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# 1One- and two-sample location problems, tests for symmetry and tests on a single distribution

## 1.1 Introduction

Many real phenomena can be represented by numerical random variables. Considering a given population and a random sample of it, for forecasting or improving the effectiveness of inferential techniques related to estimation and testing of hypothesis, it would be useful to know the functional form of the distribution of the data. Sometimes, the central interest of the statistical analysis is focused only on the symmetry or on the location of the distribution itself. Another very common statistical problem consists of comparing two independent populations in terms of central tendency. In the simpler cases the object of the analysis is a univariate population, but in some real applications we are in the presence of many variables and multivariate datasets.

The methods presented in this chapter consist of rank or permutation procedures for the tests of the hypotheses cited above. Section 1.2 is an introduction to rank and permutation tests. In Section 1.3, devoted to one-sample tests, the Kolmogorov procedure for testing whether the data are distributed according to an hypothesized cumulative distribution function (CDF), and the permutation test on the symmetry of the distribution are taken into account. Section 1.4 deals with multivariate one-sample tests, and introduces the multivariate location problem and the multivariate test on symmetry. In Section 1.5 the univariate ...

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