Maximum Likelihood Estimation of Asymmetric Double Type II Pareto Distributions
Tóm tắt
This paper considers a flexible class of asymmetric double Pareto distributions (ADP) that allows for skewness and asymmetric heavy tails. The inference problem is examined for maximum likelihood. Consistency is proven for the general case when all parameters are unknown. After deriving the Fisher information matrix, asymptotic normality and efficiency are established for a restricted model with the location parameter known. The asymptotic properties of the estimators are then examined using Monte Carlo simulations. To assess its goodness of fit, the ADP is applied to companies’ growth rates, for which it is favored over competing models.
Tài liệu tham khảo
Wang H et al (2012) Bayesian graphical lasso models and efficient posterior computation. Bayesian Anal 7(4):867–886
Armagan A, Dunson DB, Lee J (2013) Generalized double Pareto shrinkage. Stat Sin 23(1):119
Nadarajah S, Afuecheta E, Chan S (2013) A double generalized Pareto distribution. Stat Prob Lett 83(12):2656–2663
Papastathopoulos I, Tawn JA (2013) A generalised students t-distribution. Stat Probab Lett 83(1):70–77
Kotz S, Kozubowski T, Podgorski K (2012) The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance. Springer, Berlin
Punathumparambath B, Kulathinal S, George S (2012) Asymmetric type II compound laplace distribution and its application to microarray gene expression. Comput Stat Data Anal 56(6):1396–1404
Mantegna RN, Stanley HE (2007) Introduction to econophysics. Cambridge University Press, Cambridge
Fagiolo G, Napoletano M, Roventini A (2008) Are output growth-rate distributions fat-tailed? some evidence from OECD countries. J Appl Econom 23(5):639–669
Stanley M, Amaral L, Buldyrev S, Havlin S, Leschhorn H, Maass P, Salinger M, Stanley H (1996) Scaling behaviour in the growth of companies. Nature 379(6568):804–806
Komunjer I (2007) Asymmetric power distribution: theory and applications to risk measurement. J Appl Econom 22(5):891–921
Bottazzi G, Coad A, Jacoby N, Secchi A (2011) Corporate growth and industrial dynamics: evidence from French manufacturing. Appl Econ 43(1):103–116
Holly S, Petrella I, Santoro E (2013) Aggregate fluctuations and the cross-sectional dynamics of firm growth. J R Stat Soc Ser A (Stat Soc) 176(2):459–479
Zolotarev VM (1986) One-dimensional stable distributions, vol 65. American Mathematical Society, Providence
Kozubowski TJ, Rachev ST (1999) Univariate geometric stable laws. J Comput Anal Appl 1(2):177–217
Mandelbrot BB (1997) The variation of certain speculative prices. Fractals and scaling in finance. Springer, Berlin, pp 371–418
Fama EF (1965) The behavior of stock-market prices. J Bus 38(1):34–105
Mittnik S, Rachev ST, Paolella MS (1998) Stable Paretian modeling in finance: some empirical and theoretical aspects. In: Adler RJ, Feldman RE, Taqqu MS (eds) A practical guide to heavy tails: statistical techniques and applications. Birkhauser Boston Inc., Cambridge, pp 79–110
Fu D, Pammolli F, Buldyrev S, Riccaboni M, Matia K, Yamasaki K, Stanley H (2005) The growth of business firms: theoretical framework and empirical evidence. Proc Natl Acad Sci USA 102(52):18801
Clauset A, Shalizi C, Newman M (2009) Power-law distributions in empirical data. SIAM Rev 51(4):661–703
Alfarano S, Milakovic M, Irle A, Kauschke J (2012) A statistical equilibrium model of competitive firms. J Econ Dyn Control 36(1):136–149
Smith RL (1984) Threshold methods for sample extremes. In: de Oliveira J (ed) Statistical extremes and applications. Springer, New York, pp 621–638
Hosking JR, Wallis JR (1987) Parameter and quantile estimation for the generalized Pareto distribution. Technometrics 29(3):339–349
Ashkar F, Tatsambon CN (2007) Revisiting some estimation methods for the generalized Pareto distribution. J Hydrol 346(3–4):136–143
de Zea Bermudez P, Kotz S (2010) Parameter estimation of the generalized Pareto distribution—part I. J Stat Plan Inference 140(6):1353–1373
de Zea Bermudez P, Kotz S (2010) Parameter estimation of the generalized Pareto distribution—part II. J Stat Plan Inference 140(6):1374–1388
Singh V, Guo H (1995) Parameter estimation for 3-parameter generalized Pareto distribution by the principle of maximum entropy (POME). Hydrol Sci J 40(2):165–181
Newey WK, McFadden D (1994) Large sample estimation and hypothesis testing. In: Engle R, McFadden D (eds) Handbook of econometrics, vol 4. Elsevier, Amsterdam, pp 2111–2245
Lehmann E, Casella G (1998) Theory of point estimation. Springer, Berlin
Bottazzi G, Secchi A (2006) Explaining the distribution of firm growth rates. RAND J Econ 37(2):235–256
Axtell R (2001) Zipf distribution of US firm sizes. Science 293(5536):1818
Schwarz G et al (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464
Bottazzi G, Secchi A (2011) A new class of asymmetric exponential power densities with applications to economics and finance. Ind Corp Change 20(4):991–1030
Agro G (1995) Maximum likelihood estimation for the exponential power function parameters. Commun Stat Simul Comput 24(2):523–536