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Smoothed nonparametric two-sample tests

Taku Moriyama, Yoshihiko Maesono

Abstract

We propose new smoothed median and the Wilcoxon's rank sum test. As is pointed out by Maesono et al.(2016), some nonparametric discrete tests have a problem with their significance probability. Because of this problem, the selection of the median and the Wilcoxon's test can be biased too, however, we show new smoothed tests are free from the problem. Significance probabilities and local asymptotic powers of the new tests are studied, and we show that they inherit good properties of the discrete tests.

Smoothed nonparametric two-sample tests

Abstract

We propose new smoothed median and the Wilcoxon's rank sum test. As is pointed out by Maesono et al.(2016), some nonparametric discrete tests have a problem with their significance probability. Because of this problem, the selection of the median and the Wilcoxon's test can be biased too, however, we show new smoothed tests are free from the problem. Significance probabilities and local asymptotic powers of the new tests are studied, and we show that they inherit good properties of the discrete tests.

Paper Structure

This paper contains 11 sections, 7 theorems, 85 equations, 10 tables.

Table of Contents

  1. The median and the Wilcoxon's rank sum test
  2. The median and the Wilcoxon's rank sum test
  3. Local power
  4. Significance probability
  5. Smoothed median test
  6. Smoothed median test
  7. Asymptotic properties
  8. Smoothed Wilcoxon's rank sum test
  9. Smoothed Wilcoxon's rank sum test
  10. Simulation study
  11. Appendices

Key Result

Lemma 1

The difference of the distribution of $Z$ between $N=\widetilde{N}$ and $N=\widetilde{N}+1$ is

Theorems & Definitions (10)

  • Remark 1
  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 2
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Remark 3