Nonparametric Theory

Explore nonparametric statistical theory through a comprehensive series of lectures covering a wide range of topics. Delve into summation formulas for integers, prove the sign test's power, study derivatives of inverse trigonometric functions, and analyze probability distributions. Examine the Wilcoxon signed rank test, U statistics, and sample variance formulas. Investigate normal scores statistics, empirical CDFs, and probability integral transformations. Learn about Kolmogorov-Smirnov tests, correlation between variate values and ranks, and multivariate sign tests. Discover Spearman's rank correlation, Kendall's tau, and the distribution of sample medians and ranges. Gain practical experience using R for statistical analysis and visualization throughout the course.

Summation Formulas for Integers.
Prove the Sign Test is a most powerful test.
Derivatives of the inverse trigonometric functions.
Probability of 1st Quadrant for a Standard Bivariate Normal Random Variable.
Wilcoxon Signed Rank Distribution (Part 1).
Wilcoxon Signed Rank Distribution (Part 2).
Wilcoxon Signed Rank Test Statistic define as # of positive Walsh Averages.
Mean and Variance of Wilcoxon Signed Rank Test Statistics.
Interesting Formula for the Sample Variance. U Statistics.
Variance of the sample variance.
Projection method to show normality.
Integrating an inverse cdf of a normal distribution. Normal Scores Statistic.
Covariance of Empirical CDF.
Probability Integral Transformation.
Show that the Kolmogorov-Smirnov One Sample Test is Distribution Free.
Correlation between variate values and corresponding ranks (part 1).
Correlation between variate values and corresponding ranks (part 2).
Correlation between variate values and corresponding ranks (part 3): Illustration Using R..
Probability of 4th Quadrant for a Standard Bivariate Normal Random Variable.
The Univariate Sign Test.
The Multivariate Sign Test.
Probability of 1st Quadrant for a Scaled Bivariate Normal Random Variable.
Using R: The Multivariate Sign Test.
Power and Sample Size in R: Multivariate Sign Test.
Spearman's Rank Correlation.
Kendall's Tau Rank Correlation.
Distribution of the Sample Median.
CI for Population Median using Order Statistics.
Distribution of the Sample Range.
Discrete Order Statistics with Illustration using R.
Integrating over a Rectangle in Polar Coordinates.