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i) Putting m = 0, k = 1 in (2.12), the explicit formula for the single moments of order statistics of the type II exponentiated log-logistic distribution can be obtained as

r — 1

u

p=0 u=0

_av^jU/P)(p)_

[a(n — r + u+l) + p — (j/f3)\ ’

where

^rn (r — l)!(n — r)! ’

ii) Setting m = — 1 in (2.12), we deduce the explicit expression for the single moments of upper k record values for type II exponentiated log-logistic distribution in view of (2.11) and (2.6) in the form

E[X^j(r, n, -1. *)] = £[(Z<*’⁾)³'] = (ak)^ra’ jr

p~o

(-i ru/f>)M

[ak+p-(j/P)]^T

and hence for upper records

p=o

(-i YW)w

[q+p- u/nv

Recurrence relations for single moments of gos from (1.5) can be obtained in the following theorem.

2.4. Theorem. For the distribution giuen in (1.2) and 2<r<n, n>2 and k =

1,2,...,

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

+i^-E[Xⁱ-<‘(r,n,m,k)].    (2.14)

ap7r

Proof. Prom (1.5), we have

E[X³(r,n,m,k)\ = -^~^j J ^³{F{x)X'^r~^lf{x)g^r^^l{F{x))dx.    (2.15)

Integrating by parts treating [P'(x)]^7r_1/(x) for integration and rest of the integrand for differentiation, we get

E[X^j (r,n,m,k)\ = E[X^j (r — l,n,m,k)]-\--~.. f xⁱ~^l[F(x)]^lrg(^^l(F(x))dx

7r{? — 1)' Jo

the constant of integration vanishes sińce the integral considered in (2.15) is a definite integral. On using (1.3), we obtain

E[X^j(r, n, m, fc)] — E[X*(r — 1, n, m, fc)]

ap(r(r~— \)\ I”

‘jG

g-i r i

-0

[-^F(z)F’' ¹f(x)g^rm ^l(F(x))dx

a/37_r (r -

and hence the result given in (2.14).

Remark 2.1: Setting m = 0, k = 1, in (2.14), we obtain a recurrence relation for single moments of order statistics for type II exponentiated log-logistic distribution in the form

= muJ + ^(f-7+i)^- ‘

a/3(n — r + l)J