2949775128

723

By setting 2 = [F(y)]^1/,a in (3.4), we find that

GM = aa*

hi [“(c+i )+p-u/m

On substituting the above expression of G(x) in (3.3), we get J, j(a, 0, c) = aJ Y (~}-)^T(j/P)>

tiH^+n+p-um

X r x‘lF(x)]^a+c+1+ip-^u,m,af(x)dx.    (3.5)

Jo

Again by setting t = [F(:e)]¹'^/q in (3.5) and simplifying the resulting expression, we derive the relation given in (3.1).

3.2. Lemma. For the distribution as given in (1.2) and any non-negative integers a, b, c

Ji,j(a,b,c) = m ^(-lf ( l ) Ji,i(^a + (b-v)(m+ 1), 0, c +«(m + 1)) (3.6)

_    y _ptl+, / b \_Q’/ff)p_

(^m+1'l^b    V ^v J [a{c+(m+l)«+l}+p-0//3)]

^x [a{a + c+ (m+ 1)5 + 2} +p+ q - {(i +j)/^}] ’ ^m ^    ¹    ^³'⁷^

₌    _(-r<^<t! (j-fflMti/aw_

haho [^Q(^c+1)+P“ (JlP)}^b+1{a(,a + c + 1) +p + q- {(« + j)/()}]’

(3.8)

where Ji,j(a, b, c) is as given in (3.2).

Proof: When m 7^ —1, we have

(ft-(F(»)) - h_m(F(x))f = _(m‘₁₁,|(Fl₁))”^tl - (F(y)r⁺']^b

Now substituting for [h_m(F(y)) — h_m(F(x))]^b in eąuation (3.2), we get

Ji,l(a,b,c) =    ^Z(-l)” ^ ) Aj(o+ (b- v)(m + l),0,c +v(m +1)).

Making use of the Lemma 3.1, we derive the relation given in (3.7).

When m — —1, we have

J_ij(a,6,c) = g as Ł(-1)’(J)=»-

On applying L’ Hospital rule, (3.8) can be proved on the lines of (2.6).

3.3. Theorem. For type II exponentiated log-logistic distribution as given in (1-2) and 1 < r < s < n, k = 1,2,... and m ^ —1