## Math 462 & Stat 462 at Lehigh University

Math 462/Stat 462 (Nonparametric Statistics): Order and rank statistics; tests based on runs, signs, ranks, and order statistics; chi-squared and Kolmogorov-Smirnov tests for goodness-of-fit; the two-sample; confidence and tolerance intervals. Prerequisite: math231 or Math 309. (Semiparametric modeling and resampling methods (e.g. bootstrap) are also included in the courses.)

Here is a typical course description of Nonparametric Statistics at Lehigh University.

Textbooks used for Math/Stat 462 ( Nonparametric Statistics ) at Lehigh University:

•  "Theory of Rank Tests", by Hajek and Sidak
•  "Nonparametric Statistics: methods based on ranks", by E. Lehmann
•  "Distribution-Free Statistical Methods", 2nd edition, by J. S. Maritz.
•  "An Introduction to the Bootstrap", by B. Efron and Tibshirani.

Currently we have covered the following topics:

• 1. Introduction to One-Sample Statistical Problems.
• 2. Parametric procedures for one-sample problems: t-test, t-CI, paired t-test, MLE, etc.
• 3. What is Nonparametric Statistics?
• 4. Some nonparametric procedures for one-sample problems.
• 5. Signed-test for paired sample and binomial distribution.
• 6. Wilcoxon Signed-Rank test for paired sample.
• 7. The exact null distribution of Wilcoxon signed-rank statistic.
• 8. Normal approximation to null distribution of Wilcoxon statistic.
• 9. Sampling from finite populations.
• 10. The concept of scoring (Wilcoxon scores, normal scores, ....).
• 11. A class of linear rank statistics.
• 12. Confidence intervals based on Wilcoxon signed-rank statistic and Sign statistic.
• 13. Walsh's averages; Hodges-Lehmann estimator and its standard deviation.
• 14. Statistical functionals and estimating equations.
• 15. A general formula for determining the 1st order approximation of the s.d. of an estimator.
• 16. Difficulties in finding analytical forms of s.d.'s for complexed estimators.
• 17. Some resampling methods: permutation based, jackknife based, bootstrap based.
• 18. Analytical bootstrap vs simulation based bootstrap.
• 19. Comparison of Hodges-Lehmann estimator and sample median (when estimating the center of symmetry) under contaminated normal models.
• 20. Efficacies for Wilcoxon's Signed-Rank based, Sign based, Mean based and MLE based statistical procedures.
• 21. Local asymptotic powers and its relationship to asymptotic efficacies.
• 22. Estimation of CDF (Parametric vs Nonparametric).
• 23. Empirical CDF, Empirical process and Brownian Bridge Process.
• 24. Empirical CDF Based Statistical Functionals, Weak Convergence, Exact vs Asymptotic Distributions.

•

• A Casebook in Statistics and Data Analysis

•