2949775113

727

(3.18)

By setting z = [F(y)]¹/⁰ in (3.18), we find that

On substituting the above expression of J(x) in (3.17), we get

<T^j~⁰Cs-l

E[X^l (r, n, m, k)X^{3 p}(s,n,m, A;)] =

(r — l)!(s — r — l)!(m + l)^s-2

EE

p^|r(i -p) '--+²±^ł=^^a=ⁱ-_I&.

(3.19)

Again by setting t = [F(x)]^x^^Q in (3.19), we get

E[X^t(r,n,m,k)X^{j 0}(s,n,m,k)]

(m + l)^s

EE(-»'

c«{fc+(n-r)(m+l)}+P+q-{(i+j)/g}l

a(m+l)    J

p^!«! r(j -p)r(f +1 - ₉) _r    ]

p f q{fc+(n—s)(m+l)}+p+l —(j/0) 1

(3.20)

ttlfc+fn-rKTO+Dł+p+l-Cj/ffll a(m+l)    I

and hence the result given in (3.16).

Special cases

iii) Putting m = 0, k = 1 in (3.20), the explicit formula for the ratio moments of order statistics of the type II exponentiated log-logistic distribution can be obtained as

_P!₉!r(i-_P)r(i + i-₅) .r[a(n -r + l)+p + q- {(» + j)/i8}]r[q(n -s + l)+p+l- (j/P)\

; + l

r[a(n+ 1) +p + q- {(i+j)//9}]r[a(n-r + l)+p + 1 - (j/P)] iv) Putting m — —1 in (3.16), the explicit expression for the ratio moments of upper k record values for the type II exponentiated log-logistic distribution can be obtained as

^{r(i)rU + l)}

UU ^v'^q- ^r(p -p)^r(j + 1 -5)

Making use of (1.6), we can derive recurrence relations for ratio moments of gos from

(1.2).