On a Probability Distribution with Fractional Moments arising from Generalized Pearson System of Differential Equation and its Characterization
Abstract
This paper derives a probability distribution with fractional moments arising from generalized Pearson system of differential equation. The expressions for the probability density function, cumulative distribution function and moments have been obtained. The plots for the probability density function, cumulative distribution function, and survival and hazard functions are also provided. Some distributional relationships of the proposed distribution have been established. Characterization of the new distribution is given. It is hoped that the findings of the paper will be useful for researchers in different fields like economics, engineering, environmental science, finance, medical sciences and physical sciences among others, where fractional moments become required to be computed if the integer moments k>=1 do not exist.
About this article
References
Abramowitz, M., and Stegun, I. A. (1970). Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables. Dover, New York. Bondesson, L. (1979). A general result on infinite divisibility, Annals of Probability, 7 965-979.
Ahsanullah, M., Hamedani, G. G. and Shakil, M. (2011). A Note on Distributions arising from Generalized Pearsonian System of Differential Equation. Technical Report No. 481, Department of Mathematics, Statistics, and Computer Science, Marquette University, 1-9.
Chaudhry, M. A., and Ahmad, M. (1993). On a probability function useful in size modeling. Canadian Journal of Forest Research, 23(8), 1679–1683.
Chaudhry, M. A., and Zubair, S. M. (2002). On A Class of Incomplete Gamma Functions, with Applications. Chapman & Hall/CRC, Boca Raton.
Cobb, L., Koppstein, P., and Chen, N. H. (1983). Estimation and moment recursion relations for multimodal distributions of the exponential family. Journal of the American Statistical Association, 78(381), 124-130.
View more references (22)
Cottone, G., and Di Paola, M. (2009). On the use of fractional calculus for the probabilistic characterization of random variables. Probabilistic Engineering Mechanics, 24(3), 321-330.
Cottone, G., Di Paola, M., and Metzler, R. (2010). Fractional calculus approach to the statistical characterization of random variables and vectors. Physica A: Statistical Mechanics and its Applications, 389(5), 909-920.
Dunning, K., and Hanson, J. N. (1977). Generalized Pearson distributions and nonlinear programming. Journal of Statistical Computation and Simulation, Volume 6, Issue 2, 115 – 128.
Gradshteyn, I. S. and Ryzhik, I. M. (1980). Table of Integrals, Series, and Products (6th edition). Academic Press, San Diego.
Hamedani, G. G. (2011). Characterizations of the Shakil-Kibria-Singh Distribution. Austrian Journal of Statistics, Volume 40, Number 3, 201–207.
Hamedani, G. G. (2013). Characterizations of Certain Continuous Distributions., Part III Statistical Analysis – Chapter 13, Page 297, in Multiscale Signal Analysis and Modeling, Shen, Xiaoping; Zayed, Ahmed I. (Eds.), 2013, 2013, XVII, 378 p. 88 illus.
Johnson, N. L., Kotz, S., and Balakrishnan, N. (1994). Continuous Univariate Distributions, (Volume 1, Second edition). John Wiley & Sons, New York.
Kibria, B. M. G. and Shakil, M. (2011). A New Five-Parameter Burr System of Distributions based on Generalized Pearson Differential Equation. 2011 Joint Statistical Meetings Proceedings, Section on Physical and Engineering Sciences, Miami, Fl., U.S.A., July 30-August 4.
Lahcene, B. (2013). On Pearson families of distributions and its applications. African Journal of Mathematics and Computer Science Research, 6(5), 108-117.
Lebedev, N. N. (1972). Special Functions and their Applications. Dover, New York.
Lee, C., Famoye, F., and Alzaatreh, A. Y. (2013). Methods for generating families of univariate continuous distributions in the recent decades, WIREs Comput Stat 2013, 5, 219-238. doi: 10.1002/wics.1255.
NG, K.W. and Kotz, S. (1995). Kummer-Gamma and Kummer-Beta Univariate and Multivariate Distributions. Research Report, Department of Statistics, The University of Hong Kong, Hong Kong.
Pearson, K. (1895). Contributions to the mathematical theory of evolution, II: Skew variation in homogeneous material. Philosophical Transactions of the Royal Society of London, A186, 343-414.
Pearson, K. (1901). Mathematical contributions to the theory of evolution X: Supplement to a memoir on skew of variation. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 197, 343-414.
Pearson, K. (1916). Mathematical contributions to the theory of evolution, XIX: Second supplement to a memoir on skew of variation. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 216, 429-457.
Rossani, A., and Scarfone, A. M. (2009). Generalized Pearson distributions for charged particles interacting with an electric and/or a magnetic field. Physica, A, 388, 2354-2366.
Roy, L. K. (1971). An extension of the Pearson system of frequency curves. Trabajos de estadistica y de investigacion operativa, 22 (1-2), 113-123.
Shakil, M., and Kibria, B. M. G. (2010). On a family of life distributions based on generalized Pearson differential equation with applications in health statistics. Journal of Statistical Theory and Applications, 9 (2), 255-282.
Shakil, M., Singh, J. N., and Kibria, B. M. G. (2010a). On a family of product distributions based on the Whittaker functions and generalized Pearson differential equation. Pakistan Journal of Statistics, 26(1), 111-125.
Shakil, M., Kibria, B. M. G., and Singh, J. N. (2010b). A new family of distributions based on the generalized Pearson differential equation with some applications. Austrian Journal of Statistics, 39 (3), 259–278.
Shakil, M., Kibria, B. M. G. and Singh, J. N. (2011). On Some New Classes of Pearsonian Distributions, 2011 Joint Statistical Meetings Proceedings, Section on Physical and Engineering Sciences, Miami, Fl., U.S.A., July 30-August 4.
Weiss, G. H., Havlin, S., and Matan, O. (1989). Properties of noninteger moments in a first passage time problem. Journal of Statistical Physics, 55(1-2), 435-439.