# One-Sample and Two-Sample Problems

• Mayer Alvo
• Philip L. H. Yu
Chapter
Part of the Springer Series in the Data Sciences book series (SSDS)

## Abstract

In this chapter we consider several one- and two-sample problems in nonparametric statistics. Our approach will have a common thread. We begin by embedding the nonparametric problem into a parametric paradigm. This is then followed by deriving the score test statistic and finding its asymptotic distribution. The construction of the parametric paradigm often involves the use of composite likelihood. It will then be necessary to rely on the use of either linear rank statistics or U-statistics in order to determine the asymptotic distribution of the test statistic. We shall see that the parametric paradigm provides new insights into well-known problems. Starting with the sign test, we show that the parametric paradigm deals easily with the case of ties. We then proceed with the Wilcoxon signed rank statistic and the Wilcoxon rank sum statistic for the two-sample problem.

## References

1. Ansari, A. R. and Bradley, R. A. (1960). Rank-sum tests for dispersions. Annals of Mathematical Statistics, 31(4):1174–1189.
2. Daniel, W. W. (1990). Applied Nonparametric Statistics. Duxbury, Wadsworth Inc., second edition.Google Scholar
3. Fraser, D. (1957). Non Parametric Methods in Statistics. John Wiley and Sons., New York.Google Scholar
4. Gibbons, J. D. and Chakraborti, S. (2011). Nonparametric Statistical Inference. Chapman Hall, New York, 5th edition.
5. Hájek, J. and Sidak, Z. (1967). Theory of Rank Tests. Academic Press, New York.
6. Lehmann, E. (1975). Nonparametrics: Statistical Methods Based on Ranks. McGraw-Hill, New York.
7. Lindley, D. V. and Scott, W. F. (1995). New Cambridge Statistical Tables. Cambridge University Press, 2nd edition.Google Scholar
8. Mielke, Paul W., J. and Berry, Kenneth, J. (2001). Permutation Methods: A Distance Function Approach. Springer.Google Scholar
9. Siegel, S. and Tukey, J. W. (1960). A nonparametric sum of ranks procedure for relative spread in unpaired samples. Journal of the American Statistical Association, 55:429–445.
10. van der Vaart, A. (2007). Asymptotic Statistics. Cambridge University Press.Google Scholar

© Springer Nature Switzerland AG 2018

## Authors and Affiliations

• Mayer Alvo
• 1
• Philip L. H. Yu
• 2
1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
2. 2.Department of Statistics and Actuarial ScienceUniversity of Hong KongHong KongChina