Qualification Exams

Master' Degree:

The student must pass the Qualification Exam  conducted within 12 months from the date of enrollment. The Exam shall consist of two written tests: Probability (about the contents of the course MI401) and Statistical Inference (about the contents of the course MI402).Each one of these tests shall be held twice a year on dates defined by the SCPGE.

MI401 - Probability
T:60 E:30 L:0 S:0 C:6 P:3
Program: Probability spaces. Discrete and Continuous random variables. Conditional distribution. Conditional expectation. Generator functions. Convergence of random variables. Inequalities. Law of large numbers. Central limit theorem.

Bibliography: Grimmett, G.R. e Stirzaker, D.R., Probability and Random Processes. Oxford Science Publications, James, B. R. Probabilidade: um curso a nível intermediário. Projeto Euclides. IMPA.

MI402 - Statistical Inference
T:60 E:30 L:0 S:0 C:6 P:2
Program: Statistical models. Statistics and parameters. Sufficiency. Exponential family. Estimation methods: method of moments, method of least squares and maximum likelihood. Comparison of estimators: principle of optimality, minimum-variance unbiased estimators, inequality of information. Confidence intervals and hypothesis tests. Likelihood-ratio-test. Optimal tests. Neyman-Pearson lemma. Introduction to decision theory. Notions in Bayesian procedures.

Bibliography: Rohatagi, V.K., An Introduction to Probability Theory and Mathematical Statistics., John Wiley, New York, 1976. Casella, G. e Berger, R.L, Statistical Inference, Wadsworth & Brooks, California, 1990.


The Ph.D. Qualification Exam shall consist of two steps. The first one shall consist of a written test, about the contents of the courses Intermediate Probability (MI659)  and Advanced Inference (MI677).This test will be held once or twice a year, on dates defined by the SCPGE. In this first step, the student must obtain approval within 24 months from his/her enrollment. The second step of the Qualification Exam will consist of an oral dissertation about a specific subject of the student's thesis. Regarding this second step, the student must obtain approval within 36 months from his enrollment.

MI659 - Intermediate Probability
T:60 E:0 L:0 S:0 C:4 P:3
Program: Probabilistic Model for an experiment. Conditional Probability. Independence. Random variables and distribution functions. Random variables types. The distribution of a random variable. Random vectors. Independence. Distributions of variable functions and random vectors. The Jacobian method. Notions in Stieltjes Integral. Expectation. Expectation properties. Expectation of random variable functions. Moments. Expectations of random vectors. Convergence theorems. Conditional distribution of discrete random variables. Conditional distribution of random variables: general case. Formal definitions and existence theorems. Conditional expectation. Introduction to weak and strong large number law. Event sequences and Borel-Cantelli Lemma. The strong law. Characteristic functions. Convergence in distribution. Characteristic function of a random vector. Central Limit Theorem for sequences of random variables. Multivariate normal distribution. Central Limit Theory: multivariate case.

MI677 - Advanced Inference
T:60 E:0 L:0 S:0 C:4 P:3
Program: Statistical models. The statistical problem and decision theory. Statistical information on classical and Bayesian approach. Elements of estimation theory: non-biased estimators, likelihood based estimators, M-estimators, estimators by moments method, estimation with equality restrictions, minimax and Bayesian estimators, numerical procedures. Estimation by confidence intervals. Hypothesis test: asymptotic tests, relation with confidence intervals, estimation and tests with inequality relations, tests for non-fitting hypothesis, Bayesian tests.

Bibliography: Gourieroux, C. e Monfort, A. 1992, Statistics and Econometric Models, V1 e V2, Cambridge university Press, Lehmann, E. L. (1959), Testing Statistical Hypotheses, J. Wiley & Sons. DeGroot, M. H. (1970), Optimal Statistical Decisions, McGraw Hill. Lehmann, E. L. (1983), Theory of Point Estimation, J. Wiley.

Previous Qualifications Exams


2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2015 2016

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