Theoretical Statistics (8 ECTS)

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. The Fisher information metric, the Cramer-Rao-Frechet inequality for the variance of unbiased point estimators. Exponential parametric group of distributions. The Lehmann-Scheffe theorem. Methods of finding unbiased estimators of uniformly minimum variance.  Maximum Likelihood Estimators (MLE). Invariance, asymptotic properties of consistency and normality, examples of estimating MLE. The concept of estimating parameters with confidence intervals and the Pivotal Quantity concept. Methods of estimating the appropriate pivotal quantity to construct a confidence interval. Confidence interval optimization. Constructing confidence intervals using the general method. Approximate 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. Asymptotic or approximate tests.

      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.
• Crawley M.J., Στατιστική Ανάλυση με το R., Broken Hill Publishers, 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.