 # PhD Comprehensive Examinations

Comprehensive Examinations are usually offered in May of each year. All PhD students need to write all three comprehensive examinations in the first year of their program

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.
Please ask the department office if any copies of previous comprehensive examinations are available.

Graduating U of T Statistics MSc students who will continue on to our PhD program, and who have taken the appropriate prerequisite courses, have the option to write the comprehensive examinations on a trial basis

#### 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 in this exam include:

• Bernoulli trials

• Combinatorics

• Properties of standard probability distributions

• Poisson processes

• Markov chains

• Measure theory and (Lebesgue) integration

• Extension theorems

• Borel-Cantelli lemmas

• Product measures and independence, Fubini’s Theorem

• Probability distributions

• Convergence theorem (dominated convergence, monotone convergence, etc.)

• 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

• Definitions and properties

• Statistical applications

• Martingales

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 in this exam include:

• Statistical models

• Sufficiency and ancillarity

• Completeness

• 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

• Neyman-Pearson Lemma

• UMP tests

• UMP unbiased and invariant tests

• Tests based on maximum likelihood estimation (LR, score, Wald tests)

• Confidence regions

• Bayes estimators

• Minimax estimation

• Equivariance

• Least squares estimation

• Gauss-Markov Theorem

• Distribution theory for quadratic forms

• Hypothesis testing in linear models

• 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

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 on material from various undergraduate-level statistics courses.

If you plan to write the applied comprehensive exam, you may find it useful to take graduate courses such as:

• STA2101H (Methods of Applied Statistics I),

• STA2201H (Applied Statistics II),

• STA2004H (Design of Experiments),

• STA2102H (Computational Techniques in Statistics),

• STA2542H (Linear Models), and/or

• CHL5222H (Longitudinal Data Analysis).

However, none of these courses are required, and you may choose for yourself how best to prepare for this part of the comprehensive exam.

##### Applied Statistical Skills and Knowledge

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.

You should be able to:

• Choose a structure for the analysis of (possibly complex) data

• Understand and explain, in non-technical language, issues such as modeling, estimation, inference, summarisation, study type, and sources of variability

To facilitate these skills, you need to have a basic knowledge of the following applied statistics topics:

• 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

• 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.

• 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

• 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.

• Outline of generalized linear models

• Models for continuous and binary data

• Log-linear models

• Inference

• Interpretation and model checking

• 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.

• Basic sampling

• Simple random sampling

• Stratified sampling

• Cluster and systematic sampling

• Multistage sampling

• Concepts

• Ideas and inference

• 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 generally provide good perspectives 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.

#### Questions?

For further information please get in touch with our Associate Chair, Graduate Studies.