In order to ensure breadth in the course of study, the Department of Statistics and Applied Probability has set up area requirements in the disciplines of applied statistics, mathematical statistics, and probability. Students may also choose to satisfy a pure mathematics area requirement offered by the Department of Mathematics. The area requirements vary for each degree, specialization, and optional Ph.D. emphasis. Consult with the Graduate Advisor or your committee chair as to which of these area requirements are appropriate for your proposed field of study, research, and degree objective. Each area requirement consists of two parts:

- Completion of the Designated One-Year Graduate Sequence with a minimum grade of “B” or better in each class. Courses taken S/U will not satisfy this requirement
- A Qualifying Examination on both undergraduate and graduate material. The qualifying examination is offered once a year, generally in spring or summer. The examination is designed to test whether students have adequate knowledge of relevant material. Each student has up to two attempts for each qualifying exam, and must successfully pass the requisite number of exams within three years of entering the department.

A detailed description of each of the four area requirements follows:

Probability and Stochastic Processes

## 1. Mathematical Statistics Area Requirements:

### Courses:

PSTAT 207 A-B-C: Statistical Theory

### Qualifying Examination:

The examination for the mathematical statistics area requirement will be based on the topics that are usually covered in PSTAT 120 A-B-C, and PSTAT 207 A-B-C

#### These topics include:

- Univariate and multivariate distribution theory: important special distributions including the Binomial, Poisson, Hypergeometric, Negative Binomial, Normal, Gamma and Beta, sampling distributions including chi-square, t and F, and order statistics. Exponential family. Moment generating function and its uses.
- Sufficiency: factorization criterion, and minimal and complete sufficient statistics.
- Estimation: maximum likelihood and method of moments, uniformly minimum variance unbiased estimators, Cramér-Rao inequality, Rao-Blackwell theorem, consistency, and asymptotic normality.
- Hypothesis testing: Neyman-Pearson lemma, power functions, most powerful and uniformly most powerful tests. General likelihood ratio tests and their asymptotic properties and asymptotic equivalents.
- Confidence intervals and prediction intervals.
- Nonparametric tests for one-sample, two-sample, and independence problems: chi-square tests for goodness of fit and contingency tables.
- Least squares principle: linear and non-linear regression and inferences based on them.
- Bayesian inference for estimation and hypothesis testing.
- Basic decision theory for estimation and hypothesis testing.

Additional topics as announced by the Qualifying Examination committee.

### Reference Material (Mathematical Statistics)

- Casella and Berger, Statistical Inference, 2nd edition
- Bickel & Doksum, Mathematical Statistics, Vol. 1
- Ross, A First Course in Probability
- Cox and Hinkley, Theoretical Statistics
- Rao, Linear Statistical Inference and its Applications
- Lehmann, Theory of Point Estimation, Springer, also Testing Statistical Hypotheses
- Rohatigi, & Saleh,, An Introduction to Probability and Statistics, 2nd Edition

## 2. Probability and Stochastic Processes Area Requirements:

### Courses:

PSTAT 213 A-B-C: Introduction to Probability Theory and Stochastic Processes

### Qualifying Examination:

The examination for the probability area requirement will be based on the following list of topics, which are generally covered in PSTAT 120A, PSTAT 160 A-B, PSTAT 210, and PSTAT 213A-B-C

#### These topics include:

- Generating Functions
- Discrete time Markov Chains: Major examples – random walk 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: Major examples – Poisson process, birth-death processes. Backward and forward equations, generator, limiting behavior.
- Convergence of r.v.s.: Different types of convergence; ch. F., convergence in distribution, continuity theorem, LLN and CLT, some discussion of infinitely divisible and stable distributions, a.s. convergence and lemma Borel-Cantelli, convergence in probability, m.s./ Skorohod’s representation theorem, strong LLN; uniform integrability and $L^1$ convergence.
- Conditional Expectation, also with Hilbert spaces for $L^2$ variables.
- Martingales: Properties and examples, Doob and Doob-Meyer decomposition, martingale convergence theorems, stopping items, 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.

Additional topics as announced by the Qualifying Examination committee.

#### References for Probability and Stochastic Processes:

- Grimmett & Stirzaker, Probability and Random Processes
- Durrett, Probability: Theory and Example
- Resnick, Adventures in Stochastic Processes
- Resnick, A Probability Path
- Billingsley, Probability and Measure
- Breiman, Probability
- Jacod & Protter, Probability Essentials

## 3. Applied Statistics Area Requirements:

### Courses:

· PSTAT 220 A-B-C: Advanced Statistical Methods

### Qualifying Examination:

The examination for the applied statistics area requirement will be based on the following topics, which are generally covered by PSTAT 122, PSTAT 126, and PSTAT 220 A-B-C:

- Data summary exploratory data analysis
- linear models and analysis of variance
- regression and correlation
- experimental design
- generalized linear models including logistic regression and log linear models for contingency tables
- applications of regression methods in practical research
- multivariate methods

Additional topics as announced by the Qualifying Examination committee.

#### References for Applied Statistics include:

- Box, Hunter and Hunter, Statistics for Experimenters
- Cochran and Cox, Experimental Design
- Draper and Smith, Applied Regression Analysis
- McCullagh and Nelder, Generalized Linear Models. (2nd ed.)
- Johnson and Wichern, Applied Multivariate Statistical Analysis.
- Rao, Advanced Statistical Methods in Biometric Research
- Venables and Ripley, Modern Applied Statistics with S. (4th ed.)
- Yandell, Practical Data Analysis for Designed Experiments
- Neter, Wasserman & Kutner, Applied Linear Regression Models
- Faraway, Linear Models with R, 2nd Edition
- Faraway, Extending the Linear Model with R
- Seber and Lee,, Linear Regression Analysis
- Montgomery, Design and Analysis of Experiments

## 4. Pure Mathematics Area Requirements:

### Courses:

- MATH 201 A-B-C: Real Analysis
- MATH 202 A-B-C: Complex Analysis

### Qualifying Examination:

The Pure Mathematics qualifying examination is based on material found in the UCSB undergraduate courses MATH 118 A-B-C and MATH 122 A-B, and the UCSB graduate courses MATH 201 A-B-C and MATH 202 A-B-C. This exam is administered by the Mathematics Department, and students planning to take this qualifying examination should contact the Mathematics Department to enquire about any changes in the format of this examination or topics.

The examination consists of two parts;

- Topics in Real Analysis (The real number system, topology of R¬n, continuity, differentiability, Riemann integration, sequences and series, convergence processes including uniform convergence, functions of several variables, and introductions to metric spaces and to measure and integration).
- Topics in Complex Analysis (Complex numbers and functions, Cauchy integral theorem and Cauchy's integral formula and consequences, Residue calculus, elementary conformal maps, power series and Laurent series, elementary properties of analytical continuation, zeros and singularities of analytic functions).

### Reference Material (Real Analysis)

- Rudin, Principles of Mathematical Analysis, 3rd ed. (covers all topics and more). The exam excludes 7.28-7.33, 8.15-8.22, and all of Chapter 10. From Chapter 6 the exam will cover only ordinary Riemann integration, not the more general Riemann-Stieltjes integral.
- Andrew Browder, Mathematical Analysis (Chapters 1-10).
- Robert Strichartz, The Way of Analysis (excluding Chapters 11, 12 and 15).

### Reference Material (Complex Analysis)

- Brown & Churchill, Complex Variables and Applications, 6th ed. (Sections 1-80 cover these topics)
- Spiegel, Complex Variables, Schaum's Outline Series (covers all topics)
- Knopp and Konrad, Problem Book in the Theory of Functions, Vol. 1 & 2 (an old book, but still an excellent source for problems)