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Normal Approximation and Asymptotic Expansions

Normal Approximation and Asymptotic Expansions

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  • ISBN:9780898718973
  • Editorial: Siam
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  • Dimensiones: 0 cm x 0 cm
  • Nº Pág.:332
  • Idiomas:Ingles

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Although Normal Approximation and Asymptotic Expansions was first published in 1976, it has gained new significance and renewed interest among statisticians due to the developments of modern statistical techniques such as the bootstrap, the efficacy of which can be ascertained by asymptotic expansions. This is also the only book containing a detailed treatment of various refinements of the multivariate central limit theorem (CLT), including Berry-Essen-type error bounds for probabilities of general classes of functions and sets, and asymptotic expansions for both lattice and non-lattice distributions. With meticulous care, the authors develop the necessary background on: weak convergence theory, Fourier analysis, geometry of convex sets and the relationship between lattice random vectors and discrete subgroups of Rk.

Preface To The Classics Edition
Preface
List Of Symbols
Chapter 1 Weak Convergence Of Probability Measures And Uniformity Classes 1
1 Weak Convergence, 2
2 Uniformity Classes, 6
3 Inequalities for Integrals over Convex Shells, 23
Notes, 38
Chapter 2 Fourier Transforms And Expansions Of Characteristic Functions 39
4 The Fourier Transform, 39
5 The Fourier Stieltjes Transform, 42
6 Moments, Cumulants, and Normal Distribution, 44
7 The Polynomials P5 and the Signed Measures Ps, 51
8 Approximation of Characteristic Functions of Normalized Sums of Independent Random Vectors, 57
9 Asymptotic Expansions of Derivatives of Characteristic Functions, 68
10 A Class of Kernels, 83
Notes, 88
Chapter 3 Bounds For Errors Of Normal Approximation 90
11 Smoothing Inequalities, 92
12 Berry Esseen Theorem, 99
13 Rates of Convergence Assuming Finite Fourth Moments, 110
14 Truncation, 120
15 Main Theorems, 143
16 Normalization, 160
17 Some Applications, 164
18 Rates of Convergence under Finiteness of Second Moments, 180
Notes, 185

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