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Power transformations to induce normality and their applications

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  • Willa W. Chen
  • Rohit S. Deo

Abstract

Random variables which are positive linear combinations of positive independent random variables can have heavily right-skewed finite sample distributions even though they might be asymptotically normally distributed. We provide a simple method of determining an appropriate power transformation to improve the normal approximation in small samples. Our method contains the Wilson-Hilferty cube root transformation for "χ"-super-2 random variables as a special case. We also provide some important examples, including test statistics of goodness-of-fit and tail index estimators, where such power transformations can be applied. In particular, we study the small sample behaviour of two goodness-of-fit tests for time series models which have been proposed recently in the literature. Both tests are generalizations of the popular Box-Ljung-Pierce portmanteau test, one in the time domain and the other in the frequency domain. A power transformation with a finite sample mean and variance correction is proposed, which ameliorates the small sample effect. It is found that the corrected versions of the tests have markedly better size properties. The correction is also found to result in an overall increase in power which can be significant under certain alternatives. Furthermore, the corrected tests also have better power than the Box-Ljung-Pierce portmanteau test, unlike the uncorrected versions. Copyright 2004 Royal Statistical Society.

Suggested Citation

  • Willa W. Chen & Rohit S. Deo, 2004. "Power transformations to induce normality and their applications," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 66(1), pages 117-130.
  • Handle: RePEc:bla:jorssb:v:66:y:2004:i:1:p:117-130
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    File URL: http://www.blackwell-synergy.com/doi/abs/10.1111/j.1467-9868.2004.00435.x
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    Cited by:

    1. Marc Hallin & Davy Paindaveine & Miroslav Siman, 2008. "Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth," Working Papers ECARES 2008_042, ULB -- Universite Libre de Bruxelles.
    2. Zhu, Fukang & Wang, Dehui, 2010. "Diagnostic checking integer-valued ARCH(p) models using conditional residual autocorrelations," Computational Statistics & Data Analysis, Elsevier, vol. 54(2), pages 496-508, February.
    3. Kajal Lahiri & Huaming Peng & Xuguang Sheng, 2015. "Measuring Uncertainty of a Combined Forecast and Some Tests for Forecaster Heterogeneity," CESifo Working Paper Series 5468, CESifo Group Munich.
    4. Julien Chevallier & Benoît Sévi, 2011. "On the realized volatility of the ECX CO 2 emissions 2008 futures contract: distribution, dynamics and forecasting," Annals of Finance, Springer, vol. 7(1), pages 1-29, February.
    5. Daniel Preve & Anders Eriksson & Jun Yu, 2009. "Forecasting Realized Volatility Using A Nonnegative Semiparametric Model," Finance Working Papers 23049, East Asian Bureau of Economic Research.
    6. Proietti, Tommaso & Luati, Alessandra, 2015. "The generalised autocovariance function," Journal of Econometrics, Elsevier, vol. 186(1), pages 245-257.
    7. Amélie Charles & Olivier Darné, 2009. "Variance-Ratio Tests Of Random Walk: An Overview," Journal of Economic Surveys, Wiley Blackwell, vol. 23(3), pages 503-527, July.
    8. Poulin, Jennifer & Duchesne, Pierre, 2008. "On the power transformation of kernel-based tests for serial correlation in vector time series: Some finite sample results and a comparison with the bootstrap," Computational Statistics & Data Analysis, Elsevier, vol. 52(9), pages 4432-4457, May.
    9. repec:eee:ecolet:v:155:y:2017:i:c:p:104-110 is not listed on IDEAS
    10. Gonçalves, Sílvia & Meddahi, Nour, 2011. "Box-Cox transforms for realized volatility," Journal of Econometrics, Elsevier, vol. 160(1), pages 129-144, January.
    11. repec:dau:papers:123456789/4598 is not listed on IDEAS

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