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Inverse Box-Cox: The power-normal distribution

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  • Freeman, Jade
  • Modarres, Reza

Abstract

Box-Cox transformation system produces the power normal (PN) family, whose members include normal and lognormal distributions. We study the moments of PN and obtain expressions for its mean and variance. The quantile functions and a quantile measure of skewness are discussed to show that the PN family is ordered with respect to the transformation parameter. Chebyshev-Hermite polynomials are used to show that the correlation coefficient is smaller in the PN scale than the original scale. We use the Fréchet bounds to obtain expressions for the lower and upper bounds of the correlation coefficient. A numerical routine is used to compute the bounds. The transformation parameter of the PN family is used to investigate the effects of model uncertainty on the upper quantile estimates.

Suggested Citation

  • Freeman, Jade & Modarres, Reza, 2006. "Inverse Box-Cox: The power-normal distribution," Statistics & Probability Letters, Elsevier, vol. 76(8), pages 764-772, April.
    • Handle: RePEc:eee:stapro:v:76:y:2006:i:8:p:764-772
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    References listed on IDEAS

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    1. Charles N. Haas, 1997. "Importance of Distributional Form in Characterizing Inputs to Monte Carlo Risk Assessments," Risk Analysis, John Wiley & Sons, vol. 17(1), pages 107-113, February.
    2. Modarres, Reza & Nayak, Tapan K. & Gastwirth, Joseph L., 2002. "Estimation of upper quantiles under model and parameter uncertainty," Computational Statistics & Data Analysis, Elsevier, vol. 39(4), pages 529-554, June.
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    Cited by:

    1. Proietti, Tommaso & Lütkepohl, Helmut, 2013. "Does the Box–Cox transformation help in forecasting macroeconomic time series?," International Journal of Forecasting, Elsevier, vol. 29(1), pages 88-99.
    2. Klein, Ingo, 2012. "Quasi-arithmetische Mittelwerte und Normalverteilung," Discussion Papers 89/2010, Friedrich-Alexander University Erlangen-Nuremberg, Chair of Statistics and Econometrics.
    3. Kleijnen, J.P.C., 2007. "Simulation Experiments in Practice : Statistical Design and Regression Analysis," Discussion Paper 2007-09, Tilburg University, Center for Economic Research.
    4. Kleijnen, J.P.C., 2006. "White Noise Assumptions Revisited : Regression Models and Statistical Designs for Simulation Practice," Discussion Paper 2006-50, Tilburg University, Center for Economic Research.
    5. Xurxo Rigueira & María Araújo & Javier Martínez & Paulino José García-Nieto & Iago Ocarranza, 2022. "Functional Data Analysis for the Detection of Outliers and Study of the Effects of the COVID-19 Pandemic on Air Quality: A Case Study in Gijón, Spain," Mathematics, MDPI, vol. 10(14), pages 1-27, July.
    6. Tingguo Zheng & Tao Song, 2014. "A Realized Stochastic Volatility Model With Box-Cox Transformation," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(4), pages 593-605, October.
    7. Javier Martínez Torres & Jorge Pastor Pérez & Joaquín Sancho Val & Aonghus McNabola & Miguel Martínez Comesaña & John Gallagher, 2020. "A Functional Data Analysis Approach for the Detection of Air Pollution Episodes and Outliers: A Case Study in Dublin, Ireland," Mathematics, MDPI, vol. 8(2), pages 1-19, February.
    8. Tzong-Ru Tsai & Yuhlong Lio & Ya-Yen Fan & Che-Pin Cheng, 2022. "Bias Correction Method for Log-Power-Normal Distribution," Mathematics, MDPI, vol. 10(6), pages 1-19, March.
    9. Daniel A. Griffith, 2022. "Reciprocal Data Transformations and Their Back-Transforms," Stats, MDPI, vol. 5(3), pages 1-24, July.

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