Preface
Contents
1 Non-Asymptotic Bounds
1.1 Errors and Non-Asymptotic Bounds
References
2 Scale-Mixed Distributions
2.1 Introduction
2.2 Error Bounds in Sup -Norm
2.3 Special Cases
2.3.1 Scale-Mixed Normal
2.3.2 Scale-Mixed Gamma
2.3.3 Scale-Mixed F
2.4 Error Bounds Evaluated in L1-Norm
References
3 MANOVA Test Statistics
3.1 Introduction
3.2 Multivariate Scale Mixtures for TLH and TLR
3.3 Error Bounds for Approximations of TLH and TLR
3.4 Error Bound for TBNP
3.5 Error Bounds for High-Dimensional Approximations
References
4 Linear and Quadratic Discriminant Functions
4.1 Introduction
4.2 Location and Scale Mixture Expression for EPMC
4.3 General Approximation and Error Bounds
4.4 Error Bounds for EPMC
4.5 Some Related Topics
References
5 CornishโFisher Expansions
5.1 Introduction
5.2 CornishโFisher Expansion
5.3 Error Bounds for CornishโFisher Expansion
5.4 Proofs for Error Bounds
5.5 Transformations for Improved Approximations
5.6 Examples
References
6 Likelihood Ratio Tests with Box-Type Moments
6.1 Introduction
6.2 Large-Sample Asymptotic Expansions
6.3 High-Dimensional Asymptotic Expansions
6.4 Error Bound
References
7 Bootstrap Confidence Sets
7.1 Introduction
7.2 Bootstrap Procedure
7.3 Confidence Sets for Spectral Projectors: Bootstrap Validity
References
8 Gaussian Comparison and Anti-concentration
8.1 Introduction
8.2 Motivation: Prior Impact in Linear Gaussian Modeling
8.3 Gaussian Comparison
8.4 Anti-concentration Inequality
8.5 Proofs
8.5.1 Proof of Theorem8.1
8.5.2 Proof of Lemma8.1
References
9 Approximations for Statistics Based on Random Sample Sizes
9.1 Introduction
9.2 Notation and Examples
9.3 Two Transfer Propositions
9.4 Edgeworth and CornishโFisher Expansions with Student's Limit Distribution
9.5 Edgeworth and CornishโFisher Expansions with Laplace Limit Distribution
References
10 Power-Divergence Statistics
10.1 Introduction
10.2 Rates of Convergence
10.3 Refinements of Convergence Rates
10.4 Approximating the Number of Integer Points in Convex Sets
References
11 General Approach to Constructing Non-Asymptotic Bounds
11.1 Introduction
11.1.1 Notation
11.2 Results for Symmetric Functions
11.2.1 Limit Theorem with Bound for Remainder Term
11.2.2 Ideas of the Proof of Theorem 11.1
11.2.3 Asymptotic Expansions
11.3 Applications in Probability and Statistics
11.3.1 Expansion in the Central Limit Theorem for Weighted Sums
11.3.2 Expansion in the Free Central Limit Theorem
11.3.3 Expansion of Quadratic von Mises Statistics
11.3.4 Expansions for Weighted One-Sided KolmogorovโSmirnov Statistics
References