**(Revised November 2006. Subject to change.)**

***** PLEASE NOTE THAT THIS IS BEING UPDATED TO REFLECT CHANGES IN THE PhD PROGRAM *****

This web page gives basic information about the Comprehensive Examination for the PhD program in the Department of Statistics at the University of Toronto. For further information please consult the Associate Chair, Graduate Studies.

The Comprehensive Examinations are normally offered in May of each year. The Department expects PhD students to write all three comprehensive examinations in the first year of their program. (Graduating U of T Statistics MSc students who are continuing on to our PhD program, and who have taken the appropriate prerequisite courses, may write the comprehensive examinations on a trial basis if they wish.) The exam consists of three parts: Probability, Theoretical Statistics, and Applied Statistics. The three parts are offered on three separate days; each part is allotted four hours. The exams are closed-book; no aids are allowed other than a single non-programmable calculator. Students will be given at most two opportunities to pass each of the three exams, but will be required to rewrite only those exams that they fail. In appropriate cases, approval may be obtained to replace one of the three comprehensive examinations with a suitable comprehensive examination from another department.

The material covered on each of the three parts is now described. Copies of previous Comprehensive Examinations may be available from the department office.

### Probability Part

The Probability Part of the Comprehensive Exam is based on the courses STA 2111F and STA 2211S (Graduate Probability I and II). Specific topics covered include:

- Elementary probability theory
- Bernoulli trials
- Combinatorics
- Properties of standard probability distributions
- Poisson processes
- Markov chains

- Probability spaces
- measure theory and (Lebesgue) integration
- extension theorems
- Borel-Cantelli lemmas
- product measures and independence, Fubini’s Theorem

- Random variables and expectations
- probability distributions
- Radon-Nikodym derivatives and densities
- convergence theorem (dominated convergence, monotone convergence etc)

- Limit theorems
- inequalities
- weak and strong laws of large numbers for sums of i.i.d. r.v.’s
- Glivenko-Cantelli Theorem
- weak convergence (convergence in distribution)
- continuity theorem for characteristic functions
- Central Limit Theorems

- Conditional probability and expectation
- definitions and properties
- statistical applications
- martingales

- Basics of Brownian motion and diffusions

#### References

Most of the above material is covered in any one of the following texts:

- P. Billingsley (1995), Probability and measure (3rd ed.). John Wiley & Sons, New York.
- L. Breiman (1992), Probability. SIAM, Philadelphia.
- K.L. Chung (1974), A course in probability theory (2nd ed.). Academic Press, New York.
- R.M. Dudley, Real analysis and probability. Wadsworth, Pacific Grove, CA.
- R. Durrett (1996), Probability: theory and examples (2nd ed.). Duxbury Press, New York.
- B. Fristedt and L. Gray (1997), A modern approach to probability theory. Birkhauser, Boston.
- J.S. Rosenthal (2000), A first look at rigorous probability theory. World Scientific Publishing, Singapore.

### Theoretical Statistics Part

The Theoretical Statistics Part of the Comprehensive Exam is based on the course STA3000Y (Advanced Theory of Statistics). Specific topics covered include:

- Basics
- statistical models
- sufficiency and ancillarity
- completeness

- Point estimation
- unbiased estimation (i.e. Rao-Blackwell and Lehmann-Scheffe Theorems)
- method of moments and maximum likelihood estimation
- asymptotic theory of estimation
- “regular” estimators and asymptotic efficiency

- Hypothesis testing
- Neyman-Pearson Lemma
- UMP tests
- UMP unbiased and invariant tests
- tests based on maximum likelihood estimation (LR, score, Wald tests)
- confidence regions

- Decision theory
- admissibility
- Bayes estimators
- minimax estimation
- equivariance

- Linear models
- least squares estimation
- Gauss-Markov Theorem
- distribution theory for quadratic forms
- hypothesis testing in linear models

- Bayesian Inference
- Prior and posterior distributions, conjugacy
- Loss functions and Bayes decisions, including optimal estimation, hypothesis testing and sample size determination
- Large sample behaviour of posterior quantities

#### References

Main references:

- G. Casella and E.L. Lehmann (1998), Theory of Point Estimation. Springer, New York.
- E.L. Lehmann (1997), Testing Statistical Hypotheses (2nd ed.). Springer, New York.
- K. Knight (2000), Mathematical Statistics. Chapman & Hall / CRC Press, New York.

Other references:

- G. Casella and R.L. Berger (1990), Statistical Inference. Duxbury / Wadsworth, Belmont, California.
- P. Bickel and K. Doksum (1977), Mathematical Statistics: Basic Ideas and Selected Topics. Holden-Day, San Francisco.
- D.R. Cox and D. Hinkley (1974), Theoretical Statistics. Chapman and Hall, London.

### Applied Statistics Part

The Applied Statistics Part of the Comprehensive Exam is based upon material from various undergraduate-level statistics courses. Students planning to write the applied comprehensive may also find it useful to take such graduate courses as: STA2101H (Methods of Applied Statistics I), STA2201H (Applied Statistics II), STA2004H (Design of Experiments), STA2102H (Computational Techniques in Statistics), STA2209H (Lifetime Data Modelling), STA2542H (Linear Models), and/or CHL5222H (Longitudinal Data Analysis). However, none of these courses is required, and students may choose for themselves how best to prepare for this part of the comprehensive exam.

The applied statistics exam is designed to ensure that students possess sufficient applied statistical skills and knowledge, and that they can apply theoretical probability and statistics skills to solve applied problems. Students should be able to choose a structure for the analysis of (possibly complex) data. They should also be able to understand, and explain in non-technical language, such issues as modeling, estimation, inference, summarisation, study type, and sources of variability.

To facilitate these skills, students should be able to draw upon a basic knowledge of the following applied statistics topics:

**1. Experimental design.** One- and two- sample problems, one-factor ANOVA and model checking, randomized block designs, incomplete block designs, latin square designs, factorial designs, 2^k factorial and fractional factorial designs, split plot designs, principles of bias and variance reduction, blocking.

References:

- Montgomery, D.C. (1991). Design and Analysis of Experiments, 3rd Edition. Wiley. Chapters 2, 3, sections 4.1, 4.2, 4.4, 5.1, 5.2, 5.3, chapter 6, sections 7.1-7.5, chapters 9-11, section 14.2.
- Cox, D.R. and Reid, N. (2000) Chapters 1-6, except sections 3.6, 3.7.

**2. Linear models.** Simple linear regression, multiple regression, model interpretation and drawing conclusions, tests of general hypotheses, lack-of-fit, residuals and influence, model diagnostics and remedies, polynomial regression, prediction.

References:

- Weisberg, S. (1985). Applied Linear Regression, 2nd Edition. Wiley. Chapters 1-7, 9.
- Sen, A.K. and Srivastava, M.S. (1990). Regression Analysis: Theory, Methods, and Applications. Springer. Chapters 1-6, 8, 9, 11, except sections 2.12, 5.4, 5.5.

**3. Generalized linear models.** Outline of generalized linear models, models for continuous and binary data, log-linear models, inference, interpretation and model checking.

References:

- McCullagh, P. and Nelder, J.A. (1989). Generalized Linear Models, 2nd Edition. Chapman & Hall. Chapters 1-4, 6, 12, except sections 3.5, 3.8, 6.5 and 12.4.

**4. Survey sampling.** Basic sampling, simple random sampling, stratified sampling, cluster and systematic sampling, multistage sampling, concepts, ideas and inference.

References:

- Thompson, S.K. (1992). Sampling. Wiley. Chapters 1-6, 11-13.
- Lohr, S.L. (1999), Sampling Design and Analysis. Duxbury Press. Chapters 1-5.

### Books which provide good general perspective on applied statistics include:

- D.R. Cox and E.J. Snell (1981), Applied statistics: principles and examples. Chapman and Hall.
- W.N. Venables and B.D. Ripley (1999), Modern applied statistics with S-PLUS (3rd ed.). Springer.