Statistics 610a

### Statistical Inference Statistics 610a Fall 1995

Instructor: David Pollard

### Aim of the course

I hope that students who complete the course will be able to read some of the current statistical or econometrics literature, or at least understand the the standard theory behind those literatures. The course covers a range of topics similar to Statistics 242 (with the exception of the theory of linear models, which is covered in Statistics 612a), but the treatment will be more rigorous. In particular, the course will put more emphasis on the decision-theoretic interpretation of statistical procedures.

An understanding of statistical theory at the level of this course is necessary background for many of the more advanced courses offered by the Statistics Department.

### Text

In the past I have worked from a number of books, none of which I have found completely satifactory. On the suggestion of a student in the 1993 class, I have decided to follow the treatment of {Ferguson67book}, even though the book is quite old. I hope to cover at least the first five chapters of the book, but I will skip some topics that have become less important over the last thirty years.

As Ferguson has explained in his preface, a number of topics do not fit comfortably into a decision-theoretic interpretation. Nevertheless, some of those topics are vital for an understanding of modern statistics. I will supplement Ferguson's treatment with material on large-sample theory (in particular, maximum likelihood estimation, and likelihood ratio theory).

Students should not be intimidated by the references to Lebesgue integration in the early parts of the Ferguson book. I will take pains to keep the treatment accessible to anyone who understands the probability theory treated in Statistics 241a.

### Other references

Some of the supplemental material (such as the large-sample theory) will be drawn from {Lehmann59TSH}, {Lehmann83TPE}, and {IbragimovHasminskii81book}. All three books are written at a slighly more demanding theoretical level than the Ferguson book. {Hartigan83book} is a good source for elegant proofs, but it it written at a level far too advanced for this course.

I will also draw ideas from other sources, including some of my own manuscripts. For exponential families I like {Brown86book}. A few examples and problems will come from the primary literature. I will borrow examples from {BickelDoksum77book}, which is the standard texts at many universities.

For a quick overview, students might consult {Silvey75book}. For penetrating discussions and meaty examples, {Kiefer87book} is good, in places. I have used both books on several occasions as the text for the course. Unfortunately neither book seemed popular with the classes.

As with all my graduate courses, there will be no exams. The final grade will be based completely on the homework assignments. I expect to hand out about ten assignments during the course. Details of my grading system will be explained in the first class.

### Office hours and problem sessions

I am unable to specify office hours until all my class times have been set (at the 5 September meeting). I am willing to hold a regular session to discuss problem assignments and other material related to the course.

### References

{BickelDoksum77book}
Bickel, P. J. & Doksum, K. A. (1977), Mathematical Statistics: Basic Ideas and Selected Topics, Holden-Day, San Francisco.

{Brown86book}
Brown, L. D. (1986), Fundamentals of Statistical Exponential Families, Institute of Mathematical Statistics, Hayward, California.

{Ferguson67book}
Ferguson, T. S. (1967), Mathematical Statistics: A Decision Theoretic Approach, Academic Press, Boston.

{Hartigan83book}
Hartigan, J. A. (1983), Bayes Theory, Springer, New York.

{IbragimovHasminskii81book}
Ibragimov, I. A. & Has'minskii, R. Z. (1981), Statistical Estimation: Asymptotic Theory, Springer, New York.

{Kiefer87book}
Kiefer, J. C. (1987), Introduction to Statistical Inference, Springer-Verlag, New York.

{Lehmann59TSH}
Lehmann, E. L. (1959), Testing Statistical Hypotheses, Wiley, New York. (Later edition published by Chapman and Hall.)

{Lehmann83TPE}
Lehmann, E. L. (1983), Theory of Point Estimation, Wiley, New York. (Later edition published by Chapman and Hall.)

{Silvey75book}
Silvey, S. D. (1975), Statistical Inference, Vol. 7 of Monographs on Statistics and Applied Probability, Chapman and Hall, London. (Originally published by Penguin Books. Republished at some time in the 1990's by Chapman and Hall. ISBN 0-412-13820-4.)