2949775117

731

5.1. Theorem. A necessary and sufficient conditions for a random variable X to be distributed with pdf gwen by (1.1) is that

^1--^E[X^(r, n,m, A;)] = E[X^j(r — 1 ,n,m,k)\

A_a0+1

+^—E[X’~^ll(r,n,m,k)].    (5.1)

ap7_r

Proof The necessary part follows immediately from (2.14) on the other hand if the recurrence relation (5.1) is satisfied then on using (1.5), we have

jH AF(x)]^-¹nx)g^\F(x))dx = _lr^C(r~\)\ r AF(x)r^+mf(x)g^iF^))dx ⁺~a/hllr -1)1 J₀ xⁱ[F(x)]’^,r~¹Hx)9n¹(,F(x))'lx

r    f(x)9^Tm~\F(x))dx.    (5.2)

Integrating the first integral on the right-hand side of the above eąuation by parts and simplifying the resulting expression, we get

^x{^Ą:c) - ^^/(x) - ^S^^f(x)}^dx=°-

It now follows from Proposition 5.1, we get

a(lF(x) = cr[l + (x/<r)⁰]xf(x), which proves that f(x) has the form (1.1).

6. Concluding Remarks

In the study presented above, we established some new explicit expressions and recurrence relations between the single and product moments of gos from the type II exponentiated log-logistic distribution. In addition ratio and inverse moments of type II exponentiated log-logistic distribution are also established. Further, the conditional expectation of gos is used to characterize the distribution.

Acknowledgements

The author appreciates the comments and remarks of the referees which improved the original form of the paper.

References

[1]    Aggarwala, R. and Balakrishnan, N. Recurrence relations for progressiue type II right cen-sored order statistics from exponential truncated exponential distributions, Ann. Instit. Statist. Math., 48, 757-771, (1996).

[2]    Ahmad, A.A. and Fawzy, M. Recurrence relations for single moments of generalized order statistics from doubly truncated distribution, J. Statist. Plann. Inference, 177, 241-249, (2003).

[3]    Ahmad, A.A. Relations for single and product moments of generalized order statistics from doubly truncated Burr type XII distribution, J. Egypt. Math. Soc., 15, 117-128, (2007).

[4]    Ahsanullah, M. Record Statistics, Nova Science Publishers, New York, (1995).