Mathematical Statistics Topics
Cod: 23037
Department: DCET
ECTS: 10
Scientific area: Statistics
Total working hours: 260
Total contact time: 10
This learning unit aims to provide advanced knowledge and develop deep skills in statistical modeling and inference. It addresses three main components: the study of preliminary results and inferential notions; the development and analysis of multilinear regressions; and the study of simultaneous linear equations and structural equations.

Multilinear regressions
Simultaneous linear equations
Structural equations
- Know the concepts and inferential methods and to perform applications with the support of statistical software;
- Develop and apply multilinear regressions, namely to real datasets with computer support, identifying and interpreting optimal solutions for decision support;
- be familiar with the types of simultaneous linear equations and structural equations;
- Investigate new methodologies, as well as make improvements to the existing ones.
- Explore applications of these methodologies by using real datasets and taking advantage of computer support, developing and using appropriate software.

1 ) Preliminary results and advanced inferential notions: Orthogonal projections and linear equations, mean vectors and covariance matrices, moments generating functions , normal vectors, linear transformations and independence. Centered estimation , sufficient and complete statistics ; Theorems Rao- Blackwell and Blackwell- Lehman - Scheffé; Cramer - Rao Inequality , efficient estimators , estimable vectors ;
2 ) Advanced inferential notions: Techniques for point estimation , confidence intervals and hypothesis testing , including the study of the power of tests using resampling techniques; study of distributions associated with normal; numerical methods leading to optimization tasks of calculatiions - the QR method of Francis and method SVD ( Singular – Value Decomposition ) - stressing its relevance in the identification of parameters and models; statistical model selection:, Akaike ( AIC ) , Bayesian information criterion ( BIC), maximum likelihood ratio test; quality assessment of models - bootstrap and cross-validation .
3 ) Analysis of assumptions and development of multilinear regressions : Study tof the standard case of multilinear regression - adjustment and normality ; Gauss - Markov theorem ; multilinear regressions with exact linear restrictions .
4 ) Simultaneous linear equations and structural equations based on partial covariance structures and minimum parcial squares . The interaction in structural equation models.

- Hoyle, R.H. (2012). Handbook of Structural Equation Modeling. The Guilford Press.
- Rao, C. R. & Turtenburg, H. (1998) Linear Models: Least Squares and Alternatives 2nd ed Springer.
- Muller, K. E & Stevent, K. E. (2006) Linear Theory: Univariate, Multivariate and Mixed Models John Willey & Sons.
E-learning
Evaluation is made on individual basis and it involves the coexistence of two modes: continuous assessment (60%) and final evaluation (40%). Further information is detailed in the Learning Agreement of the course unit.
This Learning Unit will not be open in 2015/16.