Let $X$ be a rv having type III generalized logistic distribution with parameter $\alpha$. Then the central moment of order $\beta$ is (written in TEX) reads:
\[ \mathsf EX^\beta = \dfrac{1+(-1)^\beta}{{\rm B}(\alpha, \alpha)} \Gamma(\beta+1)
\Phi_{2\alpha}^*(-1, \beta+1, \alpha) ,\]
where $\Gamma$ denotes the Euler Gamma function, while
\[ \Phi_\mu^*(z, s, a) = \sum_{n=0}^\infty \dfrac{(\mu)_n\,z^n}{(a+n)^s\,n!}\]
stands for the Goyal-Laddha generalized Hurwitz-Lerch Zeta function. More about this special function in e.g.
1.S. P. Goyal, R. K. Laddha, On the generalized Riemann{Zeta functions
and the generalized Lambert transform, Ganita Sandesh 11(1997), 97-108.
2. H. M. Srivastava, Ram K. Saxena, Tibor K. Pogány and Ravi Saxena, Integral and Computational Representations of the Extended Hurwitz-Lerch Zeta Function, Integral Transforms and Special Functions 22 (2011), 487-506.
Obviously, odd order moments vanish, since the PDF of $X$ is an even function, while the even order moments nominator is 2.
If you need step by step calculation of the above formula, I'll send you it upon request.
1 Recommendation
2nd Feb, 2013
Partha Jyoti Hazarika
Dibrugarh University
Thank you . If possible please send the step by step calculation along with the references (in pdf format) in the following mail id
2nd Feb, 2013
Tibor K. Pogány
University of Rijeka
Paper two you can find in my contributions on researchgate. I had somewhere around only the hardcopy of the original article by Goyal and Laddha. The calculation I'll send you asap.
2nd Feb, 2013
Partha Jyoti Hazarika
Dibrugarh University
Thanks. Is there and reference (s) to get the nth order moment of Type III generalized logistic distribution ?
2nd Feb, 2013
Tibor K. Pogány
University of Rijeka
By my best knowledge, no. I am pretty sure that it is a new formula. However, since the moment generating function is well--known, the nth derivative of MGF at $t=0$ results in the same value. But calculating higher irder derivatives of the Gamma function is complicated.
2nd Feb, 2013
Partha Jyoti Hazarika
Dibrugarh University
Thanks
2nd Feb, 2013
Partha Jyoti Hazarika
Dibrugarh University
Your calculation is correct, but the restriction is very strong.
Can you help by adding an answer?
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