This is a daily summary of the topics that will be ( were) covered in each lecture. The notation WN.M dd.mm means: Week N, Lecture M. day. month. Our course will be following mainly two textbooks:
G. Casella and R. L. Berger. Statistical inference. Cengage Learning, 2021.
[ROH] Rohatgi, Vijay K. Statistical inference. Courier Corporation, 2013
The following books offer additional resources for study and learning:
D. J. Olive. Statistical theory and inference. Springer, 2014. (The university gives you free access to this book via https://link.springer.com )
[ISL]G. James, D. Witten, T. Hastie, R. Tibshirani. An Introduction to Statistical Learning with Applications in R.
[ESL]T. Hastie, R. Tibshirani, J. Fiedman. The Elements of Statistical Learning, Data mining, Inference and Prediction.
[LEH ]Lehmann, E. L., Romano, J. P., & Casella, G. (2005). Testing statistical hypotheses (Vol. 3). New York: springer.
W1.1 15.09 (Alexander) Presentation, bibliography and introduction to the syllabus of the course.
W1.2 17.09 (Alexander) Random elements: distribution function, random vectors, conditional probability and independence, different types of random variables, types of random vectors. Casella: Chapter 1: All sections, Chapter 2: 2.1, 2.2, 2.3,
W2.3 22.09 (Alexander) Expectation: expected value, variance, covariance, alternative formulae. Common distributions: Binomial, poisson, multinomial, normal, Exponential, Gamma. Characteristic functions. Casella: Chapter 3, 3.1 Discrete Distributions, 3.2 Continuous Distributions. Exercise 8 and 11 form worksheet 1.
W2.4 24.09 (Alexander) Notions of convergence: Almost surely convergence, convergence in L^\infty, convergence in probability, convergence in p-th mean, convergence in, convergence in distribution. Casella: Chapter 5: 5.5
W3.5 29.09 (Alexander) Laws of large numbers. Central limit theorem. Exercises 1 and 5 from worksheet 2.
W3.6 01.10 (Alexander) Hypothesis testing. Methodologies procedures and main concepts. Errors of type I and II. Power function. Significance level and p-value. Paired sample sign test.
W4.7 06.10 (Alexander) Sign test, Wilcoxon test, Wilcoxon-Mann-Whitney test. Exercises 9 (solved manually and with R), 1 and 5 (with R) from worksheet 3.
W4.8 08.10 (Alexander) Empirical distribution function. Notion of consistent estimator. Glivenko-Cantelli Theorem. Confidence intervals for the underlying distribution function. Kolmogorov-Smirnov test. Exercise 1 (using R) from worksheet 4. Confidence intervals for population quantiles based on order statistics. Goodness of fit test of normality: the Lilliefors-test.
W5.9 13.10 (Alexander) The Chi^2 test for goodness of fit. Exercises 2, 4 (using R) and 6 (manually and using R) from worksheet 4.
W5.10 15.10 (Alexander) The Chi^2 test for goodness of fit of absolutely continuous distributions and parametric families. The Chi^2 test of independence for contingency tables.
W6.11 20.10 (Alexander) Pearson's correlation to test the association of a bivariate normal vector. The Spearman's rank correlation to test the association between two random variables. Beginning of sufficient statistics, examples.
W6.12 22.10 (Alexander)
W7. No Classes Week of October 27th, 30th.
W8.13 03.11 Def. 5.2.1 (statistic, sample distribution), Def. 5.2.2., 5.2.3 (definition of sample mean and sample variance), Theorem 5.2.6 (expected value and variance of the sample mean, expected value of the sample variance). Maximum Likelihood estimations: Def. 6.3.1 (likelihood function). Def. 7.1.1 (point estimator). Def. 7.2.4 (maximum likelihood estimator). Definition of maximum likelihood estimator, examples of computation of MLE's. Finding method of moments estimators, difference between the method of moments and MLE. Example of method moments for a mixture model.
W8.14 05.11 (Eliana) How to compare estimators. Definition of MSE, bias and variance trade off. Parametric Hypothesis Testing. Likelihood ratio tests Casella: 8.2.1, 8.3.1. Likelihood ratio tests Example 8.2.2, 8.2.5, 8.2.3. Power function of a test. Theorem 8.2.4. Exercises Week 10 Likelihood ratio tests, restricted and unrestricted MLEs. Example 8.3.3. Power function, controlling Type I and Type II error for a test. Choice of the rejection region with a prescribed significance level. Asymptotic distribution of for the likelihood ratio.
W9.15 10.11 (Eliana) Introduction to Bayesian statistics. Prior distribution, posterior distribution. Example of Bayes rule for updating probabilities. Prior predictive distribution, posterior predictive distribution
W9.16 12.11 (Eliana) Example of Bayesian workflow through the binomial example. Estimating the probability of a female birth in the population. Uniform prior and conjugate beta prior to the binomial.
W10.17 17.11 (Eliana) Construction of the normal model with unknown mean and variance. Models with more than one paramater. Nuissance parameters, marginal posterior distribution. Improper priors, uninformative priors. Normal model with uninformative prior distribution. Solution of exercise 2 in the worksheet.
W10.18 19.11 (Eliana) Examples on how to obtain posterior estimates through simulation for the binomial model and the univariate normal model with the two parameters unknown. Example of probability of birth being female given placenta previa and measuring speed of light.
This is the last class for students with 6ECTS
W11.19 24.11 (Eliana) Def, 6.2.1 (sufficient statistic). Theorem 6.2.2. Example 6.2.4. Theorem 6.2.6 (Factorization Theorem). Def. 6.2.11 minimal sufficient statistic. Theorem 6.2.13 (Lehman, Scheffé 1950). Example 6.2.14. Exponential families.
W11.20 25.11 (Eliana)
W12.-- 01.12 No class
W11.21 03.12 (Eliana)
W12.-- 08.12 (Eliana) No class
W12.22 10.12 (Eliana) Section 3.4. Example 3.4.3, Example 3.4.4. Reparametrization of the exponential family Equation (3.4.7) Theorem 5.2.11. (pmf of a sample from an exponential family). Def. 3.4.7 (curved exponential family), Example 3.4.9, Theorem 6.2.10 (sufficient statistics for exponential family). Def. 6.2.21 (complete statistics), Example 6.2.22. Theorem 6.2.25.
W14.23 14.12 (Eliana) Neyman-Pearson UMP Lemma, Examples with joint pdf and with sufficient statistic. Karlin-Rubin Theorem for UMP one-sided hypothesis testing with a sufficient statistic. Summary of parametric hypothesis testing. Beginning of Uniformly Most Powerful tests.
W14.24 15.12 (Eliana) Example of maximum likelihood estimation for exponential families. Theorem about the sufficient statistics that characterize the MLE of the exponential family. Example for MLE in simple linear regression, and multiple linear regression. Ridge regression [ESL, 3.2.2,3.4.1], [ISL, 6.2.1 6.1.1] and Gauss-Markov Theorem. A Modern Gauss-Markov Theorem .
W12.25 17.12 (Eliana) Uniform minimum variance unbiased estimators, Cramer-Rao Lower Bound. 7.3.2, Example 7.3.8. Theorem 7.3.9. Lemma 7.3.11.