Theoretical Statistics (7 ECTS)

Basic Concepts and Terminology

Fundamental concepts and terminology of parametric statistical inference: random sample, sample space, parameter space, sampling distribution, and statistical function.

Point Estimation

Point estimation within the framework of statistical decision theory. Loss and risk functions. Criteria for evaluating estimators: unbiasedness, minimum variance, sufficiency, completeness, maximum likelihood, and efficiency. Methods for constructing uniformly minimum variance unbiased estimators (UMVUEs). Fisher information Cramer-Rao-Frechet inequality. The exponential family of distributions. The Lehmann–Scheffé theorem.

Maximum Likelihood Estimation

Maximum likelihood estimation (MLE). Invariance properties of MLEs. Asymptotic properties of maximum likelihood estimators.

Confidence Interval Estimation

The concept and interpretation of confidence intervals for parameter estimation. Methods for constructing confidence intervals. Pivotal quantities and the general pivotal method. Optimal confidence intervals. Asymptotic confidence intervals.

Parametric Hypothesis Testing

Introduction to the theory of parametric statistical hypothesis testing. Formulation of parametric hypotheses. Types of errors, test functions, and power functions. Evaluation and comparison of statistical tests based on their power functions. The Neyman–Pearson lemma and its applications to the construction of uniformly most powerful tests for simple and one-sided hypotheses. Composite hypothesis testing.

Likelihood Ratio Testing

The likelihood ratio test (LRT). Construction and properties of likelihood ratio tests. Asymptotic likelihood ratio tests and their applications.

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.