2949775114

3.6. Theorem. For type II exponentiated log-logistic distribution

(l —    ) E[X^l (r, n, m, k)X⁵ ^ (s, n, m, A;)]

= E[X^l(r, n, m, k)X^j~^(s — 1, n, m, /e)]

(3.21)

₊ (j    ^łlmⁱ(_r,„,_nl,t)r^!>(,,n,i»,H], /3 > j.

otp^s

Proof The proof is easy.

Remark 3.4 Setting m = 0, k = 1 in (3.21), we obtain a recurrence relation for Ratio moments of order statistics for type II exponentiated log-logistic distribution in the form

. <r(j ~ P)

a0(n — s + 1)

')E[XⁱT._nXi7_r!‘\ = E[Xl_nXizl_n] + Ą_r-^^-E[X‘.„Xi-²'¹] /    aB(n — s+1)

Remark 3.5 Putting m = — 1, in Theorem 3.6, we get a recurrence relation for ratio moments of upper k record values from type II exponentiated log-logistic distribution in the form

.    - w

a/3k

)£[(*,

(^k) Wy(^fc) U(r)> \-^AU(a)J

a/3k

Remark 3.6 At7_r = n — r + 1 -f J2i=_r ^m*> l<r<j<n, rm£N, k = m_n + 1 in (3.16) the product moment of progressive type II censored order statistics of type II exponentiated log-logistic distribution can be obtained.

Remark 3.7 The result is morę generał in the sense that by simply adjusting j — 0 in (3.16), we can get interesting results. For example if j — 0 = — 1 then E ^(I n m fc) ] gives the moments of quotient. For j — 0 > 0, E[X^l(r, n, m, k) X^J-^(s, n, m, /c)] represent product moments, whereas for j < 0 , it is moment of the ratio of two generalized order statistics of different powers.

4. Characterization

This Section contains characterization of type II exponentiated log-logistic distribution by using the conditional expectation of gos .

Let X(r,n,m,k), r — 1,2,... ,n be gos, then from a continuous population with cdf F(x) and pdf f(x), then the conditional pdf of X(s,n,m,k) given X(r,n,m,k) — x, 1 < r < s < n, in view of (1.5) and (1.6), is

C _i

fx(,,n,m,k)\X(r,n,m,k){y\x) = (_s _ _r J ₁)|_Cr_₁

_{i<t (4},)

4.1. Theorem. Let X be a non-negative random yariable hatńng an absolutely continuous distribution function F{x) with F(0) = 0 and 0 < F(x) < 1 for all x > 0, then