Inference for One-Shot Device Testing Under Exponential Distribution
		2.1 Inference Under Exponential Distribution
		2.2 Monte Carlo Simulation Study
		2.3 A Numerical Example: Electric Current Data
	3 Robust Inference for One-Shot Device Testing Under Gamma Distribution
		3.1 Inference Under the Gamma Distribution
		3.2 Monte Carlo Simulation Study
		3.3 A Numerical Example: Application to a Tumor Toxicological Data
	4 Robust Inference for One-Shot Device Testing Under Weibull Distribution
		4.1 Inference Under the Weibull Distribution
		4.2 Monte Carlo Simulation Study
		4.3 A Numerical Example: Glass Capacitors
	5 Robust Inference for One-Shot Device Testing Under Proportional Hazards Model
		5.1 Model Description
		5.2 Inference Under Proportional Hazards Model
		5.3 Monte Carlo Simulation Study
	6 Robust Inference for One-Shot Device Testing Under Exponential Distribution and Competing Risks
		6.1 Model Description
		6.2 Inference Under Exponential Distribution and Competing Risks
		6.3 Monte Carlo Simulation Study
	7 Notes and Comments
	References
Statistical Meaning of Mean Functions: A Novel Matrix Mean Derived from Fisher Information
	1 Introduction
	2 Properties of Fisher Information
	3 Mixtures, Mean Functions and Inequalities
	4 An Information-Based Mean Function
	5 An Inequality for Fisher Information in Sums of Random Variables
	6 General Comments
	References
The Legend of the Equality of OLSE and BLUE: Highlighted by C. R. Rao in 1967
	1 Introduction and Background
	2 What Is OLSE, What Is BLUE?
	3 Year 1967: a Good One for the OLSE = BLUE
	4 OLSE = BLUE: Conditions
	5 The Fundamental BLUE Equation
	6 The Relative Efficiency of OLSE
	7 Personal Glimpses and Conclusions
	References
Comparison of Local Powers of Some Exact Tests for a Common Normal Mean with Unequal Variances
	1 Introduction
	2 Review of Six Exact Tests for H0 Versus H1
		2.1 P-Value Based Exact Tests
		2.2 Exact Test Based on a Modified t
		2.3 Exact Test Based on a Modified F
	3 Expressions of Local Powers of the Six Proposed Tests
		3.1 Local Power of Tippett's Test [LP(T)]
		3.2 Local Power of Wilkinson's Test [LP(Wr)]
		3.3 Local Power of Inverse Normal Test [LP(INN)]
		3.4 Local Power of Fisher's Test [LP(F)]
		3.5 Local Power of a Modified t Test [LP(T1)]
		3.6 Local Power of a Modified F Test [LP(T2)]
	4 Comparison of Local Powers
	5 Conclusion
	References
Quantile Function: Overview of Collaboration with Professor C. R. Rao
	1 Introduction
	2 Bahadur's Representation
	3 Joint Asymptotic Distribution of Marginal Quantiles
		3.1 Tests of Significance for the Vector Medians
	4 Estimation of Density Quantile Function
		4.1 Location Family
	5 Concluding Remarks
	References
Distribution Theory
Some Bivariate and Multivariate Models Involving Independent Gamma Distributed Components
	1 Introduction
	2 Concerning Flexible Copulas
	3 A Bivariate Gamma-Product Model
	4 A Bivariate Gamma-Difference Model
	5 Bivariate Asymmetric Laplace Distributions
		5.1 Bivariate Asymmetric Laplace Distribution of the First Kind (BAL(I))
		5.2 Bivariate Asymmetric Laplace Distribution of the Second Kind (BAL(II))
	6 The Case in Which Addition Is Used Instead of Subtraction, Multiplication, Division, and Minimization
	7 Possible Extensions Involving Dependent Component Variables
	8 On Multivariate Versions of the Models in Sects.3–5
	9 Discussion
	References
An Absolute Continuous Bivariate Inverse Generalized Exponential Distribution: Properties, Inference and Extensions
	1 Introduction
	2 Model Description and Properties
		2.1 Model Description
		2.2 Properties
	3 Inference
	4 Data Analysis
	5 Absolute Continuous Multivariate IGE Distribution
	6 Conclusions and Some Open Problems
	References
Multivariate Analysis
The Likelihood Ratio Test of Equality  of Mean Vectors with a Doubly Exchangeable Covariance Matrix
	1 Introduction
	2 The LRT Statistic
	3 The Distribution of the l.r.t. Statistic
		3.1 The Case of Odd q and Even r
		3.2 The Case of Even q and Odd r
	4 Conclusions and Discussion
	References
Bilinear Regression with Rank Restrictions on the Mean and Dispersion Matrix
	1 Introduction
	2 The Model
	3 Preliminaries
	4 Estimation
		4.1 Estimation Under the Condition mathcalC(A)capmathcalC(Σ)={0}
		4.2 Estimation Under the Condition mathcalC(A)capmathcalC(Σ)={0}
	5 Illustrations
	References
Limiting Canonical Distribution of Two Large-Dimensional Random Vectors
	1 Introduction
	2 A Functional Relationship
	3 Fluctuations for the Eigenvalues of widetildeS1
	4 Limits and Fluctuations for the Eigenvalues of F(Ξ)
	5 Limits and Fluctuations for the Sample Canonical Correlation Coefficients
	6 Technical Proofs
		6.1 Proof of Theorem 1
		6.2 Proof of Theorem 3
		6.3 Proof of Theorem 4
		6.4 Proof of Theorem 5
		6.5 Proof of the Theorem 6
	References
Contributions to Multivariate Analysis Due to C. R. Rao and Associated Developments
	1 Introduction
	2 Rao's U-Statistic in Discriminant Analysis
	3 MANOVA Tests
	4 Asymptotic Expansion and Rao's F Approximation for Λ Statistic
	5 Growth Curve Analysis
	6 Information Criteria for Selection of Variables
	References
On Testing Structures of the Covariance Matrix: A Non-normal Approach
	1 Set-Up of the Problem
	2 Some Notation and Preliminary Results
	3 Ratio of Trace Functions
	4 Test for Uncorrelatedness
		4.1 Null Hypothesis
	5 Conclusions and Discussion
	References
Design and Sampling
The Existence of Perpendicular Multi-arrays
	1 Introduction
	2 Preliminaries
	3 Some Necessary Conditions and Existence Results
	4 Constructions from Combinatorial Designs
	5 Existence of a PMAλ(3times2,v)
	6 Asymptotic Existence on a Cyclic PMA1 (4times2,v)
	7 Conclusions
	References
Statistical Design Issues for fMRI Studies: A Beginner's Training Manual
	1 Introduction
	2 Structurally Balanced and Orthogonal Design Sequences
	3 Concept of ``Clear Zero'' Sequence
		3.1 Type I: D(f+3)=[1  0  0f  1]
		3.2 Type II: D(f+4)=[1  0  0f  0   1] Versus D*(f+4) = [1  0  1f  0   1]
		3.3 Type III: D(2s + f+2)=[1s  0  0f  0   1s] Versus D*(2s + f+2) = [1s  0  1f  0   1s]
		3.4 Type IV: D(f+5)=[1  0  0f  0   1   0] Versus D*(f+5) = [1  0  1f  0   1   0]
		3.5 Type V: D(f+6)=[0   1  0  0f  0   1   0] Versus D*(f+6) = [0   1  0  1f  0   1   0]
		3.6 Type VI: D(2s + f+4)=[0   1s  0  0f  0   1s   0] Versus D*(2s+f+4) = [0   1s  0  1f  0   1s   0]
	4 More on Orthogonal Design Sequences and Implementation of ``Clear Zero''
	5 On the Relative Status of Comparable Design Sequences
	6 Further Comparisons of Design Sequences: Case of K=3
	7 Further Comparisons of Design Sequences: Focus on ``Clear Zero'' Feature
	8 Comparisons of Symmetrized Design Sequences: More on ``Clear Zero'' Feature
	9 Concluding Remarks
	References
A Review of Rigorous Randomized Response Methods for Protecting Respondent's Privacy and Data Confidentiality
	1 Introduction
	2 Basic Elements of RR and PRAM
	3 Privacy Protection by RR
		3.1 Strict Privacy Criteria
		3.2 Privacy Characterization
	4 Comparison of RR Designs
		4.1 Admissible Designs
		4.2 Comparison of RR Designs
	5 Identification Risk Control by Post-randomization
		5.1 Identification Risk Measures
		5.2 A Post-randomization Method
	6 Discussion
	References
Shape and Circular Analysis
A Statistical Analysis of the Cardioid Radial Growth Model
	1 Introduction
	2 Radial Growth Models for Two-Dimensional Objects
	3 Fitting the FEC Growth Model for Landmark Data
		3.1 Landmark Data
		3.2 Statistical Models
	4 Estimation
		4.1 Estimation for the Radial Model
		4.2 Estimation for the Angular Model
		4.3 Overall Estimation
	5 Outline Data
	6 Three-Dimensional Version of the Growth Model
	7 Numerical Considerations
	8 Examples
		8.1 Simulated Data
		8.2 Rat Calvarial Data
	9 Conclusions
	References
A Flexible Family of Mixed Distributions for Discrete Linear and Continuous Circular Random Variables
	1 Introduction
	2 Mixed Bivariate Distributions with Discrete and Circular Marginal Distributions
		2.1 Marginal Density of Θ
		2.2 Marginal Distribution of X
		2.3 Generalizations
		2.4 Conditional Distributions
	3 Moments
		3.1 Moments of Θ
		3.2 Moments of X
		3.3 Joint Moments
	4 Linear-Circular Correlation and Regression
		4.1 Linear-Circular Correlation
		4.2 Linear-Circular Regression
	5 Parameter Estimation
		5.1 Computational Details
	6 Characterizations and Tests for Independence
	7 Application to a Real-Life Data Set
	8 Concluding Remarks
	References
Tests of Fit for Wrapped Stable Distributions Based on the Characteristic Function
	1 Introduction
	2 Tests Based on the Characteristic Function
	3 Tests for the Symmetric WSD
		3.1 The Test Statistic and Consistency for SWS Family
		3.2 Computational Details
		3.3 Estimation of Parameters
	4 Finite-Sample Comparisons with Competing Tests
	5 Real Data Applications
	6 Discussion
	References
Applications
Partial Differential Equation Models  and Riemann–Stieltjes Integrals in Measuring Sustainability
	1 Introduction
	2 Global Sustainability Models
	3 Properties of H(Ω)
	4 Real World Situations: Recycling in Pavement Construction
		4.1 Design of a Flexible Pavement Using Reclaimed Material
		4.2 Variables Influence the Sustainability of Pavements
	5 Conclusions
	References
Extreme Point Methodology in Power Calculations—The Case of Hardy–Weinberg Equilibrium
	1 Introduction
	2 Extreme Point Methodology
		2.1 Theorem
		2.2 Proof
	3 Comments
	4 Computation of Fundamental Probabilities
	5 Comparison of 4 Tests
		5.1 Comments on the Graphs
	6 Conclusion
	References
General
On the Association of Professor  C. R. Rao with the Poznań School of Mathematical Statistics and Biometry
	1 Foreword
	2 Memoirs
	References