Asymptotic Theory of Statistics and ProbabilityThis book developed out of my year-long course on asymptotic theory at Purdue University. To some extent, the topics coincide with what I cover in that course. There are already a number of well-known books on asy- totics. This book is quite different. It covers more topics in one source than areavailableinanyothersinglebookonasymptotictheory. Numeroustopics covered in this book are available in the literature in a scattered manner, and they are brought together under one umbrella in this book. Asymptotic theory is a central unifying theme in probability and statistics. My main goal in writing this book is to give its readers a feel for the incredible scope and reach of asymptotics. I have tried to write this book in a way that is accessible and to make the reader appreciate the beauty of theory and the insights that only theory can provide. Essentially every theorem in the book comes with at least one reference, preceding or following the statement of the theorem. In addition, I have p- vided a separate theorem-by-theorem reference as an entry on its own in the front of the book to make it extremely convenient for the reader to ?nd a proof that was not provided in the text. Also particularly worth mentioning is a collection of nearly 300 practically useful inequalities that I have c- lected together from numerous sources. This is appended at the very end of the book. |
Contents
Semester I Classical 1 2 3 4 7 8 11 13 15 17 21 26 27 | 1 |
Key Theorems and References | 16 |
Metrics Information Theory Convergence and Poisson | 19 |
More General Weak and Strong Laws | 35 |
Transformations | 49 |
More General Central Limit Theorems | 63 |
Moment Convergence and Uniform Integrability | 83 |
Sample Percentiles and Order Statistics | 91 |
Bayes Procedures and Posterior Distributions | 289 |
1 Proved in the text | 314 |
Testing Problems | 323 |
Asymptotic Efficiency in Testing | 347 |
Some General LargeDeviation Results | 365 |
Classical Nonparametrics | 377 |
TwoSample Problems | 401 |
Goodness of Fit | 421 |
Sample Extremes | 101 |
Central Limit Theorems for Dependent Sequences | 119 |
1 | 131 |
Accuracy of Central Limit Theorems | 141 |
Invariance Principles | 151 |
7 | 164 |
11 | 177 |
Edgeworth Expansions and Cumulants | 185 |
Saddlepoint Approximations | 203 |
1 | 226 |
Maximum Likelihood Estimates | 235 |
The Trimmed Mean | 271 |
Multivariate Location Parameter and Multivariate Medians | 279 |
Chisquare Tests for Goodness of Fit | 441 |
Goodness of Fit with Estimated Parameters | 451 |
The Bootstrap | 461 |
Jackknife | 499 |
Permutation Tests | 513 |
Density Estimation | 523 |
1 Proved in the text | 550 |
Mixture Models and Nonparametric Deconvolution | 571 |
HighDimensional Inference and False Discovery | 593 |
A Collection of Inequalities in Probability Linear Algebra | 633 |
Glossary of Symbols | 689 |
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Common terms and phrases
assume assumptions asymptotic distribution Asymptotic Theory asymptotically normal Bahadur Behrens-Fisher problem bootstrap bounds Brownian bridge CDF F central limit theorem Chapter chi-square confidence interval Consider convergence DasGupta defined Definition denote density Derive Edgeworth expansion empirical process estimate exact Example Exercise exponential family finite variance Fisher information Fn(x formula function H₁ iid observations independent invariance principle Lehmann Let X1 likelihood limiting distribution martingale Math median multivariate nonparametric normal distribution null order statistics parameter partial sums percentile Poisson posterior Prob problem proof quantile random variables Remark result saddlepoint approximation sample mean sequence Serfling simulation Springer Science+Business Media Stat stationary strong law Suppose X1 symmetric T₁ test statistic U-statistics uniformly X₁ Xn are iid Xn:n zero θη θο σ² Χη