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Maximum Likelihood Estimation of the Symmetric and Asymmetric Exponential Power Distribution

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Giulio Bottazzi
Angelo Secchi

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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.

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Paper provided by Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy in its series LEM Papers Series with number 2006/19.

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Date of creation: 29 Aug 2006
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Handle: RePEc:ssa:lemwps:2006/19

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Keywords: Maximum Likelihood estimation; Asymmetric Exponential Power; Information matrix;

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  1. 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. [Downloadable!]
  2. 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. [Downloadable!] (restricted)
  3. 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. [Downloadable!] (restricted)
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  1. 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. [Downloadable!]
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  2. Giorgio Fagiolo & Mauro Napoletano & Marco Piazza & Andrea Roventini, 2009. "Detrending and the Distributional Properties of U.S. Output Time Series," LEM Papers Series 2009/14, Laboratory of Economics and Management (LEM), Sant'Anna School of Advanced Studies, Pisa, Italy. [Downloadable!]
  3. Holly, S. & Santoro, E., 2008. "Financial Fragility, Heterogeneous Firms and the Cross Section of the Business Cycle," Cambridge Working Papers in Economics 0846, Faculty of Economics, University of Cambridge. [Downloadable!]
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