I Section (3 credits):
Complex numbers: trigonometric form, powers, roots and complex exponential. Linear algebra: vectors, matrices, vector spaces, bases, basis changes. Linear applications and linear systems: Kernel and Image. Theorems of Sylvester, Cramer, Rouchè-Capelli. Eigenvalues and eigenvectors, characteristic polynomial. Diagonalizable matrices. Symmetric matrices.
II Section (3 credits):
Functions of a vector variable: limits, continuity, differential calculus: differentiable functions, partial and directional derivatives, Taylor polynomial. Implicit functions. Free and constrained maximization, under equality and inequality constraints. Conditions of the I and II order, quadratic forms. Kuhn-Tucker theorem.
Random variables, expected value, variance. Random vectors, mean vector and variance-covariance matrix.
Parametric statistical model. Point estimators, finite sample properties (unbiasedness, mean squared error, efficiency) and large sample properties (asymptotic unbiasedness, consistency, asymptotic efficiency).
Likelihood function, maximum likelihood estimators and their properties. Score and Fisher information. Fisher scoring algorithm.
Confidence interval estimators, pivotal quantity, interval estimators based on the asymptotic properties of maximum likelihood estimators.
Testing hypotheses: test statistic, power function, type I and II errors, uniformly most powerful tests, consistency. P.value. Generalized likelihood ratio test, score test, Wald test.
Exponential family of distribution, properties of expectation and variance.
Generalized linear models, maximum likelihood estimation of model parameters, hypothesis testing for model parameters. Deviance, testing model goodness of fit.
Normal linear model (multiple linear regression, analysis of variance, general linear model).
Logistic regression, Poisson regression.
Generalized linear models will be fitted to dataset using the R-environment (R Core Team, 2019).