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Advanced Methods for Statistical Research (Probability and Inference)

Course Code: IM 521  |  Credits: 5  |  Prerequisite: PG Probability Theory, PG Inference

Syllabus Details

  • Unit I:(Probability Theory) Independence, Borel-Cantelli lemma, Kolmogorov’s zero-one law, strong law of large numbers, multivariate central limit theorem, probability and conditional expectation, martingales.
  • Unit II:(Bayesian Approach) Comparison between Bayesian and frequentist inference, inference, Bayesian computation, hierarchical Bayes, empirical Bayes inference.
  • Unit III:(Asymptotics) Convergence of the posterior density and sample techniques: types of convergence, consistency, Lindeberg-Feller theorem, asymptotic normality, rates of convergence, delta method.
  • Unit IV:(Nonparametric Curve Estimation) Histogram and kernel density estimators, non-parametric likelihood, empirical likelihood, regression.
  • Unit V:(Advanced Inference) Locally most powerful and invariant tests, simultaneous inference, false discovery rate.

Course Learning Objective

  1. Discuss the importance of the multivariate central limit theorem, the strong law of large numbers and martingales.
  2. Discuss the comparison between Bayesian and frequentist inference.
  3. Discuss the convergence of the posterior density.
  4. Apply the concepts of non-parametric likelihood and empirical likelihood.
  1. Berger, James O. Statistical Decision Theory and Bayesian Analysis. Springer, 1985
  2. Carlin, Bradley P. and Louis, Thomas A. Bayes and Empirical Bayes Methods for Data Analysis. Chapman & Hall/CRC, 2000
  3. Cox, D. R. and Hinkley, D. V. Theoretical Statistics. Chapman and Hall, 1982
  4. Davison, A. C. and Hinkley, D. V. Bootstrap Methods and Their Applications. Cambridge University Press, 1997
  5. Efron, Bradley and Tibshirani, Robert J. An Introduction to the Bootstrap. Chapman & Hall, 1993
  6. Gelman, Andrew. Bayesian Data Analysis. CRC Press, 2004
  7. Ghosh, J. K., Delampady, Mohan and Samanta, Tapas. An Introduction to Bayesian Analysis: Theory and Methods. Springer, 2006
  8. Shao, Jun. Mathematical Statistics. 2nd ed. Springer, 2003
  9. Silverman, B. W. Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC, 1998
  10. Ripley, Brian D. Stochastic Simulation. Wiley, 1987
  11. van der Vaart, A. W. Asymptotic Statistics. Cambridge University Press, 1998
  12. McLachlan, Geoffrey J. and Krishnan, Thriyambakam. The EM Algorithm and Extensions. Wiley, 1997
  13. Tanner, Martin A. Tools for Statistical Inference. Springer, 1996
  14. Wasserman, Larry. All of Statistics. Springer, 2004
  15. DasGupta, Anirban. Asymptotic Theory of Statistics and Probability. Springer, 2008
  16. Scott, David W. Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley, 1992
  17. Durrett, Rick. Probability: Theory and Examples. Cambridge University Press, 2010
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