The course is planned to be given during the second half of the Spring Term
2019 at the Department of Statistics, Stockholm University. (A more detailed
teaching plan will be posted later.)
Credits: 7.5

### Prerequisites

What is not needed is knowledge of measure theory. Basic stochastic convergence
concepts and results (with proofs) such as the Lindeberg-Levy central
limit theorem should have been covered in previous courses.

### Aim

To provide some theory concerning asymptotic methods of statistics and
probability with applications to inference problems.

### Contents

Asymptotic theory of statistics and probability is a huge topic, therefore we
will concentrate upon a few situations of interest, while also providing some
general results. A detail of particular interest is the rate of convergence (e.g.
in distribution) which is reected in the remainder term of an asymptotic
expansion. The order of this remainder term is then of importance to specify
and the notation of Op and op will be dealt with separately. This knowledge
enables us to among other things manage a rigorous proof of the Delta Theorem.
We will start with the classical problem of nding a transformation
of a statistic, leading to better behaviour for further analysis such as interval
estimation or hypothesis testing. Variance stabilizing and symmetrizing
methods will be demonstrated and on top of this there is also room for bias
corrections. This is presented in Chapter 4 of DasGupta (2008), a book
which will be the main source of information for the course. This is more of
an encyclopedia than an ordinary textbook with short and concise chapters
for each topic, but it also includes exercises. In order to fully appreciate the
contents in Chapter 4, bits and pieces from the previous chapters need to be
picked up. Then we can move forward to Edgeworth expansions (Chapter
13), which is an important tool to evaluate statistics with respect to closeness
to the normal distribution. (This is also related to the famed Berry-Esseen
inequality (bound) in Chapter 11.) For e.g. tail areas another technique is
preferable, namely saddlepoint approximation (Chapter 14). Furthermore,
some general asymptotic theory with results is covered in chapter 5 of Bickel
and Doksum (2001). In addition to the books by DasGupta and Bickel &
Doksum there are of course numerous articles and books related to these
issues.

To sign up for the course send an e-mail to per.gosta.andersson@stat.su.se Syllabus. Some asymptotic methods in statistical inference (158 Kb)