The following topics & references will prepare you for the exam.

Topics for Mathematical Statistics

- Classical parametric families of distributions. Moment generating functions. The distribution of functions of random variable, the Jacobian method.
- The Method of Moments: basic properties and asymptotics.
- Exponential families: canonical and curved. Canonical exponential families:; natural parameter space, identifiability, convexity. Location and scale families.
- Sufficient, minimal, ancillary, complete statistics. Factorization theorem, conditions for minimality or completeness.Sufficiency theory for canonical exponential families. Basu's theorem.
- Maximum Likelihood estimation: invariance of the ML estimator. General conditions for existence and uniqueness of the ML estimator. ML estimation for canonical exponential families.
- UMVU estimation: risk and loss functions, Information Inequalities. Fisher information, uniqueness of the UMVU estimator. Rao-Blackwellization of estimators. Risk-function-based optimality and sufficiency. Information inequalities for exponential families.
- Hypothesis Testing: simple and composite hypotheses, randomized and non-randomized tests, size, power, p-values. The Neyman-Pearson Lemma. Optimality of tests: UMP and UMPU tests. Generalized Likelihood Ratio tests. Asymptotics of the likelihood ratio tests.
- Asymptotics: Convergence in law and in probability, the Delta Method and Taylor approximations. Consistent roots for the likelihood equation. Asymptotic efficiency, asymptotics of the ML estimator for canonical exponential families.

**References**

1. Casella and Berger (2002), *Statistical Inference,* 2nd edition, Duxbury

2. Lehmann, E., and G. Casella (1998), *Theory of Point Estimation,* John Wiley and Sons, Inc., 2nd edition

Topics for Stochastic Processes

- Generating functions, Abel's theorem, linear difference equations
- Random walks: conditional distributions, gambler's ruin problem, hitting time probabilities, Reflection Principle, sojourn times and arcsine laws.
- Discrete-time Markov Chains: conditional distributions, transition matrices, Chapman-Kolmogorov equations, mean recurrence time. The classification of states (persistent, transient, positive, null; irreducibility, periodicity). Stopping times, the Strong Markov property. Stationary distributions and asymptotics. Reversibility. The Perron-Frobenius theorem.
- Birth processes and the Poisson process, the infintesimal approach.
- Continuous-time Markov Chains: conditional distributions, stochastic semigroups, rate matrices (generators), Kolmogorov's forward and backward equations. Levy's Dichomoty. The Ergodic theorem. Jumps chains, classification of states (instantaneous, stable, absorbing), distribution of holding times.
- Introduction to diffusions within the infinitesimal approach, the Ornstein-Uhlenbeck process, Brownian motion with drift. Gaussian processes. Brownian motion: basic properties, relation to random walks, distributional properties.

**References**

1. Grimmett, G. R., and D. R. Stirzaker (2001), *Probability and Random Processes,* Oxford Science Publications, 3rd edition.

2. Resnick, S. (1992), *Adventures in Stochastic Processes,* Birkhauser Boston

Topics for Linear Models

- Multiple and Simple Least Squares Linear Regression developed from Linear Algebra, with emphasis on the geometric interpretation and Perpendicular Projection Matrices, orthogonal matrices, t and F statistics from this viewpoint.
- Gauss Markov Theorm
- Sampling Distributions and Statistics Inference for Regression Parameters
- F tests for nested models and the general linear hypothesis
- Weighted Linear Regression
- One-way and Two-way ANOVA
- Identifiability and Estimability of Parameters
- Regression Diagnostics--residuals, plot and heteroscedasticity, leverage, outliers, influential points, Cook's distance
- Model Section and Multicollinearity--R squared, Mallows Cp, AIC, BIC, forward and backward and stepwise selection

**References**

1. Christensen, *Plane answers to Complex Questions*

2. Weisberg, *Applied Linear Regression*

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