A New Family of Distributions Based on the Generalized Pearson Differential Equation with Some Applications

Mohammad Shakil; B.M. Golam Kibria; Jai Narain Singh

doi:10.17713/ajs.v39i3.248

Authors

Mohammad Shakil Department of Mathematics, Miami Dade College, USA
B.M. Golam Kibria Department of Mathematics and Statistics, Florida International, USA
Jai Narain Singh Department of Mathematics and Computer Science, Barry, USA

DOI:

https://doi.org/10.17713/ajs.v39i3.248

Abstract

Recently, a generalization of the Pearson differential equation has appeared in the literature, from which a vast majority of continuous probability density functions (pdf’s) can be generated, known as the generalized Pearson system of continuous probability distributions. This paper derives a new family of distributions based on the generalized Pearson differential equation, which is a natural generalization of the generalized inverse Gaussian distribution. Some characteristics of the new distribution are obtained.
Plots for the cumulative distribution function, pdf and hazard function, tables
with percentiles and with values of skewness and kurtosis are provided. It
is observed that the new distribution is skewed to the right and bears most
of the properties of skewed distributions. As a motivation, the statistical applications
of the results to a problem of forestry have been provided. It is found that our newly proposed model fits better than gamma, log-normal and inverse Gaussian distributions. Since many researchers have studied the use of the generalized inverse Gaussian distributions in the fields of biomedicine, demography, environmental and ecological sciences, finance, lifetime data, reliability theory, traffic data, etc., we hope the findings of the paper will be useful for the practitioners in various fields of theoretical and applied sciences.

References

Abramowitz, M., and Stegun, I. A. (1970). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. New York: Dover.

Achcar, J. A., and Bolfarine, H. (1986). The log-linear model with generalized gamma distribution for the error: a Bayesian approach. Statistics and Probability Letters, 4, 325-332.

Barndorff-Nielson, O. E., and Halgreen, C. (1977). Infinite divisibility of the hyperbolic and generalized inverse Gaussian distributions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 38, 309-311.

Chaudhry, M. A., and Ahmad, M. (1993). On a probability function useful in size modeling. Canadian Journal of Forest Research, 23, 1679-1683.

Chaudhry, M. A., and Zubair, S. M. (2002). On a Class of Incomplete Gamma Functions with Applications. Boca Raton: Chapman & Hall/CRC.

Chou, C., and Huang, W. (2004). On characterizations of the gamma and generalized inverse gaussian distributions. Statistics and Probability Letters, 69, 381-388.

Dadpay, A., Soofi, E. S., and Soyer, R. (2007). Information measures for generalized gamma family. Journal of Econometrics, 138, 568-585.

Dunning, K., and Hanson, J. N. (1977). Generalized Pearson distributions and nonlinear programming. Journal of Statistical Computation and Simulation, 6, 115-128.

Elderton, W. P. (1953). Frequency Curves and Correlation. Washington, D.C.: Harren Press.

Erdélyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G. (1953). Higher Transcendental Functions, I. New York: McGraw-Hill.

Gneiting, T. (1997). Normal scale mixtures and dual probability densities. Journal of Statistical Computation and Simulation, 59, 115-128.

Good, I. J. (1953). The population frequencies of species and the estimation of population parameters. Biometrika, 40, 237-260.

Gradshteyn, I. S., and Ryzhik, I. M. (2000). Table of Integrals, Series, and Products (6th ed.). San Diego: Academic Press.

Johnson, N. L., Kotz, S., and Balakrishnan, N. (1994). Continuous Univariate Distributions (6th ed., Vol. 1). New York: John Wiley & Sons.

Jørgensen, B. (1982). Statistical Properties of the Generalized Inverse Gaussian Distribution. New York: Springer Verlag.

Koutrouvelis, I. A., Canavos, G. C., and Meintanis, S. G. (2005). Estimation in the threeparameter inverse Gaussian distribution. Computational Statistics & Data Analysis, 49, 1132-1147.

Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data (2nd ed.). New York: John Wiley & Sons.

Lebedev, N. N. (1972). Special Functions and their Applications. New York: Dover.

Lee, J. (2010). Parameter estimation of the extended generalized Gaussian family distributions using maximum likelihood scheme. Information Technology Journal, 9, 61-66.

Marshall, A. W., and Olkin, I. (2007). Life Distributions: Structure of Nonparametric, Semiparametric and Parametric Families. New York: Springer.

Prause, K. (1997). Modelling Financial Data Using Generalized Hyperbolic Distributions (Research Report No. 48). Universität Freiburg im Breisgau, Freiburg Center for Data Analysis and Modelling und Institut für Mathematische Stochastik.

Prause, K. (1999). The Generalized Hyperbolic Model: Estimation, Financial Derivatives, and Risk Measures. Unpublished doctoral dissertation, Universität Freiburg.

Prudnikov, A. P., Brychkov, Y. A., and Marichev, O. I. (1986). Integrals and Series (Vols. 1, 2, and 3). Amsterdam: Gordon and Breach Science Publishers.

Rice, S. (1999). Tree measurements: An outdoor activity for learning the principles of scaling. American Biology Teacher, 61, 677-679.

Seshadri, V., and Wesolowski, J. (2001). Mutual characterizations of the gamma and generalized inverse Gaussian laws by constancy of regression. Sankhya, A, 63, 107-112.

Shakil, M., Singh, J. N., and Kibria, B. M. G. (2010). On a family of product distributions based on the Whittaker functions and generalized Pearson differential equation. Pakistan Journal of Statistics, 26, 111-125.

Shannon, C. E. (1948). A mathematical theory of communication. Bell System Technical Journal, 27, 379-423, 623-656.

Silva, R. S., Lopes, H. F., and Migon, H. S. (2006). The extended generalized inverse Gaussian distribution for log-linear and stochastic volatility models. Brazilian Journal of Probability and Statistics, 20, 67-91.

Steutel, F. W., and Harn, V. K. (2004). Infinite Divisibility of Probability Distributions on the Real Line. New York: Marcel Dekker, Inc.

Stuart, A., and Ord, J. K. (1994). Kendall’s Advanced Theory of Statistics (6th ed.). London: Edward Arnold.

Wesolowski, J. (2002). The Matsumoto-Yor independence property for GIG and gamma laws revisited. Mathematical Proceedings of Cambridge Philosophical Society, 133, 153-161.

Wise, M. G. (1975). Skew distributions in biomedicine including some with negative powers of time. In Model Building and Model Selection, Statistical Distributions in Scientific Work.