2949775111

725

where,

^r,an (r — l)!(s — r — l)!(ra — s)!

ii)Putting m = — 1 in (3.10), we deduce the explicit expression for the product moments of upper k record values for the type II exponentiated log-logistic distribution in view of (3.9) and (3.8) in the form

(-i y^+qu/P)(_r)

[ak+p-(j/0)Y~^r

= (“^fc)^v+i E E

p=0 9=0

and hence for upper records

.    im w

[ak + p + q - {(i + j)/0}Y

E(Xⁱ X^j ) = a^s(T^i+j Y' y'__( ¹)^P+<?(j/P)(,_P){^l/P)(g)_

^{u{r) u(a)}    iśźilB+P-U/W-rla+p + g-Ui+Jmi

Remark 3.1 At j — 0 in (3.10), we have

E[X^l(r, n, m, k) =

aa¹ C_a-1

(r — l)!(s — i— l)!(m + 1)^;

hEE E (-!)⁵⁺“⁺”

?=0u=0 v=0

(</«,

7—«[^Q7r-u+q-(i/P)Y Making use of (3.12) in (3.13) and simplifying the resułting expression, we get

(3.12)

B[X‘(r,n,m,k) = -T.',^Cr~\s    , EEM)”'

¹    (r-l)!(m +1)-¹ fr-fr;'

_x ( r- 1 \    (i/0),

\ u J [a_7r_„ + q - (i/0)} ’

as obtained in (2.12).

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

(1.2).

3.4. Theorem. For the given type II exponentiated log-logistic distribution and n € N, meS, l<r<s<n — 1

■,*)]

^1---j^— ^ E[X^l(r, n, m, k)X^ (s, n, m, fc)] = E[X^l{r, n, m, k)X^:’(s — 1, i

a (3 7<

Proof: Prom (1.6), we have

E[X^t(r,n,m,k)X^{j 0}(s,n,m,k)]. E[X*(r, n, m, k)X^j (s,n,m, A:)] = -

(3.14)

C_s-1

(r — l)!(s — r — 1)!

x f x^t[F(x)]^rnf(x)gy^l(F{x))I(x)dx,    (3.15)

Jo

where

I(x) = jH    (»)) - h_m(F(x))]-^r-¹f(y)dy.