Mathematical statistics (CEU), topics for the final (2016)

1. Statistical space, statistical sample. Basic statistics, empirical distribution function, Glivenko-Cantelli theorem. Ordered sample, Kolmogorov-Smirnov theorems.
2. Sufficiency, Neyman-Fisher factorization. Completeness, minimal sufficient statistics, exponential family.
3. Theory of point estimation: unbiased estimators, efficiency, consistency.
4. Fisher information (regularity conditions, additivity). Cramer-Rao inequality, Rao-Blackwell-Kolmogorov theorem..
5. Methods of point estimation: maximum likelihood estimation (asymptotic properties), method of moments, Bayes estimation. Interval estimation: confidence intervals (for the normal population mean).
6. Theory of hypothesis testing, UMP tests, Neyman-Pearson lemma for simple alternative and its extension to composite hypotheses.
7. Parametric inference: z, t, F, chi-square, Welch tests.
8. Nonparametric inference: chi-square, Kolmogorov-Smirnov tests.
9. Theory of least squares, regression analysis, correlation, Gauss-Markov theorem,

ML-estimation in linear models (with deterministic predictors and Gaussian error).