MATH 6070

Textbook: No textbook

Prerequisite: Math 5010, 5080, 5090

Office Hour: Tuesday 12:00 or by appointment

Outline of the Course:
  1. Elements of probability (3 lectures): Distribution and density functions, moment generating function, characteristic function, stable distribution, multivariate normal density, convergence of random variables, Slutsky's theorem, uni- and multivariate central limit theorems, rate of convergence in the central limit theorem (Berry Esséen theorem, Edgeworth expansions), conditional expected value.

  2. Estimation in parametric models (3 lectures): Method of moments, likelihood method, least-squares, least-absolute deviations, complete and sufficient statistics, variance reduction, uniformly minimum variance unbiased estimator, exponential family, information number, Cramér-Rao inequality, asymptotic properties of the maximum likelihood estimator, efficiency, super efficiency, sequential methods.

  3. Hypothesis testing in parametric models (2 lectures): Errors, power funtion, Neyman-Pearson lemma, randomization, likelihood method, asymptotics for the likelihood ratio, X2-test, multinomial models, sequential methods.

  4. Nonparametric methods (3 lectures): Probability integral transformation, order statistics, empirical distribution and quantile functions, elements of stochastic processes, Kolmogorov-Smirnov, Cramér-von Mises, Anderson-Darling and Kuiper statistics, comparison of distribution functions, power of tests, alternatives.

  5. More on nonparametric methods (3 lectures): M-statistics, L-statistics, U-statistics, rank statistics.

  6. Some important tests (3 lectures): Hypothesis testing, general methods, X2-tests, parameter estimated processes, tests for exponentiality, Total Time on Test, tests for normality (Shapiro-Wilks and Shapiro-Francia), comparison of moments, tests for multivariate normality.

  7. Resampling methods (2 lectures): Jackknife, bootstrap, randomization.

  8. Curve estimation (2 lectures): Empirical distribution function, confidence bands, smoothing, density estimation (histogram, kernel, orthogonal expansions), choice of the smoothing parameter, nonparametric regression, splines, weighted least-squares, local smoothing.

  9. Analysis of incomplete data (2 lectures): Censoring, product-limit estimator, confidence bands, tests in censorship models, proprotional hazards, Cox model.

  10. Elements of decision theory (1 lecture): Censoring, product-limit estimator confidence bands, tests in censorship mod Loss function, decision rule, risks, convex sets, Bayes method, minimax method.
Midterms: Midterms will be the 7th and 14th weeks of the semester

Final: The final will be during the last week of the semester

Computation of the final grade:
Theoretical Problems  10%
Applied Problems      30%
Midterms              30%
Final                 30%

A : 91-100     C : 60-64
A-: 85-90      C-: 55-59
B+: 80-84      D+: 50-54
B : 75-79      D : 40-49
B-: 70-74      D-: 24-39
C+: 65-69      E : 0-24


Homework:

Each Tuesday (Thursday, if there is no class on Tuesday) you must submit the solution of one problem from the list of theoretical problems for that week's topic. No credit for late solutions.
Theoretical Problems Topic 1
Theoretical Problems Topic 2
Theoretical Problems Topic 3
Theoretical Problems Topic 4
Theoretical Problems Topic 5
Theoretical Problems Topic 6
Theoretical Problems Topic 7
Theoretical Problems Topic 8
Theoretical Problems Topic 9

Applied Problem 1
Applied Problem 2   water.dat
Applied Problem 3   sound.dat
Applied Problem 4   myoinositol.dat
You should use statistical packages (SAS, C+ or similar) to solve the applied problems. It is assumed that you are familiar with at least one package. The solutions must be submitted as reports. This means that they are typed, the problems and the solutions are clearly explained. All figures and tables must be done on the computer.

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