Statistics

This course aims to develop your understanding of the statistical concepts of parameter estimation and hypothesis testing that are fundamental for real life applications of statistics as well as for numerous further courses in the curriculum of the statistics track. Review of probability - PDF, CDF, PMF, mean, variance, moments - Common probability distributions - Upper percentage points of distributions - Moment generating functions - Probability distributions of functions of random variables - Distribution of maximum and minimum of random variables Random samples, sample mean and sample variance - Distributions derived from the normal distribution - Chi-Square distribution - t-distribution - F-distribution Central Limit Theorem and its significance for Statistics Parameter estimation - Introduction - Parameter estimation as part of statistical inference - Examples for the procedure of parameter estimation - Discrete and continuous parametric models Criteria for quality of estimators - Bias, standard error, mean squared error - Estimated standard error - Consistency of estimators Constructing good estimators - Method of moments - Maximum likelihood method Asymptotic properties of estimators - Review of Law of Large Numbers - Consistency of method of moments and maximum likelihood estimators - Fisher information - Asymptotic normality of maximum likelihood estimators - Cramer-Rao bound and efficient estimators Confidence intervals for estimators - Review of concept of confidence intervals - Large sample confidence intervals for maximum likelihood estimators - Pivotal quantities - Construction of exact confidence intervals - Asymptotically pivotal quantities - Construction of approximate confidence intervals Hypothesis testing - Purpose and philosophy of hypothesis testing - Role of hypothesis testing in statistical inference - Null hypothesis and its interpretation - Simple and composite hypotheses Fisher-type tests - p-values and critical values - Connection between confidence intervals and Fisher-type tests Neyman-Pearson tests - Alternative hypothesis - One-sided and two-sided tests and their p-values - Type I and type II errors - Power and size of a test - Constructing good tests: Neyman-Pearson lemma - Overview of frequently used tests