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.
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

BorelCantelli lemmas

Product measures and independence, Fubini’s Theorem

Probability distributions

RadonNikodym derivatives and densities

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

Inequalities

Weak and strong laws of large numbers for sums of i.i.d. r.v.’s

GlivenkoCantelli Theorem

Weak convergence (convergence in distribution)

Continuity theorem for characteristic functions

Central Limit Theorems

Definitions and properties

Statistical applications

Martingales
No additional information
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. RaoBlackwell and LehmannScheffe Theorems)

Method of moments and maximum likelihood estimation

Asymptotic theory of estimation

“Regular” estimators and asymptotic efficiency

NeymanPearson Lemma

UMP tests

UMP unbiased and invariant tests

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

Confidence regions

Admissibility

Bayes estimators

Minimax estimation

Equivariance

Least squares estimation

GaussMarkov 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. HoldenDay, 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 undergraduatelevel 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),

STA2209H (Lifetime Data Modelling),

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 nontechnical 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

Onefactor 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
Reading List:

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 7.17.5, chapters 911, section 14.2.

Cox, D.R. and Reid, N. (2000) Chapters 16, except sections 3.6, 3.7.

Simple linear regression

Multiple regression

Model interpretation and drawing conclusions

Tests of general hypotheses

Lackoffit

Residuals and influence

Model diagnostics and remedies

Polynomial regression

Prediction
Reading List:

Weisberg, S. (1985). Applied Linear Regression, 2nd Edition. Wiley. Chapters 17, 9.

Sen, A.K. and Srivastava, M.S. (1990). Regression Analysis: Theory, Methods, and Applications. Springer. Chapters 16, 8, 9, 11, except sections 2.12, 5.4, 5.5.

Outline of generalized linear models

Models for continuous and binary data

Loglinear models

Inference

Interpretation and model checking
Reading List:

McCullagh, P. and Nelder, J.A. (1989). Generalized Linear Models, 2nd Edition. Chapman & Hall. Chapters 14, 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
Reading List:

Thompson, S.K. (1992). Sampling. Wiley. Chapters 16, 1113.

Lohr, S.L. (1999), Sampling Design and Analysis. Duxbury Press. Chapters 15.
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 SPLUS (3rd ed.). Springer.
Questions?
For further information please get in touch with our Associate Chair, Graduate Studies.