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Maximum Likelihood Estimation of the Symmetric and Asymmetric Exponential Power Distribution

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  • Giulio Bottazzi
  • Angelo Secchi

Abstract

We introduce a new 5-parameter family of distributions, the Asymmetric Exponential Power (AEP), able to cope with asymmetries and leptokurtosis and at the same time allowing for a continuous variation from non-normality to normality. We prove that the Maximum Likelihood (ML) estimates of the AEP parameters are consistent on the whole parameter space, and when sufficiently large values of the shape parameters are considered, they are also asymptotically efficient and normal. We derive the Fisher information matrix for the AEP and we show that it can be continuously extended also to the region of small shape parameters. Through numerical simulations, we find that this extension can be used to obtain a reliable value for the errors associated to ML estimates also for samples of relatively small size ( 100 observations). Moreover we find that at this sample size, the bias associated with ML estimates, although present, becomes negligible.

Suggested Citation

  • Giulio Bottazzi & Angelo Secchi, 2006. "Maximum Likelihood Estimation of the Symmetric and Asymmetric Exponential Power Distribution," LEM Papers Series 2006/19, Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy.
  • Handle: RePEc:ssa:lemwps:2006/19
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    References listed on IDEAS

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    1. Newey, Whitney K. & McFadden, Daniel, 1986. "Large sample estimation and hypothesis testing," Handbook of Econometrics,in: R. F. Engle & D. McFadden (ed.), Handbook of Econometrics, edition 1, volume 4, chapter 36, pages 2111-2245 Elsevier.
    2. Giulio Bottazzi, 2004. "Subbotools User's Manual," LEM Papers Series 2004/14, Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy.
    3. DiCiccio T.J. & Monti A.C., 2004. "Inferential Aspects of the Skew Exponential Power Distribution," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 439-450, January.
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    Cited by:

    1. Ascari, Guido & Fagiolo, Giorgio & Roventini, Andrea, 2015. "Fat-Tail Distributions And Business-Cycle Models," Macroeconomic Dynamics, Cambridge University Press, vol. 19(02), pages 465-476, March.
    2. Giorgio Fagiolo & Mauro Napoletano & Marco Piazza & Andrea Roventini, 2009. "Detrending and the Distributional Properties of U.S. Output Time Series," Economics Bulletin, AccessEcon, vol. 29(4), pages 3155-3161.
    3. Giulio Bottazzi & Angelo Secchi & Federico Tamagni, 2014. "Financial constraints and firm dynamics," Small Business Economics, Springer, vol. 42(1), pages 99-116, January.
    4. Sandro Sapio, 2012. "Modeling the distribution of day-ahead electricity returns: a comparison," Quantitative Finance, Taylor & Francis Journals, vol. 12(12), pages 1935-1949, December.
    5. Giovanni Dosi & Marco Grazzi & Chiara Tomasi & Alessandro Zeli, 2010. "Turbulence underneath the big calm? Exploring the micro-evidence behind the flat trend of manufacturing productivity in Italy," LEM Papers Series 2010/03, Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy.
    6. Giorgio Fagiolo & Mauro Napoletano & Andrea Roventini, 2008. "Are output growth-rate distributions fat-tailed? some evidence from OECD countries," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 23(5), pages 639-669.
    7. Giorgio Fagiolo & Lucia Alessi & Matteo Barigozzi & Marco Capasso, 2010. "On the distributional properties of household consumption expenditures: the case of Italy," Empirical Economics, Springer, vol. 38(3), pages 717-741, June.
    8. Sean Holly & Emiliano Santoro, 2007. "Financial Fragility, Heterogeneous Firms and the Cross Section of the Business Cycle," Money Macro and Finance (MMF) Research Group Conference 2006 96, Money Macro and Finance Research Group.

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    Keywords

    Maximum Likelihood estimation; Asymmetric Exponential Power; Information matrix;

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