On Moment Generating Function of Generalized Order Statistics from Erlang-Truncated Exponential Distribution

Affiliation(s)

Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, India.

Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, India.

ABSTRACT

In this paper explicit expressions and some recurrence relations are derived for marginal and joint moment generating functions of generalized order statistics from Erlang-truncated exponential distribution. The results for*k*-th record values and order statistics are deduced from the relations derived. Further, a characterizing result of this distribution on using the conditional expectation of function of generalized order statistics is discussed.

In this paper explicit expressions and some recurrence relations are derived for marginal and joint moment generating functions of generalized order statistics from Erlang-truncated exponential distribution. The results for

Cite this paper

A. Kulshrestha, R. Khan and D. Kumar, "On Moment Generating Function of Generalized Order Statistics from Erlang-Truncated Exponential Distribution,"*Open Journal of Statistics*, Vol. 2 No. 5, 2012, pp. 557-564. doi: 10.4236/ojs.2012.25071.

A. Kulshrestha, R. Khan and D. Kumar, "On Moment Generating Function of Generalized Order Statistics from Erlang-Truncated Exponential Distribution,"

References

[1] A. R. El-Alosey, “Random Sum of New Type of Mixtures of Distributions,” International Journal of Statistics and Systems, Vol. 2, No. 1, 2007, pp. 49-57.

[2] U. Kamps, “A Concept of Generalized Order Statistics,” B. G. Teubner, Stuttgart, 1995.

[3] U. Kamps and E. Cramer, “On Distribution of Generalized Order Statistics,” Statistics, Vol. 35, No. 3, 2001, pp. 269-280.

[4] M. Ahsanullah and M. Z. Raqab, “Recurrence Relations for the Moment Generating Functions of Record Values from Pareto and Gumble Distributions,” Stochastic Modelling and Applications, Vol. 2, No. 2, 1999, pp. 35-48.

[5] M. Z. Raqab and M. Ahsanullah, “Relations for Marginal and Joint Moment Generating Functions of Record Values from Power Function Distribution,” Journal of Applied Statistical Sciences, Vol. 10, No. 1, 2000, pp. 27-36.

[6] M. Z. Raqab and M. Ahsanullah, “On Moment Generating Function of Records from Extreme Value Distribution,” Pakistan Journal of Statistics, Vol. 19, No. 1, 2003, pp. 1-13.

[7] J. Saran and A. Pandey, “Recurrence Relations for Marginal and Joint Moment Generating Functions of Generalized Order Statistics from Power Function Distribution,” Metron, Vol. LXI, No. 1, 2003, pp. 27-33.

[8] E. K. Al-Hussaini, A. A. Ahmad and M. A. Al-Kashif, “Recurrence Relations for Moment and Conditional Moment Generating Functions of Generalized Order Statistics,” Metrika, Vol. 61, No. 2, 2005, pp. 199-220.

[9] E. K. Al-Hussaini, A. A. Ahmad and M. A. Al-Kashif, “Recurrence Relations for Joint Moment Generating Functions of Generalized Order Statistics Based on Mixed Population,” Journal of Statistical Theory and Applications, Vol. 6, 2007, pp. 134-155.

[10] R. U. Khan, B. Zia and H. Athar, “On Moment Generating Function of Generalized Order Statistics from Gompertz Distribution and Its Characterization,” Journal of Statistical Theory and Applications, Vol. 9, 2010, pp. 363-373.

[11] S. M. Ruiz, “An Algebraic Identity Leading to Wilson’s Theorem,” The Mathematical Gazette, Vol. 80, No. 489, 1996, pp. 579-582.

[12] Z. Grudzień and D. Szynal, “On the Expected Values of k-th Record Values and Associated Characterizations of Distributions,” Proceedings of the 4th Pannonian Symposium on Mathematical Statistics, Reidel, 4-10 September 1983, pp. 119-127.

[13] M. Ahsanullah, “Record Statistics and the Exponential Distribution,” Pakistan Journal of Statistics, Vol. 3A, 1987, pp. 17-40.

[14] N. Balakrishnan, H. J. Malik and S. E. Ahmed, “Recurrence Relations and Identities for Moments of Order Statistics,” II: Specific distributions, Communications in Statistics—Theory and Methods, Vol. 17, 1988, pp. 2657- 2694.

[1] A. R. El-Alosey, “Random Sum of New Type of Mixtures of Distributions,” International Journal of Statistics and Systems, Vol. 2, No. 1, 2007, pp. 49-57.

[2] U. Kamps, “A Concept of Generalized Order Statistics,” B. G. Teubner, Stuttgart, 1995.

[3] U. Kamps and E. Cramer, “On Distribution of Generalized Order Statistics,” Statistics, Vol. 35, No. 3, 2001, pp. 269-280.

[4] M. Ahsanullah and M. Z. Raqab, “Recurrence Relations for the Moment Generating Functions of Record Values from Pareto and Gumble Distributions,” Stochastic Modelling and Applications, Vol. 2, No. 2, 1999, pp. 35-48.

[5] M. Z. Raqab and M. Ahsanullah, “Relations for Marginal and Joint Moment Generating Functions of Record Values from Power Function Distribution,” Journal of Applied Statistical Sciences, Vol. 10, No. 1, 2000, pp. 27-36.

[6] M. Z. Raqab and M. Ahsanullah, “On Moment Generating Function of Records from Extreme Value Distribution,” Pakistan Journal of Statistics, Vol. 19, No. 1, 2003, pp. 1-13.

[7] J. Saran and A. Pandey, “Recurrence Relations for Marginal and Joint Moment Generating Functions of Generalized Order Statistics from Power Function Distribution,” Metron, Vol. LXI, No. 1, 2003, pp. 27-33.

[8] E. K. Al-Hussaini, A. A. Ahmad and M. A. Al-Kashif, “Recurrence Relations for Moment and Conditional Moment Generating Functions of Generalized Order Statistics,” Metrika, Vol. 61, No. 2, 2005, pp. 199-220.

[9] E. K. Al-Hussaini, A. A. Ahmad and M. A. Al-Kashif, “Recurrence Relations for Joint Moment Generating Functions of Generalized Order Statistics Based on Mixed Population,” Journal of Statistical Theory and Applications, Vol. 6, 2007, pp. 134-155.

[10] R. U. Khan, B. Zia and H. Athar, “On Moment Generating Function of Generalized Order Statistics from Gompertz Distribution and Its Characterization,” Journal of Statistical Theory and Applications, Vol. 9, 2010, pp. 363-373.

[11] S. M. Ruiz, “An Algebraic Identity Leading to Wilson’s Theorem,” The Mathematical Gazette, Vol. 80, No. 489, 1996, pp. 579-582.

[12] Z. Grudzień and D. Szynal, “On the Expected Values of k-th Record Values and Associated Characterizations of Distributions,” Proceedings of the 4th Pannonian Symposium on Mathematical Statistics, Reidel, 4-10 September 1983, pp. 119-127.

[13] M. Ahsanullah, “Record Statistics and the Exponential Distribution,” Pakistan Journal of Statistics, Vol. 3A, 1987, pp. 17-40.

[14] N. Balakrishnan, H. J. Malik and S. E. Ahmed, “Recurrence Relations and Identities for Moments of Order Statistics,” II: Specific distributions, Communications in Statistics—Theory and Methods, Vol. 17, 1988, pp. 2657- 2694.