Rules and Regulations
Ph.D. Comprehensive Examination for Students in Mathematics
The Ph.D. comprehensive examination for students in mathematics consists of a basic component and an advanced component.
 Basic Comprehensive Component

 Students in mathematics (excluding statistics) must satisfy the basic component of the comprehensive examination in two subjects selected from the following list:
 Algebra
 Analysis
 Topology
 Continuous applied mathematics (c.a.m.)
 Discrete applied mathematics (d.a.m)
 Probability
 Statistics
The choice of subjects will be made by the student’s supervisor, in consultation with the student.
 The basic comprehensive component is satisfied by passing the final examinations in the two oneterm core courses for each of the selected subjects:
Algebra
MAT 5141/MATH5107 and MAT 5142/MATH 5109
Analysis* MAT 5125/MATH5007 and MAT 5126/MATH 5008
Probability MAT 5170/STAT5708 and MAT 5171/STAT 5709
Statistics MAT 5190/STAT5600 and MAT 5191/STAT 5501
Topology MAT 5151/MATH5205 and MAT 5152/MATH 5206
c.a.m. MAT 5131/MATH5405 and MAT 5133/MATH 5406
d.a.m. MAT 5105/MATH5818 and MAT 5107/MATH 5819
* For analysis, in addition it is required that the student have demonstrated competence in complex analysis at the advanced undergraduate level.
 In the event of a failure in one or more attempts of the general comprehensive examinations, a student may either be asked to withdraw from the program, or be required to retake one or more of the examinations.
 In any event, the basic comprehensive component must be met within 20 months of the student’s initial registration in the case of fulltime students and within 40 months in the case of parttime students.
 All final examination papers in all core courses will be kept by the department for 20 months. A student who has passed the final examination of a core course with a grade of B or better, within this time period, may request to have this be counted as a grade of “pass” towards his or her basic comprehensive component.
The above core courses are offered at Carleton University and the University of Ottawa on an alternating basis.
Each final examination is a formal threehour written examination reflecting the respective syllabus. Graduate students can take a final examination even if they are not registered in the course; these students have to inform the Instructor of the course by November 15, or March 15 of the given academic year if they intend to write an examination in December or in April, respectively.
All comprehensive examinations will be set and graded by examiners from both universities. A grade of “pass” (a grade of B or better) or “fail” will be assigned for each examination. In addition to the final examination grade, each student registered in the course will be given a grade based on his/her overall performance in the course. Such a grade will be the sole responsibility of the professor teaching the course. A passing grade in the course does not ensure a “pass” for the comprehensive requirement, or vice versa.
 Students in mathematics (excluding statistics) must satisfy the basic component of the comprehensive examination in two subjects selected from the following list:
 Advanced Comprehensive Component

 The advanced comprehensive component will be prescribed by the student’s supervisory committee. It will take the form of a special examination (written and/or oral) based on a syllabus prescribed by the student’s supervisory committee and given to the student six months before the examination. At that time, the form of the advanced comprehensive component will also be specified.
 The advanced comprehensive component must be completed within 20 months of admission to the Ph.D. program in the case of fulltime students and within 40 months in the case of parttime students.
 A student changing his / her area of specialization will be required to complete the advanced comprehensive component in the new area within a time period specified by the student’s new supervisory committee.
Ph.D. Comprehensive Examination for Students in Mathematics with Specialization in Statistics
The Ph.D. comprehensive examination for students in mathematics with specialization in statistics consists of a General requirement, an Area requirement and a Special examination. There are two options, Option A and Option B.
 Option A  General Requirement

 Students in mathematics with a specialization in statistics, must satisfy a general comprehensive requirement in two subjects selected from the following list:
 Analysis
 Algebra
 Topology
 Probability
 continuous applied mathematics (c.a.m.)
 discrete applied mathematics (d.a.m)
The choice of subjects will be made by the student’s supervisor, in consultation with the student.
 The general comprehensive requirement is satisfied by passing the final examinations in the two oneterm core courses for each of the selected subjects:
Algebra
MAT 5141/MATH5107 and MAT 5142/MATH 5109
Analysis* MAT 5125/MATH5007 and MAT 5126/MATH 5008
Probability MAT 5170/STAT5708 and MAT 5171/STAT 5709
Topology MAT 5151/MATH5205 and MAT 5152/MATH 5206
c.a.m. MAT 5131/MATH5405 and MAT 5133/MATH 5406
d.a.m. MAT 5105/MATH5818 and MAT 5107/MATH 5819
* For analysis, in addition it is required that the student have demonstrated competence in complex analysis at the advanced undergraduate level. The examinations will each be set and graded by examiners from both universities. A grade of “pass” or “fail” will be assigned for each examination.
Note: The evaluation procedure for the general comprehensive requirement is independent of the assignment of grades for the related courses. Students need not be registered in the course to write the examination. A passing grade in the course does not ensure a “pass” for the comprehensive requirement, or vice versa.
 In the event of a failure in one or more attempts of the general comprehensive examinations, a student may be asked to withdraw from the program, or shall be required to retake one or more of the examinations.
 Students must attempt to satisfy these regulations as soon as possible after their initial registration. In any event, the general comprehensive requirement must be satisfied within 20 months of the student’s initial registration in the case of fulltime students and within 38 months in the case of parttime students.
 All final examination papers in all core courses will be kept by the departments for 20 months, and a student who enters the Ph.D. program after completing one or more of these courses within this time period may request that his/her examination(s) be remarked for the general comprehensive requirement.
 Students in mathematics with a specialization in statistics, must satisfy a general comprehensive requirement in two subjects selected from the following list:
 Option A  Area requirement

 The area requirement consists of one threehour, written, closedbook examination in the area of mathematical statistics.
 There are two examination sessions annually, one in October and one in February. Students are encouraged to attempt the examination early in their programs. In any event, the Area examination must be passed within 18 months of initial registration (36 months for part time students).
 Students must notify the director of the Institute of their intent to write the examination at least one month in advance.
 The Area examination is prepared and evaluated by at least two members of the Institute, appointed by the Director in consultation with the appropriate advisory committee of the Institute.
 Option B  General requirement

 Students in mathematics with a specialization in statistics, must satisfy a general comprehensive requirement in one subject other than their area of specialization, selected from the following list:
 Analysis
 Algebra
 Topology
 Probability
 Continuous applied mathematics (c.a.m.)
 Discrete applied mathematics (d.a.m)
The choice of subjects will be made by the student’s supervisor, in consultation with the student.
 The general comprehensive requirement is satisfied by passing the final examinations in the two oneterm core courses for the selected subject:
Algebra
MAT 5141/MATH5107 and MAT 5142/MATH 5109
Analysis* MAT 5125/MATH5007 and MAT 5126/MATH 5008
Probability MAT 5170/STAT5708 and MAT 5171/STAT 5709
Topology MAT 5151/MATH5205 and MAT 5152/MATH 5206
c.a.m. MAT 5131/MATH5405 and MAT 5133/MATH 5406
d.a.m. MAT 5105/MATH5817 and MAT 5107/MATH 5819
*For analysis, in addition it is required that the student have demonstrated competence incomplex analysis at the advanced undergraduate level. The examinations will each be set and graded by examiners from both universities. A grade of “pass” or “fail” will be assigned for each examination.
Note: The evaluation procedure for the general comprehensive requirement is independent of the assignment of grades for the related courses. Students need not be registered in the course to write the examination. A passing grade in the course does not ensure a “pass” for the comprehensive requirement, or vice versa.
 In the event of a failure in one or more attempts of the general comprehensive examinations, a student may be asked to withdraw from the program, or shall be required to retake one or more of the examinations.
 Students must attempt to satisfy these regulations as soon as possible after their initial registration.
In any event, the general comprehensive requirement must be satisfied within 20 months of the student’s initial registration in the case of fulltime students and within 38 months in the case of parttime students.
 All final examination papers in all core courses will be kept by the departments for 20 months, and a student who enters the Ph.D. program after completing one or more of these courses within this time period may request that his/her examination(s) be remarked for the general comprehensive requirement.
 Students in mathematics with a specialization in statistics, must satisfy a general comprehensive requirement in one subject other than their area of specialization, selected from the following list:
 Option B  Area requirement

 The area requirement consists of two threehour, written, closedbook examinations, one in mathematical statistics and one in applied statistics.
 There are two examination sessions annually, one in October and one in February.Students are encouraged to attempt the examination early in their programs.
In any event, the Area examination must be passed within 18 months of initial registration (36 months for parttime students).
 Students must notify the Director of the Institute of their intent to write the examination at least one month in advance.
 The Area examination is prepared and evaluated by at least two members of the Institute, appointed by the Director in consultation with the appropriate advisory committee of the Institute.
 Option A and B  Special Examination

 The Special examination consists of an examination related to the student’s proposed thesis.
 The format of the examination is decided by the student’s advisory committee, in consultation with the student. The examination is scheduled, prepared and evaluated by this committee.
 The student should request a list of topics for the examination from his advisory committee at least 6 months before the examination. In case of failure, the examination may be repeated once, subject to approval of the student’s advisory committee.
In any event, the Special examination must be passed within 2 years of initial registration (4 years for parttime students).
Mathematical Statistics
The area exam in mathematical statistics will cover statistical inference (estimation and testing), linear models, and multivariate theory. Students writing this examination are expected to perform at a satisfactory level in each of the two categories in order to receive a passing grade.
 References

Statistical Inference (Mathematical Statistics I and II):
 G. Casella and R.L. Berger, Statistical Inference, 2nd ed. Duxbury (or Brooks/Cole Cengage Learning), 2002.
 P.J. Bickel and K.A. Doksum, Mathematical Statistics: Basic Ideas and Selected Topics, Vol. I, 2nd ed., Prentice Hall, 2001.
 G.G. Roussas, A First Course in Mathematical Statistics, AddisonWesley, 1973.
 E.L. Lehmann and J.P. Romano, Testing Statistical Hypotheses, 3d ed., Springer, 2005.
Casella and Berger:
Chapter 5 (5.45.5), Chapter 6 (6.16.3), Chapter 7, Chapter 8, Chapter 9 (9.19.2), Chapter 10
Bickel and Doksum:
Chapter 1 (1.11.6), Chapter 2 (2.12.3), Chapter 3 (3.13.3), Chapter 4 (4.14.5 and 4.9), Chapter 5 (5.15.4)
Roussas:
Chapter 8, Chapter 10 (10.1 only), Chapter 11, Chapter 12, Chapter 13, Chapter 15 (15.115.4 only), Chapter 16 (16.116.2)
E.L. Lehmann and J.P. Romano:
Chapter 1 (1.11.9), Chapter 3 (3.13.9), Chapter 4 (4.14.2, 4.44.7, 4.9)
Multivariate Analysis:
 Inge Koch, Analysis of Multivariate and HighDimensional Data, 2014, Cambridge University Press.
 Theodore W. Anderson, An Introduction to Multivariate Statistical Analysis, 3rd Edition, 2003, Wiley.
 Y.L., Tong, The Multivariate Normal Distribution, 1990, Springer.
 R.A. Johnson and D.W. Wichern, Applied Multivariate Statistical Analysis, 6th Edition, 2008, Pearson
Linear Models:
 C.E. McCulloch, S.R. Searle and J.M. Neuhaus, Generalized, Linear, and Mixed Models; 2008, Wiley
 Ronald Christensen, Plane Answers to Complex Questions, 2002, Springer
 C.R. Rao, Linear Statistical Inference and Its Applications, 1973, Wiley
Applied Statistics
The area exam in applied statistics will cover three of the following topic areas:
experimental design; categorical data analysis; time series; sampling theory; and reliability theory. Students in conjunction with their advisor, will choose three of these topic areas and write an examination based upon these three. Students writing the examination are expected to perform at a satisfactory level in each of their three chosen categories in order to receive a passing grade.
 References

 W.M. John Statistical Design and Analysis of Experiments, MacMillan, 1971.
 Kempthorne Design of Experiments, John Wiley, 1952.
 Bishop, Fienberg and HollandDiscrete Multivariate Analysis, M.I.T. Press, 1975.
 J. Brockwell and R.A. Davis Time Series: Theory and Methods, Springer Series in Statistics, 1987.
 G. Cochran Sampling Techniques, 3rd. edition, Wiley, 1977.
 Lawless Statistical Models and Methods for Lifetime Data, Wiley, 1982.
 STAT 5502 Lecture Notes by J.N.K. Rao
 STAT 5505 Lecture Notes by S. Mills
 STAT 5602 Lecture Notes by J.N.K. Rao
 Experimental Design
 STAT 5505 Lecture Notes by S. Mills
 W.M. John
 Chapter 2
 Chapter 3
 Chapter 4 (omitting 4.84.9)
 Chapter 5 (omitting 5.9)
 Chapter 6
 Chapter 7 (omitting 7.87.13)
 Chapter 8 (omitting 8.58.6, 8.98.16)
 Chapter 11 (omitting 11.15)
 Chapter 12 (omitting 12.312.5, 12.912.10)
 Chapter 13 (omitting 13.3, 13.513.7, 13.9)
 O. Kempthorne
 Chapter 8: 8.2, 8.3
Topics: Review of matrix theory and linear model theory; Randomization theory for completely randomized and randomized block designs; analysis with treatment errors; efficiency of randomized block design; Orthogonality of factors; randomization theory for Latin squares and mutually orthogonal latin squares; cross over designs, split plot designs; Combinatorial properties of balanced incomplete block designs; intra and interblock analysis for BIB; group
divisible designs and partially balanced incomplete block designs; Factorial experiments; confounding and fractional replication.2. Categorical Data Analysis

 STAT 5602 Lecture Notes by J.N.K. Rao
 Bishop, Fienberg, Holland
 Chapter 2: 2.42.5
 Chapter 3: 3.33.5
 Chapter 4: 4.2, 4.4
 Chapter 8: 8.2
 Chapter 10: 10.6
 Chapter 11: 11.3
 Chapter 13: 13.4
 Chapter 14: 14.3, 14.614.9
Topics: Asymptotic theory for chisquared and G2. Maximum likelihood estimation for exponential families, two fundamental equivalence theorems, Haberman’s approach, ordinal data (twoway tables), multiarray tables and loglinear models, model selection; weighted least squares approach, tests of symmetry and quasisymmetry; logic models; measures of association.
3. Time Series

 Brockwell & Davis
 Chapter 3
 Chapter 4: 4.14.4
 Chapter 5
 Chapter 7
 Chapter 8
 Brockwell & Davis
 Reliability Theory
 Lawless
 Chapter 1: 1.4
 Chapter 2: 2.32.4
 Chapter 6: 6.16.5; 6.7
 Chapter 7: 7.27.4
 Chapter 8: 8.2
 Lawless
Topics: Nonparametric estimation of survival function; parametric models and maximum likelihood estimation; exponential and Weibull regression models; nonparametric hazard function models and associated inference; rank tests with chisquared data.
5. Sampling Theory and Methods

 STAT 5502 Lecture notes by J.N.K. Rao
 Cochran
 Chapter 5A
 Chapter 6
 Chapter 9: 9A.19A.4; 9A.69A.11; 9A.13
 Chapter 10: 10.4; 10.6; 10.910.10
 Chapter 11: 11.611.10; 11.1311.14; 11.1711.20
 Chapter 12: 12.212.13
Topics: Unified theory for standard errors; unequal probability sampling with and without replacement, stratification; Ratio estimation; domain estimation; post stratification, multistage sampling unified theory; nonlinear statistics: jacknife, balanced repeated replication, linearization; twophase sampling; nonresponse and importation; response array.
6. Modern Computational/Applied Statistics

 References:
 [1] Efron, B., and Tibshirani, R. “An Introduction to the Bootstrap”, Chapman and Hall, 1993.
 [2] Hastie, T. and Tibshirani, R. “Generalized additive models”, Chapman and Hall, 1990.
 [3] Ripley, B. “Pattern Recognition and Neural Networks”, Cambridge University Press, 1996.
 Lecture notes relevant to:
 Chapters 7, 8, 9, 10, 11, 12, 13, 14, 22 of [1]
 Chapter 2 of [2]
 Chapters 5, 6, 7 of [3]
 References:
Topics: Resampling and computer intensive methods: bootstrap, jackknife, and datasplitting methods with applications to bias estimation, variance estimation, confidence intervals, and regression analysis; smoothing methods in curve estimation; statistical classification and pattern recognition: error counting methods, optimal classifiers, combined classifiers, use of bootstrap in estimating the bias of the misclassification error rate, nearest neighbour classifiers, neural network classifiers, and tree classifiers.
Share: Twitter, Facebook
Short URL:
https://carleton.ca/math/?p=10983