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Statistics II PROFESSOR DIEGO ATTILIO MANCUSO; PROFESSOR SILVIA FACCHINETTI COURSE AIMS To introduce the foundations of statistical inference, which are necessary in order to make decisions on the basis of sample information. COURSE CONTENT MODULE I: Probability theory topics – Overview and additions Probability spaces. Random variables in one and several dimensions. Jensen's inequality and the Chebyshev–Markov inequality. Sequences of random variables: convergence and law of large numbers. Transformations and convolutions of random variables. Stochastic Processes. – Review of random variables discrete: binomial, hypergeometric and Poisson; geometric and Pascal; binomial-Poisson mixture; Poisson stochastic process. continuous: negative exponential and Erlang; Gamma and Beta; Poissongamma and beta-binomial mixtures; normal distribution in p dimensions; lognormal and Pareto distributions. truncated distributions. – Completion of probability theory Moment-generating functions and cumulants. Central limit theorem. Delta method. MODULE II: Mathematical statistics topics – Order Statistics. Cochran's theorem. – Sampling and random variable samples: Student's t and Fisher's F random variables. – Statistic. Likelihood function. Sufficient, minimal sufficient and complete statistics. Exponential family. – Point estimation. Properties of estimators. Method for determining estimators: method of moments and maximum likelihood method. Sufficient statistics and minimum variance estimators. – Interval estimation. Construction of confidence intervals. – Hypothesis testing. Neyman-Pearson lemma. UMP tests. Tests based on likelihood ratio. Significance tests. Regression analysis topics – Analysis of variance with criterion of classification. – Overview of the regression function: descriptive aspects. – Regression functions for bivariate normal random variables. – Analysis of simple regression. Gauss-Markov theorem. Maximum likelihood estimates for the normal regression model, and related tests. Estimating the conditional expectation. Calculating the prediction interval. – Simple regression with errors in both variables. – Multiple regression analysis: (a) using the ordinary least squares method, (b) using the weighted least squares method. The problem of multicollinearity. READING LIST Suggested reading Module I G. CICCHITELLI, Probabilità e statistica, Maggioli editore, Rimini, 2001. D. ZAPPA - S. FACCHINETTI, Appunti di Statistica II, EDUCatt, Milan, 2013. B.V. FROSINI, Complementi sulle variabili casuali, EDUCatt, Milan, 2012. Module II D. ZAPPA - S. FACCHINETTI, Appunti di Statistica II, EDUCatt, Milan, 2013. B.V. FROSINI, Analisi di regressione, EDUCatt, Milan, 2011. TEACHING METHOD Lectures. ASSESSMENT METHOD Written examination. NOTES Further information can be found on the lecturer's webpage http://www2.unicatt.it/unicattolica/docenti/index.html or on the Faculty notice board. at