2949775120

716

or any other type of treatment. It has also been used in hydrology to model stream flow and precłpitation, and in economics to model the distribution of wealth or income.

Kamps [24] introduced the concept of generalized order statistics (gos) as follows: Let Xi,X2... be a seąuence of independent and identically distributed (iid) random vari-ables (rv) with absolutely continuous cumulative distribution function (cdf) F(x) and pdf, f(x), x G (a, /3). Let n G N, n > 2, k > 0, m G 3?, be the parameters such that

7r = k + (n — r)(m + 1)> 0, for all r G {1,2,..., n — 1}, where M_r = YlijZr ^m3- Then X(l,n,m, k),... ,X(n,n,m, k), r = 1,2,... n are called gos if their joint pdf is given by

(i.4)

on the cone F^-1(0) < xi < x% < ... < x_n < F^-1( 1).

The model of gos contains as special cases, order statistics, record values, seąuential order statistics.

Choosing the parameters appropriately (Cramer, [181), ^we S^et the variant of the gos given in Table 1.

Table 1: Yariants of the generalized order statistics

	■4Ć II £	Ir	mr
i) Seąuential order statistics		(n — r + l)or	I + 1
ii) Ordinary order statistics	1	n — r + 1	0
ii) Record values	1	1	-1
iv) Progressively type II censored order statistics	Rn + 1	n-r + 1 + Yjj=r	Rr
v) Pfeifer’s record values	0n	Pr	0r ~ Pr+1 ~ 1

For simplicity we shall assume mi — mi — ... — m_n-1 = m.

The pdf of the i—th gos, X(r,n,m,k), 1 < r < n, is

/_x(nn,»,«w =    (i.B)

and the joint pdf of X(r, n, m, k) and X(s,n,m,k), 1 < r < s < n, is

fx(= (_r _ _Ł)^’ _r _ WCWĆ ‘(^l¹))

x[8m(F(!/))-ftm(F(a_:))]*-’-¹[F(»)]¹--¹/(i,), x < y, (1.6)

where

F(x) = 1 -F(x), C_r-1 = i li = fc+(n—i)(m+l),

t=i

hm(x) = |__|n(⁺1__x)j    m= — i

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