Asymptotic statistics

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Preface
This book grew out of courses that I gave at various places, including a graduate course in
the Statistics Department of Texas A&M University, Master's level courses for mathematics
students specializing in statistics at the Vrije Universiteit Amsterdam, a course in the DEA
program (graduate level) of Universite de Paris-sud, and courses in the Dutch AIO-netwerk
(graduate level).
The mathematical level is mixed. Some parts I have used for second year courses for
mathematics students (but they find it tough), other parts I would only recommend for a
graduate program. The text is written both for students who know about the technical
details of measure theory and probability, but little about statistics, and vice versa. This
requires brief explanations of statistical methodology, for instance of what a rank test or
the bootstrap is about, and there are similar excursions to introduce mathematical details.
Familiarity with (higher-dimensional) calculus is necessary in all of the manuscript. Metric
and normed spaces are briefly introduced in Chapter 18, when these concepts become
necessary for Chapters 19,20, 21 and 22, but I do not expect that this would be enough as a
first introduction. For Chapter 25 basic knowledge of Hilbert spaces is extremely helpful,
although the bare essentials are summarized at the beginning. Measure theory is implicitly
assumed in the whole manuscript but can at most places be avoided by skipping proofs, by
ignoring the word "measurable" or with a bit of handwaving. Because we deal mostly with
i.i.d. observations, the simplest limit theorems from probability theory suffice. These are
derived in Chapter 2, but prior exposure is helpful.
Sections, results or proofs that are preceded by asterisks are either of secondary
importance or are out of line with the natural order of the chapters. As the chart in Figure 0.1
shows, many of the chapters are independent from one another, and the book can be used
for several different courses.
A unifying theme is approximation by a limit experiment. The full theory is not developed
(another writing project is on its way), but the material is limited to the "weak topology"
on experiments, which in 90% of the book is exemplified by the case of smooth parameters
of the distribution of i.i.d. observations. For this situation the theory can be developed
by relatively simple, direct arguments. Limit experiments are used to explain efficiency
properties, but also why certain procedures asymptotically take a certain form.
A second major theme is the application of results on abstract empirical processes. These
already have benefits for deriving the usual theorems on M-estimators for Euclidean
parameters but are indispensable if discussing more involved situations, such as M-estimators
with nuisance parameters, chi-square statistics with data-dependent cells, or semiparamet-
ric models. The general theory is summarized in about 30 pages, and it is the applications

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