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Testing a differential condition and local normality of densities

Author

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  • Mynbayev, Kairat
  • Aipenova, Aziza

Abstract

In this paper, we consider testing if a density satisfies a differential equation. This result can be applied to see if a density belongs to a particular family of distributions. For example, the standard normal density f(t) satisfies the differential equation f'(t)+tf(t)=0. If a density satisfies this equation at that point t, then it is called locally standard normal at that point. Thus, there is a practical need to test whether a density satisfies a certain differential equation. We consider a more general differential equation F(x)=0 involving f(x). We can test the null hypothesis H0: f satisfies the equation F(x)=0 against the alternative hypothesis Ha: F(x)≠0. The testing procedure is accompanied by an asymptotic normality statement.

Suggested Citation

  • Mynbayev, Kairat & Aipenova, Aziza, 2013. "Testing a differential condition and local normality of densities," MPRA Paper 87045, University Library of Munich, Germany, revised 2014.
    • Handle: RePEc:pra:mprapa:87045
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    References listed on IDEAS

    as
    1. Davidson, James, 1994. "Stochastic Limit Theory: An Introduction for Econometricians," OUP Catalogue, Oxford University Press, number 9780198774037.
    2. Thorsten Thadewald & Herbert Buning, 2007. "Jarque-Bera Test and its Competitors for Testing Normality - A Power Comparison," Journal of Applied Statistics, Taylor & Francis Journals, vol. 34(1), pages 87-105.
    3. Szekely, Gábor J. & Rizzo, Maria L., 2005. "A new test for multivariate normality," Journal of Multivariate Analysis, Elsevier, vol. 93(1), pages 58-80, March.
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    JEL classification:

    • C14 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Semiparametric and Nonparametric Methods: General

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