Theoretical Statistics (8 ECTS)

Terminology and definition of basic introductory concepts of parametric statistical inference (random sample, sampling space, parametric space, sample distribution, estimating statistical function). Point estimation in decision making theory (loss function, risk function).  Criteria for estimator evaluation: Unbiasedness, Minimum Variance, Sufficiency, completeness, maximum Likelihood, efficiency. Methods of finding unbiased estimators of uniformly minimum variance. Fisher information, Cramer-Rao-Frechet inequality. The exponential family of distributions. Lehmann-Scheffe theorem. Maximum Likelihood Estimators (MLE). Invariance and asymptotic properties of the MLE. The concept of estimating parameters with confidence intervals. Methods of constructing confidence intervals. Pivotal quantity and the general method. Optimal confidence intervals. Asymptotic confidence intervals. Introduction to theory of parametric statistical hypothesis testing (defining the parametric hypothesis, types of errors, control function, power function). Evaluating statistical tests based on the power function. The Neyman-Pearson lemma and its applications in finding a uniformly powerful statistical test of simple hypotheses. Composite hypothesis testing. Likelihood Ratio test (LRT) and asymptotic LRT.

Recommended Reading

• Φερεντίνος Κ. και Παπαϊωάννου Τ. (2000) Μαθηματική Στατιστική, 2^η Έκδοση, Εκδόσεις Σταμούλη, Αθήνα.
• Κολυβά-Μαχαίρα Φ., Μαθηματική Στατιστική, Εκδόσεις Ζήτη, 1998.
• Φουσκάκης Δ., Ανάλυση Δεδομένων με τη Χρήση της R., Εκδόσεις Τσότρας, 2013.
• CrawleyM.J., Στατιστική Ανάλυση με το R., BrokenHillPublishers, 2013.
• Ρούσσας Γ. (1994) Στατιστική Συμπερασματολογία, Τόμος Ι - Εκτιμητική, 2^η Έκδοση, Εκδόσεις Ζήτη, Θεσσαλονίκη.
• Ρούσσας Γ. (1994) Στατιστική Συμπερασματολογία, Τόμος ΙΙ – Έλεγχοι Υποθέσεων, 2^η Έκδοση, Εκδόσεις Ζήτη, Θεσσαλονίκη.
• Bickel P.J. and Doksum K.A. (2007): Mathematical Statistics, vol.I, 2^nd Edition – Updated Printing, Pearson Prentice Hall.
• Casella G. and Berger R. (2002): Statistical Inference, 2^nd Edition, Duxbury.
• Mood A.M., Graybill F.A. and Boes D.C. (1974): Introduction to the Theory of Statistics, 3^rd Edition, McGraw-Hill Book Company.

The courses outline can be found here.