Estimating the Parameters of the Generalized Lambda Distribution: Which Method Performs Best?

Generalized lambda distribution (GLD) is a flexible distribution that can represent a wide variety of distributional shapes. This property of the GLD has made it very popular in simulation input modeling in recent years, and several fitting methods for estimating the parameters of the GLD have been proposed. Nevertheless, there appears to be a lack of insights about the performances of these fitting methods in estimating the parameters of the GLD for a variety of distributional shapes and input data. Our primary goal in this article is to compare the goodness-of-fits of the popular fitting methods in estimating the parameters of the GLD introduced in Freimer et al. (1988), i.e., Freimer–Mudholkar–Kollia–Lin (FMKL) GLD, and provide guidelines to the simulation practitioner about when to use each method. We further describe the use of the genetic algorithm for the FMKL GLD, and investigate the performances of the suggested methods in modeling the daily exchange rates of eight currencies.

[1]  D. Würtz,et al.  The Generalized Lambda Distribution as an Alternative Model to Financial Returns , 2009 .

[2]  Juha Karvanen,et al.  Characterizing the generalized lambda distribution by L-moments , 2008, Comput. Stat. Data Anal..

[3]  J. Filliben.  The Probability Plot Correlation Coefficient Test for Normality , 1975 .

[4]  T. Stengos,et al.  Information-Theoretic Distribution Test with Application to Normality , 2009 .

[5]  William H. Asquith,et al.  L-moments and TL-moments of the generalized lambda distribution , 2007, Comput. Stat. Data Anal..

[6]  Domenico Alessandro Lampasi,et al.  Generalized lambda distribution for the expression of measurement uncertainty , 2006, IEEE Transactions on Instrumentation and Measurement.

[7]  V. Beena,et al.  Measuring inequality and social welfare from any arbitrary distribution , 2010 .

[8]  David Allingham,et al.  Bayesian estimation of quantile distributions , 2009, Stat. Comput..

[9]  C. Corrado.  Option Pricing Based on the Generalized Lambda Distribution , 2001 .

[10]  A. Öztürk,et al.  A Study of Fitting the Generalized Lambda Distribution to Solar Radiation Data. , 1982 .

[11]  Paul H. Kupiec,et al.  Techniques for Verifying the Accuracy of Risk Measurement Models , 1995 .

[12]  R. Wilcox.  Comparing the variances of two independent groups. , 2002, The British journal of mathematical and statistical psychology.

[13]  A. Öztürk,et al.  Least Squares Estimation of the Parameters of the Generalized Lambda Distribution , 1985 .

[14]  Thomas E. Wehrly.  Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstrap Methods , 2002, Technometrics.

[15]  Anil K. Bera,et al.  Efficient tests for normality, homoscedasticity and serial independence of regression residuals: Monte Carlo Evidence , 1981 .

[16]  E. Dudewicz,et al.  COMPARISON OF GLD FITTING METHODS: SUPERIORITY OF PERCENTILE FITS TO MOMENTS IN L2 NORM , 2003 .

[17]  B. P. Murphy,et al.  Handbook of Methods of Applied Statistics , 1968 .

[18]  R Willink,et al.  Representing Monte Carlo output distributions for transferability in uncertainty analysis: modelling with quantile functions , 2009 .

[19]  B. L. Joiner,et al.  Some Properties of the Range in Samples from Tukey's Symmetric Lambda Distributions , 1971 .

[20]  K. Pearson.  Contributions to the Mathematical Theory of Evolution , 1894 .

[21]  K. Pearson.  Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material , 1895 .

[22]  Edward J. Dudewicz,et al.  Fitting the generalized lambda distribution to data: a method based on percentiles , 1999 .

[23]  K. K. Achary,et al.  On Approximating Lead Time Demand Distributions Using the Generalised λ-type Distribution , 1996 .

[24]  Fitting generalized lambda distribution to solar radiation in Colombo, Sri Lanka , 2013 .

[25]  Todd C. Headrick,et al.  On simulating multivariate non-normal distributions from the generalized lambda distribution , 2006, Comput. Stat. Data Anal..

[26]  Stuart Jay Deutsch,et al.  A Versatile Four Parameter Family of Probability Distributions Suitable for Simulation , 1977 .

[27]  Averill M. Law,et al.  Simulation Modeling and Analysis , 1982 .

[28]  Marshall Freimer,et al.  a study of the generalized tukey lambda family , 1988 .

[29]  R Core Team,et al.  R: A language and environment for statistical computing. , 2014 .

[30]  R. Hogg.  Adaptive Robust Procedures: A Partial Review and Some Suggestions for Future Applications and Theory , 1974 .

[31]  N. L. Johnson,et al.  Systems of frequency curves generated by methods of translation. , 1949, Biometrika.

[32]  Mansooreh Mollaghasemi,et al.  Simulation input data modeling , 1994, Ann. Oper. Res..

[33]  Juha Karvanen,et al.  GENERATION OF CORRELATED NON-GAUSSIAN RANDOM VARIABLES FROM INDEPENDENT COMPONENTS , 2003 .

[34]  Gautam Mitra,et al.  Robust solutions and risk measures for a supply chain planning problem under uncertainty , 2008, J. Oper. Res. Soc..

[35]  D. Brillinger,et al.  Handbook of methods of applied statistics , 1967 .

[36]  K. K. Achary,et al.  Response to Shore , 1996 .

[37]  F. Mosteller,et al.  Low Moments for Small Samples: A Comparative Study of Order Statistics , 1947 .

[38]  Steve Su,et al.  Numerical maximum log likelihood estimation for generalized lambda distributions , 2007, Comput. Stat. Data Anal..

[39]  T. Stengos,et al.  Information-Theoretic Distribution Tests with Application to Symmetry and Normality , 2004 .

[40]  John S. Ramberg,et al.  Fitting a distribution to data using an alternative to moments , 1979, WSC '79.

[41]  H. L. MacGillivray,et al.  Theory & Methods: A Starship Estimation Method for the Generalized λ Distributions , 1999 .

[42]  A. Tarsitano.  FITTING THE GENERALIZED LAMBDA DISTRIBUTION TO INCOME DATA , 2004 .

[43]  A. Negiz,et al.  Statistical monitoring of multivariable dynamic processes with state-space models , 1997 .

[44]  Maxence Bigerelle,et al.  Application of the generalized lambda distributions in a statistical process control methodology , 2006 .

[45]  John A. Nelder,et al.  A Simplex Method for Function Minimization , 1965, Comput. J..

[46]  M. D. Martínez-Miranda,et al.  Computational Statistics and Data Analysis , 2009 .

[47]  D. Darling,et al.  A Test of Goodness of Fit , 1954 .

[48]  S. Su.  Fitting GLDs and Mixture of GLDs to Data Using Quantile Matching Method , 2010 .

[49]  Steve Su,et al.  Fitting Single and Mixture of Generalized Lambda Distributions to Data via Discretized and Maximum Likelihood Methods: GLDEX in R , 2007 .

[50]  Surajit Pal.  Evaluation of Nonnormal Process Capability Indices using Generalized Lambda Distribution , 2004 .

[51]  Pandu R. Tadikamalla,et al.  A Probability Distribution and its Uses in Fitting Data , 1979 .

[52]  A. Lakhany,et al.  Estimating the parameters of the generalized lambda distribution , 2000 .