Structure

This exam is a four-four exam on statistical theory. It is assumed that all candidates will have a background corresponding to Statistics 581, 582, 583, and 570, 571, 572. The exam will typically consist of 4-8 questions on the following topics:

• Linear estimation theory and multivariate normal distributions
• Asymptotic theory
• Models
• Estimation
• Testing Principles
• Dependent data
• Resampling methods

A study guide for each of these topics and references are given below.

Time

This exam is given once a year. Under the agreement worked out with Biostatistics in 1997, this exam will be given in September starting in the year 1999. [In the past it had been given in August.]

Study Guide and References

LINEAR ESTIMATION AND MULTIVARIATE DISTRIBUTIONS: linear estimation , quadratic forms, linear combinations of normals, marginals, conditionals, least squares estimates, normal equations, Gauss-Markov theorem, residual sum of squares, diagonalizing a matrix, projections, conditions for quadratic forms to be chi-square distributed, Cochran's theorem, standard linear models.

• Seber, G.A.F. Linear Regression Analysis. Wiley, Chapters 1-3.
• Scheffé, H. Analysis of Variance. Wiley, Chapters 1-2, appendix.
• Rao, C. R. Linear Statistical Inference and Its Applications, Wiley.

ASYMPTOTIC THEORY: weak and strong laws of large numbers; CLT's of Lévy, Lindeberg, Feller, and multivariate type; g'-theorem; Mann-Wald theorem; applications to samle means, variances, correlations, chi-squared tests of fit, medians; Pitman efficiency; elementary empirical processes. (See also maximum likelihood estimates and likelihood ratio tests in ESTIMATION and TESTING below).

• Bickel and Doksum, Mathematical Statistics, Holden-Day, Appendix.
• Rao, C.R., Linear Statistical Inference and Its Applications, Wiley, pages 110 - 130.
• Serfling, R. Approximation Theorems in Mathematical Statistics, Wiley.

MODELS: the iid model, exponential families and their marginal and conditional distributions, sufficiency and completeness, factorization theorem, Basu's theorem.

• Bickel and Doksum, Mathematical Statistics, Holden-Day.
• Lehmann, E. L. Testing Statistical Hypotheses, 2nd ed. Springer-Verlag
• Lehmann, E. L. Theory of Point Estimation, Springer-Verlag, NY.

ESTIMATION: linear, unbiased, equivariant, Bayes, maximum likelihood, Cramér-Rao bound, method of moments, M-estimates, Rao-Blackwell, Lehmann-Scheffé, UMVU estimates, Pitman estimators, Bayes, admissible, minimax, conjugate priors, properties of score function, asymptotics of maximum likelihood estimates; EM algorithm.

• Lehmann, E. L. Theory of Point Estimation, Springer-Verlag, NY.
• Rao, C.R., Linear Statistical Inference and Its Applications, Wiley, Chapters 5 and 6.
• Ferguson, T., Mathematical Statistics, Academic Press, Chapter 5.

TESTING: Neyman-Pearson, monotone likelihood ratio, UMP, UMP unbiased, UMP invariant, likelihood ratio, Wald, and Rao (score) tests; chi-square tests, locally most powerful; simple nonparametric tests; permutation tests.

• Lehmann, E. L. Testing Statistical Hypotheses, 2nd ed. Springer-Verlag
• Rao, C.R., Linear Statistical Inference and Its Applications, Wiley, Chapters 5 and 6.
• Ferguson, T., Mathematical Statistics, Academic Press, Chapter 5.

PRINCIPLES: sufficiency, ancillarity, unbiasedness, invariance, likelihood (full, partial, conditional), Bayes, robustness.

• Cox and Hinkley, Theoretical Statistics, Chapman and Hall.
• Berger, J. Statistical Decision Theory, Springer-Verlag.
• Silvey, S. D., Statistical Inference, Chapman and Hall.

Notes: Additions and changes, July 1992

Since this syllabus was written in about 1984-1985, the course sequence 581-2-3 has evolved and changed to some degree. There is currently less emphasis on some of the classical topics such as sufficiency, completeness, and (especially) optimal testing, with correspondingly more emphasis on likelihood methods - estimation and testing, robustness (with respect to distributional form, dependence, 2#2), resampling methods (bootstrap and jackknife), and inference for dependent data.