2949775126

721

Remark 2.2: Putting m = — 1 , in Theorem 2.4, we get a recurrence relation for single moments of upper k record values from type II exponentiated log-logistic distribution in the form

(i ~ ^k)^E l(x&,⁾³']=)i+

Inverse moments of gos from type II exponentiated log-logistic distribution can be obtain by the following Theorem.

2.5. Theorem. For type II exponentiated log-logistic distribution as giuen in (1.2) and 1 <r <n, k = 1,2,...,

E[X^{j 0}(r, n, m, fc)] = ^ -

^p!r(j-p)nr„ (l+Łt!^

Proof. From (1.5), we have

>j.    (2.16)

(r — l)!(m + l)^r'

x f x’~⁰[F(x)]'^1r~^u~¹ f(x)dx.

Jo

Now letting t = [p’(x)]¹ in (2.17), we get 1

(2.17)

s[x^»(_wąi)] ₌ ^Lvy;(--i)'^ł'(^¹ ) Z*)

(r-l)l(m+l)"^^    V “ JpwU-t

Kfl(-^-_{r +} B-r _{+ M+}^ł’ ^{+ 1}-»/«,!)

\m+l    a(m+l) /

Since

^(—1)“^ ^ ^ B(a+ fc, c) = B(fc, c + 6)

where B(a, b) is the complete beta function.

Therefore,

(2.18)

a^j-⁰Cr-1

(m + l)^r

^p!r(i

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

E(-i)^p

_P=o

tY «{fc+(n-r)(m+l)}+p+l-(J/g)\

H_Z

p/"q{fc+n(m+l)}+p+l —(j/ff) \

V “(m+D    )

(2.19)

and hence the result given in (2.16).

Special Cases

iii) Putting m = 0, k = 1 in (2.19), we get inverse moments of order statistics from type II exponentiated log-logistic distribution as;

oS-l>n\ y, (-¹)”^r(j)r[“(^re- r + l)+p + l- 07/3)] ⁽ⁿ - ^r)! ho p!r(j — p)r[a(n+ 1) + p+ 1 - (j//3)]