Chalabi, Yohan / Y. and Scott, David J and Wuertz, Diethelm (2012): An asymmetry-steepness parameterization of the generalized lambda distribution.
Download (759Kb) | Preview
The generalized lambda distribution (GLD) is a versatile distribution that can accommodate a wide range of shapes, including fat-tailed and asymmetric distributions. It is defined by its quantile function. We introduce a more intuitive parameterization of the GLD that expresses the location and scale parameters directly as the median and inter-quartile range of the distribution. The remaining two shape parameters characterize the asymmetry and steepness of the distribution respectively. This is in contrasts to the previous parameterizations where the asymmetry and steepness are described by the combination of the two tail indices. The estimation of the GLD parameters is notoriously difficult. With our parameterization, the fitting of the GLD to empirical data can be reduced to a two-parameter estimation problem where the location and scale parameters are estimated by their robust sample estimators. This approach also works when the moments of the GLD do not exist. Moreover, the new parameterization can be used to compare data sets in a convenient asymmetry and steepness shape plot. In this paper, we derive the new formulation, as well as the conditions of the various distribution shape regions and moment conditions. We illustrate the use of the asymmetry and steepness shape plot by comparing equities from the NASDAQ-100 stock index.
|Item Type:||MPRA Paper|
|Original Title:||An asymmetry-steepness parameterization of the generalized lambda distribution|
|Keywords:||Quantile distributions; generalized lambda distribution; shape plot representation|
|Subjects:||C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C16 - Specific Distributions|
|Depositing User:||Yohan Chalabi|
|Date Deposited:||03. Apr 2012 12:42|
|Last Modified:||19. Feb 2013 06:14|
W. H. Asquith. L-moments and TL-moments of the generalized lambda distribution. Computational Statistics & Data Analysis, 51(9):4484– 4496, May 2007.
M. Bigerelle, D. Najjar, B. Fournier, N. Rupin, and A. Iost. Application of lambda distributions and bootstrap analysis to the prediction of fatigue lifetime and confidence intervals. International Journal of Fatigue, 28(3):223–236, Mar. 2006.
C. J. Corrado. Option pricing based on the generalized lambda distribution. Journal of Futures Markets, 21(3):213–236, 2001.
B. Dengiz. The generalized lambda distribution in simulation of M/M/1 queue systems. Journal of the Faculty of Engineering and Architecture of Gazi University, 3:161–171, 1988.
J. J. Filliben. The probability plot correlation coefficient test for normality. Technometrics, 17(1):111–117, Feb. 1975.
B. Fournier, N. Rupin, M. Bigerelle, D. Najjar, and A. Iost. Application of the generalized lambda distributions in a statistical process control methodology. Journal of Process Control, 16(10):1087 – 1098, 2006.
B. Fournier, N. Rupin, M. Bigerelle, D. Najjar, A. Iost, and R. Wilcox. Estimating the parameters of a generalized lambda distribution. Computational Statistics & Data Analysis, 51(6):2813 – 2835, 2007.
M. Freimer, G. Kollia, G. Mudholkar, and C. Lin. A study of the generalized Tukey lambda family. Communications in Statistics-Theory and Methods, 17(10):3547–3567, 1988.
W. Gilchrist. Statistical modelling with quantile functions. CRC Press, 2000.
J. Hastings, Cecil, F. Mosteller, J. W. Tukey, and C. P. Winsor. Low moments for small samples: A comparative study of order statistics. The Annals of Mathematical Statistics, 18(3):pp. 413–426, 1947.
D. Hogben. Some properties of Tukey’s test for non-additivity. Unpublished Ph.D. thesis, Rutgers-The State University, 1963.
B. L. Joiner and J. R. Rosenblatt. Some properties of the range in samples from Tukey’s symmetric lambda distributions. Journal of the American Statistical Association, 66(334):pp. 394–399, 1971.
Z. Karian and E. Dudewicz. Fitting statistical distributions: the generalized lambda distribution and generalized bootstrap methods. Chapman & Hall/CRC, 2000.
Z. Karian and E. Dudewicz. Comparison of GLD fitting methods: superiority of percentile fits to moments in L2 norm. Journal of Iranian Statistical Society, 2(2):171–187, 2003.
Z. A. Karian and E. J. Dudewicz. Fitting the generalized lambda distribution to data: a method based on percentiles. Communications in Statistics - Simulation and Computation, 28(3):793–819, 1999.
J. Karvanen and A. Nuutinen. Characterizing the generalized lambda distribution by L-moments. Computational Statistics & Data Analysis, 52(4):1971–1983, Jan. 2008.
J. Karvanen, J. Eriksson, and V. Koivunen. Maximum likelihood estimation of ICA model for wide class of source distributions. In Neural Networks for Signal Processing X, 2000. Proceedings of the 2000 IEEE Signal Processing Society Workshop, volume 1, pages 445 –454 vol.1, 2000.
R. King and H. MacGillivray. Fitting the generalized lambda distribution with location and scale-free shape functionals. American Journal of Mathematical and Management Sciences, 27(3-4):441–460, 2007.
R. A. R. King and H. L. MacGillivray. A starship estimation method for the generalized λ distributions. Australian & New Zealand Journal of Statistics, 41(3):353–374, 1999.
A. Lakhany and H. Mausser. Estimating the parameters of the generalized lambda distribution. Algo Research Quarterly, 3(3):47–58, 2000.
D. Najjar, M. Bigerelle, C. Lefebvre, and A. Iost. A new approach to predict the pit depth extreme value of a localized corrosion process. ISIJ international, 43(5):720–725, 2003.
A. Negiz and A. Çinar. Statistical monitoring of multivariable dynamic processes with state-space models. AIChE Journal, 43(8):2002–2020, 1997.
A. Ozturk and R. Dale. A study of fitting the generalized lambda distribution to solar-radiation data. Journal of Applied Meteorology, 21(7): 995–1004, 1982.
A. Ozturk and R. Dale. Least-squares estimation of the parameters of the generalized lambda-distribution. Technometrics, 27(1):81–84, 1985.
S. Pal. Evaluation of nonnormal process capability indices using generalized lambda distribution. Quality Engineering, 17(1):77–85, 2004.
J. S. Ramberg and B. W. Schmeiser. An approximate method for generating asymmetric random variables. Commun. ACM, 17(2):78–82, 1974.
S. Shapiro, M. Wilk, and H. Chen. A comparative study of various tests for normality. Journal of the American Statistical Association, 63(324): 1343–&, 1968.
S. S. Shapiro and M. B. Wilk. An analysis of variance test for normality (complete samples). Biometrika, 52(Part 3–4):591–611, Dec. 1965.
H. Shore. Comparison of generalized lambda distribution (GLD) and response modeling methodology (RMM) as general platforms for distribution fitting. Communications In Statistics-Theory and Methods, 36 (13-16):2805–2819, 2007.
S. Su. A discretized approach to flexibly fit generalized lambda distributions to data. Journal of Modern Applied Statistical Methods, 4(2): 408–424, 2005.
S. Su. Numerical maximum log likelihood estimation for generalized lambda distributions. Computational Statistics & Data Analysis, 51(8): 3983–3998, May 2007.
A. Tarsitano. Fitting the generalized lambda distribution to income data. In COMPSTAT’2004 Symposium, pages 1861–1867. Physica-Verlag/Springer, 2004.
A. Tarsitano. Comparing estimation methods for the FPLD. Journal of Probability and Statistics, 2010.
J. W. Tukey. The practical relationship between the common transformations of percentages or fractions and of amounts. Technical Report Technical Report 36, Statistical Research Group, Princeton, 1960. 32
J. W. Tukey. The future of data analysis. The Annals of Mathematical Statistics, 33(1):1–67, Mar. 1962.
J. Van Dyke. Numerical investigation of the random variable y = c(u^\lambda − (1 − u)^\lambda). Unpublished working paper, National Bureau of Standards, 1961.
Available Versions of this Item
- An asymmetry-steepness parameterization of the generalized lambda distribution. (deposited 03. Apr 2012 12:42) [Currently Displayed]