# Oral Exam in Applied Probability and Statistics

Oral Exam in Applied Probability and Statistics

Oral Exam in Applied Probability and Statistics

This topic of the oral exam is based on the course Math 691, Stochastic Processes with Applications.

Topics on which questions can be based are as follows:

- Poisson processes: homogeneous, non-homogeneous (NHPP), mixed, compound-Poisson and doubly stochastic Poisson processes. Illustrative applications.
- Renewal theory: renewal quantity and the renewal function, the renewal equation. Basic asymptotic results. The key renewal theorem and its applications. Stationary, delayed and alternating renewal processes. Renewal reward processes, the inspection paradox, computing the renewal function.
- Discrete time Markov Chains (DTMC): Markov chains in discrete time with finite or countably many states and a time-homogeneous transition matrix. Recurrence and transience. Decomposing the state space into equivalence classes. Hitting times and absorption probabilities. Convergence to equilibrium and invariant (stationary) distribution. Time reversal. Typical applications including extinction probability in Galton-Watson processes.
- Continuous time Markov Chains (CTMC): The transition probability function, forward and backward Chapman-Kolmogoroff equations. Infinitesimal generators, class structure, embedded DTMC, hitting times and absorption probabilities. Asymptotic long-run behavior. Time reversibility. Birth and death processes.

**Primary References: **There is no single best reference for material on this exam. The following is a list of good references:

- `Stochastic Processes' by S.M. Ross. 2nd Edition, Wiley, ISBN 0-471-12062-6.
- `Adventures in Stochastic Processes' by S. Resnick. Birkhauser, ISBN 0-817-63591-2.

**Secondary references:** (indicative of a diverse range of applications that aid in developing probabilistic intuition and insight).

- `Introduction to Probability Models' by S.M. Ross. 10th Edition, Academic Press, ISBN 0-123-75686-3.
- `Stochastic Processes & Models' by David Stirzaker. Oxford University Press, ISBN 0-198-56813-4

This topic of the oral exam is based on the course Math 659, Survival Analysis.

Topics on which questions can be based are as follows:

- Basic quantities and models: survival, hazard, and mean residual functions. Common parametric models, regression models, competing risks.
- Censoring and truncation: right censoring, left or interval censoring, truncation, likelihood construction, counting processes.
- Nonparametric estimation: survival and cumulative hazard function estimators, their asymptotic properties. Point wise confidence interval and confidence bands for survival functions. Mean and median survival time. Survival function estimators for left truncated and right censored data.
- Semiparametric estimation: cumulative hazard function and survival function estimators. Comparison with nonparametric estimators.
- Hypothesis testing: one sample tests, tests for two or more samples.
- Proportional hazards regression: partial likelihoods. Local tests, estimation of survival function.
- Additive hazards regression: Aalen's model, Lin and Ying's model.
- Regression diagnostics: Cox-Snell residuals, martingale residuals, deviant residuals, graphical checks.
- Inference for parametric regression models: Weibull and log logistic distributions, other parametric models, diagnostic tests for parametric models.

**Primary References:**

- `Survival Analysis: Techniques for Censored and Truncated Data' by John P. Klein and Melvin L. Moeschberger. Springer Verlag, ISBN 0-387-95399-X. Chapters 1-12.
- `Counting Processes and Survival Analysis' by Thomas R. Fleming and David P. Harrington. 2nd Edition, Wiley Series in Probability and Statistics, ISBN 0-471-76988-6. Chapters 1-8.

The first reference contains mostly concepts and methodology. The second reference provides rigorous theoretical results.

This topic of the oral exam is based on the course Math 707, Generalized Linear Models.

Topics on which questions can be based are as follows:

- Statistical inference in simple linear regression.
- Least squares approach, estimable functions. Gauss-Markov theorem. Estimation under linear constraints.
- Multivariate normal and related distributions, distribution of quadratic forms, sampling from a multivariate normal distribution.
- Exponential family of distributions and their properties.
- The generalized linear model (GLM). Link functions, likelihood equations for GLMs, quasi-likelihoods. GLMs with exponential and gamma families.
- Maximum likelihood and least squares estimates, hypothesis testing, standardized residuals, influential observations, leverage, Cook's distance.
- Multiple linear regression, non linear regression, intrinsically linear regression models.
- Binary variables, general logistic regression model, goodness-of-fit, residuals, diagnostics, and applications. Nominal and ordinal logistic regression.
- Poisson regression, maximum likelihood estimates, examples of Poisson regression.

**Primary References:**

- `Generalized Linear Models: with Applications in Engineering and the Sciences' by Myers, Montgomery, Vining and Robinson. 2nd Edition, Wiley, ISBN 0-470-45463-6. Chapters 1-6.
- `An Introduction to Generalized Linear Models' by Annette J. Dobson and Adrian G. Barnett. 3rd Edition, Chapman & Hall/CRC Press, ISBN 1-584-88950-0. Chapters 1-9.
- `Probability and Statistics' by Morris H. DeGroot and Mark J. Schervish. Addison Wesley, ISBN 0-201-52488-0. Chapter 10, sections 10.1-10.3 and 10.5-10.9.

The overlap of topics between these references should emphasize the underlying concepts and broaden understanding.