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Graduate Student Information

Section 5 Preliminary Requirements

As of Fall 2024, before Advancing to Candidacy, PhD student must complete two requirements.

  • Qualifying Exam in either Statistics or Probability.
  • Research Project under the direction of a faculty

5.1 Qualifying Exams

These exams are given in September. You can choose which exam you would like to take. Anyone who is hoping to complete the Financial Mathematics Emphasis should take the Probability and Stochastic Processes exam.

5.1.1 Mathematical Statistics

The Mathematical Statistics qualifying exam covers topics mainly from PSTAT 207ABC with some from PSTAT 220ABC, including:

  • Distributions of random variables including transformations, mixtures, and order statistics
  • Sufficiency
    • Factorization theorem
    • Minimal sufficient statistics
    • Complete Experiments
  • Estimation
    • Minimum Variance Unbiased Estimates, Rao–Blackwell and Lehmann–Scheffe Theorems
    • Maximum likelihood and method of moments
    • Fisher Information and the Cramer–Rao lower bound
  • Hypothesis testing
    • Neyman–Pearson lemma and uniformly most powerful tests
    • Monotone Likelihood Ratios
    • Most Powerful Unbiased tests
    • Generalized Likelihood Ratio Tests
  • Confidence intervals and prediction intervals.
  • Asymptotic methods
    • Consistent estimators
    • Asymptotic normality and the delta method
  • Bayesian Methods
    • Prior and posterior distributions
    • Bayesian estimators
    • Credible intervals
  • Regression Models
    • Best Linear Unbiased Estimators
    • Projections

5.1.2 Probability and Stochastic Processes

The Probability and Stochastic Processes qualifying exam covers topics from PSTAT 213ABC including:

  • Generating Functions
  • Discrete time Markov Chains including random walks and branching processes. Chapman-Kolmogorov equations, classification of states, decomposition of the state space, invariant measures and stationary distributions, limit theorems, time reversibility.
  • Continuous time Markov Chains including Poisson processes and birth-death processes. Backward and forward equations, generators, limiting behavior.
  • Convergence of random variables: Different types of convergence; characteristic functions, convergence in distribution, continuity theorem, Law of Large Numbers and Central Limit Theorem, a.s. convergence and Borel-Cantelli, convergence in probability, Skorohod’s representation theorem, strong law of large numbers; uniform integrability and \(L^1\) convergence.
  • Conditional Expectation and Hilbert spaces for \(L^2\) variables.
  • Martingales: Doob and Doob-Meyer decomposition, martingale convergence theorems, stopping times, optional sampling, optional stopping theorems and applications, maximal inequalities.
  • Brownian motion: Brownian Motion as the scaling limit of a random walk, martingale and Markov properties, quadratic variation. Hitting times. Conditional distributions and Brownian bridge. Stochastic integral, Ito formula, applications to related processes including Ornstein-Uhlenbeck process and Geometric Brownian motion.

5.1.3 Policies and Procedures

Each exam is written and graded by a committee of at least three faculty who are most familiar with the content of the exam. Typically, the chair of this committee will be one of the instructors of the relevant core course sequence that year. The exam committee will be responsible for specifying the details regarding the exact time and place where the exam will be given as well as issues like what materials you may be allowed to consult during the exam.

The exam committee will grade all of the exam papers, and then make a recommendation to the entire faculty as to who deserves to pass the exam. The results of the exams will be approved by all of the faculty.

There are three possible exam results: no pass, MA Pass, and PhD Pass. If you are a PhD student and your exam was evaluated to be below the standard for a PhD Pass, then you will likely be awarded an MA Pass. This means you will have to retake the exam to proceed with the doctoral degree, but it is sufficient to fulfill the requirements for a masters degree.

5.1.3.1 Disabled Students Accommodations

If you have a letter on file with DSP which stipulates that you need extra time or any other accommodations for exams, please let the GPA know so that they can make arrangements for your qualifying exam. The department is committed to the accessibility for all students, and we encourage you to discuss any needs you have with the exam committee or Graduate Advisors.

5.1.3.2 Retaking Qualifying Exams

You can retake any exam where you did not receive a passing grade, but you may only take an exam in the same subject area 2 times.

Typically, students that do not pass a qualifying exam will be encouraged to take additional coursework, and then take the exam again in the following year. You may use the course PSTAT 510 Readings for Area Examinations as a way to register for units while retaking a core class that you have already taken for credit. You can also just audit that class. In either case, you need to first talk to the instructor and get explicit permission to attend the course as well as their expectations about whether you will be completing the homework assignments, exams, and/or quizzes.

5.2 Research Projects

Students will produce their own report on a research project that they have worked on with a faculty member or team in the department.

The report should demonstrate:

  • Familiarity with the tools and techniques of their area of statistics, data science, applied probability, or finance.
  • Knowledge of some of the current research questions in the chosen area.
  • Facility with the form and style of research papers in this field. This will include careful citations of previous published research or contributions from other team members to the project work.

These projects could take a number of different forms. Some examples of the type of work that is expected

  • A statistical analysis of a real data set requiring novel applications of statistical techniques,
  • Development of an innovative computational tool,
  • Review of a current area of research with a comparison of the competing approaches in the literature,
  • New theory or argument on a question of current relevance.

Faculty Advisor

The recommendation is that you find a research-track faculty advisor to work with on your research project. Begin by consulting with the appropriate faculty member in order to find an interesting and relevant subject to pursue. Then it is recommended that students enroll in PSTAT 596 with the faculty advisor for at least one quarter to work on this project.

You are also encouraged to work within research groups on these projects. These groups may include other faculty, post-docs, and/or students. Your research report should highlight your contributions to the collaboration.

5.3 Time Limit for Preliminary Requirements

The expectation is that the qualifying exam and research project can be completed in your first two years as a PhD student.

The Department Time to Degree limit for Preliminary Requirements (exam and the research project) is 3 years. Any student who has not completed these requirements by the start of their fourth year will be placed on Academic Monitoring, and they will be expected to submit an Academic Plan outlining how they expect to complete the requirements.

5.4 Exceptions and Petitions

Graduate students have the right to request exceptions to department policies if they feel they have a compelling case.

Some requests dealing with the Preliminary Requirements and Qualifying Exams are:

  1. Additional Exam Attempts Students are given 2 attempts to complete each qualifying exam. Students may submit a request to the Director of Graduate Studies if they would like to be given a third attempt to take an exam. This request should include sufficient justification that you will be successful in an additional attempt and that the circumstances of the first two attempts did not properly demonstrate your knowledge and potential.

  2. Requests to Forgo Core Courses The department requires all PhD students take the core courses and to complete a qualifying exam. Under extraordinary circumstances, students who are able to pass a qualifying exam may request to be exempt from taking the core course. These exceptions are only granted when you have taken these courses at other institutions. The request should be made to the Director of Graduate Studies along with documentation and justification.

  3. Alternative Exam Conditions Students may submit a request to the DGS and to the Exam Committee if they wish to set up an alternative exam. This could be a qualifying exam scheduled outside of the normal annual schedule, or an exam structured differently such as an oral exam, etc. The faculty are unlikely to approve such a request unless there is a very compelling reason because of the required workload and the need to make the exam results equitable across years.

  • Please note that any student with a DSP letter will automatically receive the required accommodation without any further petition. If you have a DSP letter, you just need to let the GPA know when scheduling the exam so that the proper venue can be arranged.
  1. Regrade a Qualifying Exam Students have the right to ask the Examination Committee to reconsider their grade on a qualifying exam. This request should be made to the Chair of the appropriate exam committee, and it should include why you think the original exam paper was graded incorrectly. Qualifying exam grades will only be changed if the Exam Committee agrees that there was a clearly an error in the original evaluation of the exam paper.
  • Students may further petition the DGS if they wish to have the entire faculty reconsider their exam results.