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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