Abstract
Being able to formally test for symmetry hypotheses is an important topic in many fields, including environmental and physical sciences. In this paper, one concentrates on a large family of nonparametric tests of symmetry based on Cramér–von Mises statistics computed from empirical distribution and characteristic functions. These tests possess the highly desirable property of being universally consistent in the sense that they detect any kind of departure from symmetry as the sample size becomes large. The asymptotic behaviour of these test statistics under symmetry is deduced from the theory of first-order degenerate V-statistics. The issue of computing valid p-values is tackled using the multiplier bootstrap method suitably adapted to V-statistics, yielding elegant, easy-to-compute and quick procedures for testing symmetry. A special focus is put on tests of univariate symmetry, bivariate exchangeability and reflected symmetry; a simulation study indicates the good sampling properties of these tests. Finally, a framework for testing general symmetry hypotheses is introduced.
Highlights
In many scientific fields, a natural or experimentally-controlled phenomenon is observed and a dataset is collected
While this paper focuses on the two above-mentioned notions of bivariate symmetry, other definitions have been proposed, e.g., joint symmetry and spherical symmetry
This paper focuses on consistent nonparametric tests of symmetry based on Cramér–von Mises functionals of empirical distribution and characteristic functions
Summary
A natural or experimentally-controlled phenomenon is observed and a dataset is collected. This paper focuses on consistent nonparametric tests of symmetry based on Cramér–von Mises functionals of empirical distribution and characteristic functions. These tests are attractive since they do not require any assumptions on the form of the underlying distribution and provide universally-consistent procedures. As will be seen, these test statistics for symmetry can be expressed as V-statistics This representation allows for the derivation of their asymptotic behaviour and, most importantly, suggests a resampling method based on the multiplier bootstrap for the computation of p-values. Describe a general family of Cramér–von Mises test statistics for symmetry hypotheses based on empirical distributions and characteristic functions.
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