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Empirical behaviour of tests for the beta distribution and their application in environmental research

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Abstract

The beta distribution is used in different models of environmental research. The power of the test for beta distribution of Raschke [Biased transformation and its application in goodness-of-fit tests for the beta and gamma distribution. Commun. Statist. B–Computa. Simula. 38 (2009): 1870–1890] is researched here. The power of the Kolmogorov–Smirnov, Kuiper, Cramér-von Mises, Watson and Anderson–Darling tests are researched for different sample sizes, levels of significance and parameters of the beta distribution. The limitation to these tests is discussed including the differences between previous publications. The empirical behaviour is investigated by extensive Monte Carlo simulations. The most powerful test for the beta distribution is the Anderson–Darling test for the considered constellations of alternative distribution, contamination or scaling. The second best test is the Cramér-von Mises test, followed by the Watson test. The analysis of relative humidity data of meteorology and of runoff coefficients of the hydrology demonstrates the advantages of the new tests and the necessity to test an assumption of beta distribution.

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Authors and Affiliations

  1. Gustav-Freytag-Str.24, Leipzig, Germany

    Mathias Raschke

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  1. Mathias Raschke
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Correspondence to Mathias Raschke.

Appendix

Appendix

Data of relative humidity of the Haarweg-Wageningen weather station

May 2007: 0.4, 0.44, 0.5, 0.55, 0.58, 0.62, 0.65, 0.69, 0.72, 0.72, 0.73, 0.75, 0.77, 0.8, 0.81, 0.81, 0.83, 0.83, 0.85, 0.85, 0.85, 0.85, 0.86, 0.86, 0.87, 0.87, 0.89, 0.92, 0.94, 0.94, 0.97

May 2008: 0.39, 0.4, 0.42, 0.43, 0.43, 0.43, 0.44, 0.46, 0.48, 0.49, 0.51, 0.52, 0.53, 0.54, 0.56, 0.59, 0.62, 0.64, 0.66, 0.73, 0.75, 0.76, 0.83, 0.85, 0.88, 0.91, 0.92, 0.92, 0.95, 0.97, 0.98

See Tables 2, 3, 4, 5.

Table 3 Share of rejections, in %, of the different tests for different distributions for X, sample sizes n and levels of significance
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Table 4 Share of rejections, in %, of the different tests for contaminated samples
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Table 5 Share of rejections, in %, of the different tests for scaled beta distributed variable X with scale parameter s for different sample sizes n, parameters of Eq. 1 and levels of significance
Full size table

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Raschke, M. Empirical behaviour of tests for the beta distribution and their application in environmental research. Stoch Environ Res Risk Assess 25, 79–89 (2011). https://doi.org/10.1007/s00477-010-0410-3

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