Asymptotic Tools

Spring 2020

Though we may do things differently in spring 2020, a previous version of the course (FA 2011) covered all sections except:

- 4.4: Univariate extensions of the Central Limit Theorem
- 8.3: Asymptotics of the Wilcoxon rank-sum test
- 10.3: Multivariate and multi-sample U-statistics

Many exercises require students to do some computing, based on the
notion that computing skills should be emphasized in
*all* statistics courses whenever possible, provided that the
computing enhances the understanding of the subject matter.
The study of large-sample
theory lends itself very well to computing, since frequently the
theoretical large-sample results we prove do not give any
indication of how well asymptotic approximations work for finite
samples. Thus, simulation for the purpose of checking the
quality of asymptotic approximations for small samples is very
important in understanding the limitations of the results being
learned.
Of course, all computing activities will force students to choose
a particular computing environment. Occasionally, hints are
offered in the notes using R
(http://www.r-project.org), though
these exercises can be completed using other packages or
languages, provided that they possess the necessary statistical
and graphical capabilities.

Credit where credit is due:
These notes originally evolved as an accompaniment to the
book *Elements of Large-Sample Theory* by the late
Erich Lehmann; the strong influence of that great book,
which shares the philosophy of these notes regarding
the mathematical level at which an introductory
large-sample theory course
should be taught, is still very much evident here.
I am fortunate to have had the chance to correspond with
Professor Lehmann several times about his book, as my
students and I provided lists of
typographical
errors that we
had spotted. He was extremely gracious and I treasure the letters that
he sent me, written out longhand and sent through the mail
even though we were already well
into the era of electronic communication.

I have also drawn on many other
sources for ideas or for exercises. Among these are
the fantastic and concise *A Course in Large Sample Theory*
by Thomas Ferguson,
the comprehensive and beautifully written
*Asymptotic Statistics* by A. W. van der Vaart,
and the classic probability textbooks *Probability and Measure* by
Patrick Billingsley and *An Introduction to Probability Theory and
Its Applications, Volumes 1 and 2* by William Feller. Arkady Tempelman
at Penn State helped with some of the Strong-Law material in
Chapter 3, and it was Tom Hettmansperger who originally
convinced me to design this course at Penn State back in 2000 when I was a new
assistant professor. My goal in doing so was to teach a course that
I wished I had had as a graduate student, and I hope that these notes
help to achieve that goal.