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Relative entropy measures of asymmetry with applications. (English) Zbl 1193.62003

Summary: We propose a measure of asymmetry of a probability density functions (pdf) by considering the relative entropy between itself and its (appropriately defined) mirror image. The measure is shown to be useful for detecting asymmetry in distributions of categorical or continuous random variables. Asymmetries of a pdf near its center and away from the center are investigated. This measure leads to generalizations of asymmetric categorical models. Comparisons (using examples) with the asymmetry measures of H.L. MacGillivray [Ann. Stat. 14, 994–1011 (1986; Zbl 0604.62011)] show that the proposed measures are useful for non-monotonic asymmetry. For square contingency tables with same row and column classifications, the sampling distributions of the measures are studied asymptotically. Applications are discussed for two-way tables and in linear regression models. Monte Carlo simulations show that the proposed measures/tests have good size and power properties when compared with competitors, even for smaller samples. Two illustrative examples are analyzed.

MSC:

62B10 Statistical aspects of information-theoretic topics
62E10 Characterization and structure theory of statistical distributions
62H17 Contingency tables
62J05 Linear regression; mixed models
65C05 Monte Carlo methods
94A17 Measures of information, entropy

Citations:

Zbl 0604.62011
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