| Instructor: | Charles Geyer (5-8511, charlie@stat.umn.edu) |
|---|---|
| Textbook: | Asymptotic Statistics by Aad W. van der Vaart |
There is a lot more to asymptotics than what is taught in measure-theoretic probability (Math 8651-2) and PhD level theoretical statistics (Stat 8111-2).
This course picks up where the others leave off. There's so much to asymptotics that we could have many, many courses like this. So we won't do much more than survey what is out there.
It is assumed that the student knows about the law of large numbers (LLN) and the central limit theorem (CLT) for independent and identically distributed (IID) sequences of scalar or vector valued random variables. These are covered in Math 8651-2 in, for example, Fristedt and Gray Sections 12.2 (strong LLN), 15.2 (weak LLN), and 15.3 (CLT). It is also assumed that the student knows about the asymptotics of maximum likelihood as covered in Stat 8112. But there's a lot more to asymptotics than that.
small sample asymptotics, well explained by Field and Ronchetti.
Cramér styleregularity conditions for maximum likelihood (covered in Stat 8112) can be considerably weakened. The log likelihood need not be differentiable for
Le Cam styletheory to hold. (An application is the sample median as the MLE for the double exponential distribution.) These are well explained by van der Vaart and by Yang and Le Cam.
That's probably too much for one course. We'll cover as much as we can. We'll be guided by student interests (other topics can also be added).
None of these are required. All will be on reserve in the math library (I hope).
Basic properties of strong mixing conditionsby Richard C. Bradley (in Dependence in Probability and Statistics above)
Recent advances in the central limit theorem and its weak invariance principle for mixing sequences of random variables (A survey)by Magda Peligrad (in Dependence in Probability and Statistics above)
None of these are required. All except the last two are available from JSTOR. (But students should not print these on laser printers. Dana in the department office has printouts. Photocopy those.)
On the Asymptotics of Constrained M-Estimationby Charles J. Geyer (Annals of Statistics, 1994, vol. 22, no. 4, pp. 1993-2010)
Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditionsby Steven G. Self and Kung-Yee Liang (Journal of the American Statistical Association, 1987, vol. 82, no. 398, pp. 605-610)
On the assumptions used to prove asymptotic normality of maximum likelihood estimatesby L. Le Cam (Annals of Mathematical Statistics, 1970, vol. 41, no. 3, pp. 802-828)
On the distribution of the likelihood ratioby Herman Chernoff (Annals of Mathematical Statistics, 1954, vol. 25, no. 3, pp. 573-578)
Discussion of the paper by Tierneyby Kung-Sik Chan and Charles J. Geyer (Annals of Statistics, 1994, vol. 22, no. 4, pp. 1747-1758)
Geometric ergodicity and hybrid Markov chainsby Gareth O. Roberts and Jeffrey S. Rosenthal (Electronic Communications in Probability, 1997, vol. 2, pp. 13-25) Available on the web at http://www.emis.de/journals/EJP-ECP/EcpVol2/paper2.pdf
Central limit theorems for empirical and U-processes of stationary mixing sequencesby Miguel A. Arcones and Bin Yu ( Journal of Theoretical Probability, 1994, vol. 7, no. 1, pp. 47--71).