2949775123

718

Ma,0) = _a^f\ , (-^^CP)

[“(<* + 1) +p- 0//3)]

1 j and j = 0,1,2,.

(2.1)

where

Mm = {r,⁽“⁺ⁱ⁾"(“⁺ⁱ-ⁱ⁾' s„°.

and

Jj(a,b)= [ x’[F(x)]“f(x)gl_l(F(x))dx.    (2.2)

Jo

Proof From (2.2), we have

Jj(a, 0) = f x^j[F(x)]^a f{x)dx.    (2.3)

Jo

By making the substitution z = [F(x)]¹^^Q in (2.3), we get

Jj (a, 0) = aa⁰ [°° (1 - z)^j/0 Jo

= co* Y,(-1YM)<p) r

p=o    ■'<>

and hence the result given in (2.1).

2.2. Lemma. For type II exponentiated log-logistic distribution as given in (1.2) and any non-negative and finite integers a and b

Jj(a, b)

_1_

(m + l)¹

^5Z(^-1)“( ^bu ) Jj(a+u(m+ 1),0

(2.4)

^aa‘ Y' Y't ii^ll+'‘ ( ⁶ i_U/P)w_

(^m+ ¹)^Ł “S“S' ’ V “ / [a{a + (m + l)ti + l}+p-(j/P)]’

m # -1    (2.5)

(2.6)

m = —1,

= a^b+Vf>t V__

■^w« + i )+P-u/m^b+i’

where Jj(a,b) is as given in (2.2).

Proof: On expanding y^(F(x)) = [^y(l — (F(rr))^m+1)]^b binomially in (2.2), we get when m ^ —1

f(x)dx

=    E/-¹)" (t(“+^u<^m+!)■ °)-

Making use of Lemma 2.1, we establish the result given in (2.5)

and when m — —1 that

J_J(o,6) = iasELo(-D”(‘)=0-

Since (2.5) is of the form | at m = -1, therefore, we have