Flexible statistical models: Methods for the ordering and comparison of theoretical distributions

Rigby, Robert and Stasinopoulos, Dimitrios and Voudouris, Vlasios (2015): Flexible statistical models: Methods for the ordering and comparison of theoretical distributions.

Abstract

Statistical models usually rely on the assumption that the shape of the distribution is fixed and that it is only the mean and volatility that varies. Although the fitting of heavy tail distributions has become easier due to computational advances, the fitting of the appropriate heavy tail distribution requires knowledge of the properties of the different theoretical distributions. The selection of the appropriate theoretical distribution is not trivial. Therefore, this paper provides methods for the ordering and comparison of continuous distributions by making a threefold contribution. Firstly, it provides an ordering of the heaviness of distribution tails of continuous distributions. The resulting classification of over 30 important distributions is given. Secondly it provides guidance on choosing the appropriate tail for a given variable. As an example, we use the USA box-office revenues, an industry characterised by extreme events affecting the supply schedule of the films, to illustrate how the theoretical distribution could be selected. Finally, since moment based measures may not exist or may be unreliable, the paper uses centile based measures of skewness and kurtosis to compare distributions. The paper therefore makes a substantial methodological contribution towards the development of conditional densities for statistical model in the presence of heavy tails.

• Item Type: MPRA Paper
• Language: English
• Keywords: centile measures, heavy tails, distributions
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; C - Mathematical and Quantitative Methods > C4 - Econometric and Statistical Methods: Special Topics > C46 - Specific Distributions ; Specific Statistics

References

Ali, M. (1974). Stochastic ordering and kurtosis measure. Journal of the American Statistical Association, 69:543–545.

Andrews, D., Bickel, P., Hampel, F., Huber, P., Rogers, W., and Tukey, J. (1972). Robust estimation of location: Survey and advances. Technical report, Princeton University Press, Princeton, NJ.

Bahadur, R. and Savage, L. (1956). The non-existence of certain statistical procedures in non- the non-existence of certain statistical procedures in nonparametric problems. Annals of Statistics, 27:1115–1122.

Balanda, K. P. and MacGillivray, H. L. (1988). Kurtosis: A critical review. The American Statistician, 42:111–119.

Cole, T. J. and Green, P. J. (1992). Smoothing reference centile curves: the lms method and penalized likelihood. Statistics in Medicine., 11:1305–1319.

Crowder, M. J., Kimber, A. C., Smith R. L. and Sweeting, T. J. (1991). Statistical Analysis of Reliability Data. Chapman and Hall, London.

Davidson, R. (2012). Statistical inference in the presence of heavy tails. The Econometrics Journal, 15:31–53.

Dunn, P. K. and Smyth, G. K. (1996). Randomised quantile residuals. J. Comput. Graph. Statist., 5:236–244.

Fernandez, C. and Steel, M. F. J. (1998). On bayesian modelling of fat tails and skewness. J. Am. Statist. Ass., 93:359–371.

Fernandez, C., Osiewalski, J. and Steel, M. J. F. (1995). Modeling and inference with v-spherical distributions. J. Am. Statist. Ass., 90:1331–1340.

Haavelmo, T. (1943). The statistical implications of a system of simultaneous equations. Econometrica, 11(1):1–12.

Haavelmo, T. (1944). The probability approach in econometrics. Econometrica, 12:1–118.

Harter, H. L. (1967). Maximum-likelihood estimation of the parameters of a four parame- ter generalized gamma population from complete and censored samples. Technometrics, 9:159–165.

Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist., 3:1163–1174.

Johnson, N. L. (1949). Systems of frequency curves generated by methods of translation. Biometrika, 36:149–176.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions, Volume I, 2nd edn. Wiley, New York.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995). Continuous Univariate Distributions, Volume II, 2nd edn. Wiley, New York.

Jones, M. C. (2005). In discussion of Rigby, R. A. and Stasinopoulos, D. M. (2005) Generalized additive models for location, scale and shape,. Applied Statistics, 54:507–554.

Jones, M. C. and Pewsey, A. (2009). Sinh-arcsinh distributions. Biometrika, 96:761–780.

Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution, Lecture Notes in Statistics No.9. Springer-Verlag, New York.

Jørgensen, B. (1997). The Theory of Dispersion Models. Chapman and Hall: London.

Lopatatzidis, A. and Green, P. J. (2000). Nonparametric quantile regression using the gamma distribution. Private Communication.

MacGillivray, H. (1986). Skewness and asymmetry: measures and orderings. Annals of Statistics, 14:994–1011.

Mandelbrot, B. (1997). Fractals and scaling in finance: discontinuity, concentration, risk:selecta volume E. Springer Verlag.

McDonald, J. B. (1991). Parametric models for partially adaptive estimation with skewed and leptokurtic residuals. Economic Letters, 37:273–278.

McDonald, J. B. (1996). Probability distributions for financial models. In Maddala, G. S. and Rao, C. R., editors, Handbook of Statistics, Vol. 14, pages 427–460. Elsevier Science.

McDonald, J. B. and Newey, W. K. (1988). Partially adaptive estimation of regression models via the generalized t distribution. Econometric Theory, 4:428–457.

McDonald, J. B. and Xu, Y. J. (1995). A generalisation of the beta distribution with applications. Journal of Econometrics, 66:133–152.

Nelson, D. B. (1991). Conditional heteroskedasticity in asset returns: a new approach. Econometrica, 59:347–370.

Nolan, J. P. (2012). Stable Distributions - Models for Heavy Tailed Data. Birkhauser, Boston. In progress, Chapter 1 online at academic2.american.edu/∼jpnolan.

Pokorny, M. and Sedgwick, J. (2010). Profitability trends in Hollywood: 1929 to 1999: somebody must know something. Economic History Review, 63:56–84.

Rigby, R. A. and Stasinopoulos, D. M. (2004). Smooth centile curves for skew and kurtotic data modelled using the Box-Cox power exponential distribution. Statistics in Medicine, 23:3053–3076.

Rigby, R. A. and Stasinopoulos, D. M. (2005). Generalized additive models for location, scale and shape, (with discussion). Appl. Statist., 54:507–554.

Rigby, R. A. and Stasinopoulos, D. M. (2006). Using the Box-Cox t distribution in gamlss to model skewness and kurtosis. Statistical Modelling, 6:209–229.

Rosenberger, J. and Gasko, M. (1983). Comparing location estimators: Trimmed means, medians and trimean. In Hoaglin, D., Mosteller, F., and Tukey, J., editors, Understanding Robust and Exploratory Data Analysis, pages 297–338. John Wiley, New York.

Stasinopoulos, D. M., Rigby, R. A. and Akantziliotou, C. (2008). Instructions on how to use the gamlss package in r, second edition. Technical Report 01/08, STORM Research Centre, London Metropolitan University, London.

Voudouris, V. Gilchristand, R., Rigby, R., Sedgwick, J., and Stasinopoulos, D. (2012). Modelling skewness and kurtosis with the bcpe density in gamlss. Journal of Applied Statistics.

Yee, T. W. and Wild, C. J. (1996). Vector generalized additive models. Journal of the Royal Statistical Society, Series B (Methodological), 58(3):481–493.