Generalized Order Statistics from Generalized Exponential Distributions in Explicit Forms

Author(s)
Hazem I. El Shekh Ahmed

ABSTRACT

The generalized order statistics which introduced by [1] are studied
in the present paper. The Gompertz distribution is widely used to describe the
distribution of adult deaths, and some related models used in the economic
applications [2]. Previous works concentrated on formulating approximate
relationships to characterize it [3-5]. The main aim of this paper is to obtain the
distribution of single, two, and all generalized order statistics from Gompertz
distribution with some special cases. In addition the conditional distribution
of two generalized order statistics from the same distribution is obtained. The Gompertz
distribution has a continuous probability density function with location
parameter a and shape parameter b, , where x restricted by the interval . The *n*th moment generated
function of the Gompertz distributed random variable X is given on the form: where,

is the generalized
integro-exponential function [6]. In this paper we shall obtain joint
distribution, distribution of product of two generalized order statistics from
the Gompertz distribution, and then derive some useful formulas of these
distributions as special cases.

Cite this paper

H. Ahmed, "Generalized Order Statistics from Generalized Exponential Distributions in Explicit Forms,"*Open Journal of Statistics*, Vol. 3 No. 2, 2013, pp. 129-135. doi: 10.4236/ojs.2013.32014.

H. Ahmed, "Generalized Order Statistics from Generalized Exponential Distributions in Explicit Forms,"

References

[1] U. Kamps, “Characterizations of the Exponential Distribution by Weighted Sums of Iid Random Variables,” Statistical Papers, Vol. 31, No. 1, 1990, pp. 233-237. doi:10.1007/BF02924695

[2] P. Jodrá, “A Closed-Form Expression for the Quantile Function of the Gompertz-Makeham Distribution,” Mathematics and Computers in Simulation, Vol. 79, No. 10, 2009, pp. 3069-3075. doi:10.1016/j.matcom.2009.02.002

[3] D. Kunimura, “The Gompertz Distribution-Estimation of Parameters,” Actuarial Research Clearing House, Vol. 2, 1998, pp. 65-76.

[4] F. M. Bass, “A New Product Growth Model for Consumer Durables,” Management Science, Vol. 15, No. 5, 1969, pp. 215-227. doi:10.1287/mnsc.15.5.215

[5] J. Pollard and E. Valkovics, “The Gompertz Distribution and Its Applications,” Genus, Vol. 48, No. 34, 1992, pp. 15-29.

[6] A. Lenart, “The Gompertz Distribution and Maximum Likelihood Estimation of Its Parameter—A Revision,” Max Planck Institute for Demographic Research, Rostock, 2012.

[7] R. U. Khan and D. Kumar, “On Moments of Lower Generalized Order Statistics from Exponintial Pareto Distribution and Its Characterization,” Applied Mathematical Sciences, Vol. 4, No. 55, 2010, pp. 2711-2722.

[8] U. Kamps and U. Gather, “Characteristic Properties of Generalized Order Statistics from Exponential Distributions,” Applicationes Mathematicae, Vol. 24, No. 4, 1997, pp. 383-391.

[9] U. Kamps, “A Concept of Generalized Order Statistics,” Elsevier Journal of Statistical Planning and Inference, Vol. 48, No. 1, 1995, pp. 1-23.

[10] U. Kamps, “Subranges of Generalized Order Statistics from Exponential Distributions,” Fasciculi Mathematici, Vol. 28, 1998, pp. 63-70.

[11] M. Ragab, “Generalized Exponential Distribution: Moments of Order Statistics,” Journal of Theoretical and Applied Statistics, Vol. 38, No. 1, 2004, pp. 29-41.

[12] R. D. Gupta and D. Kundu, “Generalized Exponential Distributions,” Australian and New Zealand Journal of Statistics, Vol. 41, No. 2, 1999, pp. 173-188. doi:10.1111/1467-842X.00072

[13] G. Qiu and J. Wang, “Some Comparison between Generalized Order Statistics,” Applied Mathematics—A Journal of Chinese Universities Series B, Vol. 22, No. 3, 2007, pp. 325-333. doi:10.1007/s11766-007-0310-6

[14] M. Garg, “On Generalized Order Statistics from Kumara-swamy Distribution,” Tamsui Oxford Journal of Mathematical Sciences, Vol. 25, No. 2, 2009, pp. 153-166.

[15] P. Samuel, “Characterization of Distributions by Conditional Expectation of Generalized Order Statistics,” Statistical Papers, Vol. 49, No. 1, 2008, pp. 101-108. doi:10.1007/s00362-006-0364-1

[1] U. Kamps, “Characterizations of the Exponential Distribution by Weighted Sums of Iid Random Variables,” Statistical Papers, Vol. 31, No. 1, 1990, pp. 233-237. doi:10.1007/BF02924695

[2] P. Jodrá, “A Closed-Form Expression for the Quantile Function of the Gompertz-Makeham Distribution,” Mathematics and Computers in Simulation, Vol. 79, No. 10, 2009, pp. 3069-3075. doi:10.1016/j.matcom.2009.02.002

[3] D. Kunimura, “The Gompertz Distribution-Estimation of Parameters,” Actuarial Research Clearing House, Vol. 2, 1998, pp. 65-76.

[4] F. M. Bass, “A New Product Growth Model for Consumer Durables,” Management Science, Vol. 15, No. 5, 1969, pp. 215-227. doi:10.1287/mnsc.15.5.215

[5] J. Pollard and E. Valkovics, “The Gompertz Distribution and Its Applications,” Genus, Vol. 48, No. 34, 1992, pp. 15-29.

[6] A. Lenart, “The Gompertz Distribution and Maximum Likelihood Estimation of Its Parameter—A Revision,” Max Planck Institute for Demographic Research, Rostock, 2012.

[7] R. U. Khan and D. Kumar, “On Moments of Lower Generalized Order Statistics from Exponintial Pareto Distribution and Its Characterization,” Applied Mathematical Sciences, Vol. 4, No. 55, 2010, pp. 2711-2722.

[8] U. Kamps and U. Gather, “Characteristic Properties of Generalized Order Statistics from Exponential Distributions,” Applicationes Mathematicae, Vol. 24, No. 4, 1997, pp. 383-391.

[9] U. Kamps, “A Concept of Generalized Order Statistics,” Elsevier Journal of Statistical Planning and Inference, Vol. 48, No. 1, 1995, pp. 1-23.

[10] U. Kamps, “Subranges of Generalized Order Statistics from Exponential Distributions,” Fasciculi Mathematici, Vol. 28, 1998, pp. 63-70.

[11] M. Ragab, “Generalized Exponential Distribution: Moments of Order Statistics,” Journal of Theoretical and Applied Statistics, Vol. 38, No. 1, 2004, pp. 29-41.

[12] R. D. Gupta and D. Kundu, “Generalized Exponential Distributions,” Australian and New Zealand Journal of Statistics, Vol. 41, No. 2, 1999, pp. 173-188. doi:10.1111/1467-842X.00072

[13] G. Qiu and J. Wang, “Some Comparison between Generalized Order Statistics,” Applied Mathematics—A Journal of Chinese Universities Series B, Vol. 22, No. 3, 2007, pp. 325-333. doi:10.1007/s11766-007-0310-6

[14] M. Garg, “On Generalized Order Statistics from Kumara-swamy Distribution,” Tamsui Oxford Journal of Mathematical Sciences, Vol. 25, No. 2, 2009, pp. 153-166.

[15] P. Samuel, “Characterization of Distributions by Conditional Expectation of Generalized Order Statistics,” Statistical Papers, Vol. 49, No. 1, 2008, pp. 101-108. doi:10.1007/s00362-006-0364-1