Resampling Techniques

Course Code: IM 523; ST 812 | Credits: 5 | Prerequisite: None

Unit I:(Review) Statistical model, parameters, estimates and statistics; sampling distribution; point and interval estimates; estimate of the mean; pivot; t-statistic; confidence interval; maximum likelihood estimate; normal distribution; the central limit theorem and its ramifications; measures of accuracy of point and interval estimates; normal approximation and its limitations; shortcomings of analytic derivations, with examples; estimates of the distribution function: parametric and non-parametric MLE.
Unit II:(Jackknife and Bootstrap) Purpose of resampling, with examples of estimating the variance, the sampling distribution and other features of a statistic; jackknife: bias reduction, estimation of variance, delete-1 and delete-d jackknives; bootstrap: parametric and non-parametric bootstrap, estimation of variance and of the distribution function.
Unit III:(Consistency and Comparisons) Comparison between bootstrap approximation and normal approximation; notions of variance consistency and distributional consistency; jackknife distributional inconsistency; bootstrap distributional consistency; comparison between bootstrap and jackknife; resampling in non-i.i.d. models and the need for other resampling schemes.
Unit IV:(Resampling in Linear Models) Residual bootstrap and weighted bootstrap, concept of robust and efficient resampling schemes; introduction to estimating equations and the generalized bootstrap.

Discuss the importance of the sampling distribution and of point and interval estimates.
Discuss the importance of the normal approximation and its limitations.
Discuss the purpose of resampling.
Apply jackknife and bootstrap methods, including resampling in non-i.i.d. models and in linear models.

Bose, Arup and Chatterjee, Snigdhansu. U-Statistics, Mm-Estimators and Resampling. Texts and Readings in Mathematics 75. Hindustan Book Agency, 2018
Davison, A. C. and Hinkley, D. V. Bootstrap Methods and Their Applications. Cambridge University Press, 1997
Efron, Bradley. The Jackknife, the Bootstrap and Other Resampling Plans. CBMS-NSF Regional Conference Series in Applied Mathematics 38, 1982
Efron, Bradley and Tibshirani, Robert J. An Introduction to the Bootstrap. Chapman & Hall/CRC, 1993
Shao, Jun and Tu, Dongsheng. The Jackknife and Bootstrap. Springer, 1995
Barbe, Philippe and Bertail, Patrice. The Weighted Bootstrap. Lecture Notes in Statistics 98, 1995
Gine, Evarist. Lectures on Some Aspects of the Bootstrap. Lecture Notes in Mathematics 1665. Springer, 1997
Hall, Peter. The Bootstrap and Edgeworth Expansion. Springer, 1992
Mammen, Enno. When Does Bootstrap Work? Asymptotic Results and Simulations. Lecture Notes in Statistics 77. Springer, 1992