. – Mathematical statistics II
PROFESSOR LUCIO BERTOLI BARSOTTI
Text under revision. Not yet approved by academic staff.
COURSE AIMS
To introduce the main paradigms of statistical inference, with particular attention
to point and interval estimation and hypothesis tests.
COURSE CONTENT
- Multivariate random variables. Bivariate random variables, discrete and
continuous. Marginal. Conditional. Correlation and independence. Conditional
expected value. Regression function. Multinomial random variable. Bivariate
normal random variable. Multivariate normal.
- Transformations of multivariate random variables. General case. Sum. Linear
combinations of the components of a multivariate normal.
- Parametric families: location-scale family and exponential family.
- Sampling and random variables. Sample space. Determination of the distribution
function of a random variable when using simple random sampling. Likelihood
function.
- Statistical information. Overview. Sample average and variance. Exact and
approximate distributions of sample moments. Fisher information. Cramér–Rao
information inequality. Sufficiency and ancillarity. Neyman Fisher factorization
criterion. Subordinate and equivalent statistics. Minimal sufficiency.
Completeness. Sufficient statistics and the exponential family. Rao-Blackwell
theorem.
- Parametric estimation. Identifiability. Estimator and estimation. Comparison of
estimators based on the mean square error. Unbiasedness, consistency, efficiency
and asymptotic efficiency.
- Estimation methods. Method of moments. Maximum likelihood method (ML).
ML estimation in the exponential family case. Asymptotic optimality of ML
estimators.
- Confidence intervals (CIs). Construction in the general case. Pivotal quantity
method. Calculation of minimum sample size needed for a certain CI size.
Applications to cases of determination of exact CIs in normal sampling.
Asymptotic confidence intervals: applications to cases of continuous and discrete
random variables. Determination of exact CIs in the case of discrete random
variables: general procedure. Confidence intervals for comparing two normal
populations on the basis of two simple random samples of different sizes:
confidence interval for the difference in means and confidence interval for the
comparison of variances.
- Statistical hypothesis testing theory. Exact parametric tests for the mean and
variance in normal distribution. Asymptotic tests for parameters in non-normal
models. Nonparametric tests: Kolmogorov-Smirnov test. Chi-squared test. Rank
tests.
READING LIST
L. BERTOLI - BARSOTTI, Corso di Statistica Matematica, Quaderni del Dipartimento di Matematica,
Statistica, Informatica e Applicazioni Università di Bergamo, Serie Didattica, n. 3, 2005.
A.M. MOOD - F.A. GRAYBILL - D.C. BOES, Introduzione alla Statistica, Mc-Graw-Hill, 1991.
N. WEISS, Calcolo delle Probabilità, Pearson PBM, 2008.
L. PACE, A. SALVAN, Introduzione alla Statistica Vol. 2 – Inferenza, Verosimiglianza, Modelli,
Cedam, 2001.
PELOSI, SANDIFER, CERCHIELLO, GIUDICI, Introduzione alla Statistica 2nd edition, Mc-Graw-Hill,
2009.
TEACHING METHOD
Theory lessons plus a cycle of exercises (first unit only).
ASSESSMENT METHOD
Preliminary written examination followed by an oral 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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