Methods of Non-Parametric
Statistics
Lecture Notes
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\[\frac {12}{N(N+1)}\sum ^k_{i=1}\frac {1}{n_i} \Bigg [R_i-\frac {n_i(N+1)}{2}\Bigg ]^2 =\frac {12}{N(N+1)}\sum ^k_{i=1}\frac {R^2_{i}}{n_i}-3(N+1)\]
Contents
1 Introduction
1.1 Parametric Inference, Recalled
1.2 When Parametric Inference Will Not Do
1.3 Population and Sample
1.4 Hypothesis Tests, Recalled
1.5 The Rationale of \(H_0\) and \(\alpha \)
1.6 Types of Hypotheses
1.7 A First Non-Parametric Test
1.8 Practice Problems
2 Some Tests Based on the Binomial Distribution
2.1 Sign Test
2.2 Confidence Interval for the Median
2.3 Order Statistics
2.4 Quantile Functions
2.5 Percentile and Quartile
2.6 Quartile Test
2.7 Practice Problems
3 Some Tests Based on Ranks
3.1 Ranks
3.2 Mid Ranks
3.3 The Wilcoxon Signed-Rank Test
3.4 Zeros and Ties in the Signed-Rank Test
3.5 General Two Sample Problem
3.6 The Wilcoxon- Mann-Whitney Test
3.7 Wilcoxon Rank Sum
3.8 Wilcoxon- Mann- Whitney Test
3.9 Ties in the Rank-Sum Test
3.10 Practice Problems
4 Tests for Three or More Samples
4.1 Kruskal-Wallis Test
4.2 Assumption of Kruskal-Wallis Test
4.3 The Friedman Test
4.4 Multiple Comparisons
4.5 Practice Problems
5 Non-Parametric Measures of Correlation
5.1 Association and Correlation
5.2 Rank correlation
5.3 Spearmans Rank Correlation
5.4 Computing the Coefficient \((r_s)\) Spearman correlation
5.5 Tied Ranks
5.6 Tests
5.7 Kendall’s Tau
5.8 Practice Problems
6 Kolmogorov-Smirnov (K-S) Test
6.1 The K-S One Sample Test (goodness-of-fit test)
6.2 Rationale Of K-S
6.3 Two Sample Test
6.4 The K Sample K-S Statistic
6.5 Practice Problems
7 Two Further Tests
7.1 The Runs Test for Randomness
7.2 The Median Test
7.3 Practice Problems
8 Non-Parametric Methods for Trend
8.1 The Mann–Kendall Test
8.2 The Variance of \(S\), With Ties
8.3 The Theil–Sen Slope
8.4 Serial Correlation, and Why It Breaks the Test
8.5 A Warning Not Found in the Textbooks
8.6 Testing Many Series at Once
8.7 Seasonality
8.8 Practice Problems
1 Introduction
1.1 Parametric Inference, Recalled
1.2 When Parametric Inference Will Not Do
1.3 Population and Sample
1.4 Hypothesis Tests, Recalled
1.5 The Rationale of \(H_0\) and \(\alpha \)
1.6 Types of Hypotheses
1.7 A First Non-Parametric Test
1.8 Practice Problems
2 Some Tests Based on the Binomial Distribution
2.1 Sign Test
2.2 Confidence Interval for the Median
2.3 Order Statistics
2.4 Quantile Functions
2.5 Percentile and Quartile
2.6 Quartile Test
2.7 Practice Problems
3 Some Tests Based on Ranks
3.1 Ranks
3.2 Mid Ranks
3.3 The Wilcoxon Signed-Rank Test
3.4 Zeros and Ties in the Signed-Rank Test
3.5 General Two Sample Problem
3.6 The Wilcoxon- Mann-Whitney Test
3.7 Wilcoxon Rank Sum
3.8 Wilcoxon- Mann- Whitney Test
3.9 Ties in the Rank-Sum Test
3.10 Practice Problems
4 Tests for Three or More Samples
4.1 Kruskal-Wallis Test
4.2 Assumption of Kruskal-Wallis Test
4.3 The Friedman Test
4.4 Multiple Comparisons
4.5 Practice Problems
5 Non-Parametric Measures of Correlation
5.1 Association and Correlation
5.2 Rank correlation
5.3 Spearmans Rank Correlation
5.4 Computing the Coefficient \((r_s)\) Spearman correlation
5.5 Tied Ranks
5.6 Tests
5.7 Kendall’s Tau
5.8 Practice Problems
6 Kolmogorov-Smirnov (K-S) Test
6.1 The K-S One Sample Test (goodness-of-fit test)
6.2 Rationale Of K-S
6.3 Two Sample Test
6.4 The K Sample K-S Statistic
6.5 Practice Problems
7 Two Further Tests
7.1 The Runs Test for Randomness
7.2 The Median Test
7.3 Practice Problems
8 Non-Parametric Methods for Trend
8.1 The Mann–Kendall Test
8.2 The Variance of \(S\), With Ties
8.3 The Theil–Sen Slope
8.4 Serial Correlation, and Why It Breaks the Test
8.5 A Warning Not Found in the Textbooks
8.6 Testing Many Series at Once
8.7 Seasonality
8.8 Practice Problems