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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.
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