Asymptotic generalized bivariate extreme with random index

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by  M. A. Abd Elgawad

Asymptotic generalized bivariate extreme with random index
M. A. Abd Elgawada,b, A. M. Elsawaha,c,d, , Hong Qina and Ting Yana
∗
7 a Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
1
0 b Department of Mathematics, Faculty of Science, Benha University, Benha 13518, Egypt
2
c Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt
n
a
d Division of Science and Technology, BNU-HKBU United International College, Zhuhai 519085, China
J

7
1

]
T Abstract
S
. In many biological, agricultural, military activity problems and in some quality
h
at control problems, it is almost impossible to have a fixed sample size, because some
m
observations are always lost for various reasons. Therefore, the sample size itself is
[
considered frequently to be a random variable (rv). The class of limit distribution

1
v functions (df’s) of the random bivariate extreme generalized order statistics (GOS)
2
8 from independent and identically distributed rv’s are fully characterized. When the
6
random sample size is assumed to be independent of the basic variables and its df is
4
0
assumed to converge weakly to a non-degenerate limit, the necessary and sufficient
.
1
conditions for the weak convergence of the random bivariate extreme GOS are ob-
0
7
tained. Furthermore, when the interrelation of the random size and the basic rv’s
1
: is not restricted, sufficient conditions of the convergence and the forms of the limit
v
i
X df’s are deduced. Illustrative examples are given which lend further support to our
r theoretical results.
a
Keywords: Weak convergence; Random sample size; Generalized order statistics; Gen-
eralized bivariate extreme.
1 Introduction
The concept of generalized order statistics (GOS) have been introduced by Kamps (1995).
It’s enable a unified approach to ascendingly ordered random variables (rv’s) as ordinary
order statistics (oos), sequential order statistics (sos), order statistics with non integral
sample size, progressively type II censored order statistics (pos), record values, kth record
∗Correspondingauthor. E-mail: [e-mail],[e-mail],a [e-mail]
1
valuesandPfeifer’srecords. Letγn = k > 0,γr = k+n−r+ jn=−r1mj >0,r =1,2,...,n−1,
and m˜ = (m1, m2, ,mn 1) n−1. Then the rv’s Xr:nP X(r,n,m˜,k),r = 1,2,...,n,
··· − ∈ ℜ ≡
are called GOS based on the distribution function (df) F with density function f which
are defined by their probability density function (pdf)
n n 1
f1(,m˜2,,.k..),n:n(x1,x2,...,xn)= γj − (1−F(xj))γj−γj+1−1f(xj)
j=1 j=1
Y Y
  
(1 F(x ))γn 1f(x ),
n − n
× −
where F 1(0) x ... x F 1(1).
− 1 n −
≤ ≤ ≤ ≤
In this work, we consider a wide subclass of GOS , by assuming γ γ = m+1> 0.
j j+1
−
This subclass is known as m GOS. Clearly many important practical models of m GOS
− −
are included such as oos, order statistics with non integer sample size and sos. The
marginal df’s of the rth and r`th m GOS (Nasri- Roudsari, 1996 and Barakat, 2007) are
−
(m,k) (m,k)
represented by Φ (x) = I (r,N r+1) and Φ (x) = I (N R +1,R ),
r:n Lm(x) − r`:n Lm(x) − r r
respectively, where r`= n r+1,L (x) = 1 (1 F(x))m+1,I (a,b) = 1 xta 1(1
− m − − x β(a,b) 0 − −
t)b 1dt denotes the incomplete beta ratio function, N = ℓ +n 1,R = ℓ+Rr 1 and
− r
− −
ℓ = k . Moreover, by usingthe results of Kamps(1995), we can write explicitly the joint
m+1
df’s of the rth and sth m GOS, m = 1,1 r < s n, as:
− 6 − ≤ ≤
F(x) F(y)
Φ(m,k)(x,y) = C⋆ ξmηγs 1(1 ξm+1)r 1(ξm+1 ηm+1)s r 1dηdξ, x y,
r,s:n n − − − − − − ≤
Z0 Zξ
(1.1)
whereC⋆ = (m+1)2Γ(N+1) andΓ(.)istheusualgammafunction. RecentlyBarakat
n Γ(N s+1)(r 1)!(s r 1)!
− − − −
et al. (2014a) studied the limit df’s of joint extreme m GOS, for a fixed sample size.
−
Moreover, the asymptotic behavior for bivariate df of the lower-lower (l-l), upper-upper
(u-u) and lower-upper (l-u) extreme m GOS in Barakat et al. (2014b).
−
In the last few years much efforts had been devoted to investigate the limit df’s of
independent rv’s with random sample size. The appearance of this trend is naturally
because many applications require the consideration of such problem. For example, in
many biological, agricultural and in some quality control problems, it is almost impossible
to have a fixed sample size because some observations always get lost for various reasons.
Therefore, the sample size n itself is considered frequently to be a rv ν , where ν is
n n
independentofthebasicvariables(i.e.,theoriginalrandomsample)orinsomeapplications
the interrelation of the basic variables and the random sample size is not restricted. Limit
theorems for extremes with random sample size indexes have been thoroughly studied in
the above mentioned two particular cases :
2
1. The basic variables and sample size index are independents (see, Barakat, 1997).
2. The interrelation of the basic variables and the random sample size is not restricted
(see, Barakat and El Shandidy, 1990, Barakat, 1997 and Barakat et al., 2015a).
Our aim in this paper is to characterize the asymptotic behavior of the bivariate df’s
of the (l-l), (u-u) and (l-u) extreme m GOS with random sample size. When the random
−
sample size is assumed to be independent of the basic variables and its df is assumed to
converge weakly to a non-degenerate limit, the necessary and sufficient conditions for the
weak convergence of the random bivariate extreme m GOS are obtained. Furthermore,
−
when the interrelation of the random size and the basic rv’s is not restricted, sufficient
conditions of the convergence and the forms of the limit df’s are deduced. An illustrative
examples are given which lend further supportto our theoretical results. Throughout this
paper the convergence in probability and the weak convergence, as n , respectively,
→ ∞
p w
denoted as ” ” and ” ”.
−n→ −n→
2 Asymptotic random bivariate extreme under m GOS
−
2.1 Random sample size and basic rv’s are independents
Inthissubsectionwedealwiththeweakconvergence ofbivariatedf’softhe(u-u),(l-l)and
(l-u) extreme m GOS are fully characterized in Theorems 2.1, 2.2 and 2.3, respectively.
−
When the sample size itself is a rv ν , which is assumed to be independent of the basic
n
variables X ,r = 1,2,...,n.
r:n
Theorem 2.1. Consider the following three conditions :
Φ(m,k)(x ,y ) = P(Z(n) < x) =P(Z(n) <x,Z(n) < y) w Φˆ(m,k)(x,y),x y, (i)
r`,s`:n n n r`,s`:n r`:n s`:n −n→ r,s ≤
νn w
H (nz) = P( < z) H(z), (ii)
n
n −n→
Φ(m,k)(x ,y )= P(Z(n) < x) = P(Z(n) < x,Z(n) < y) w Ψˆ(m,k)(x,y)
r`,s`:νn n n r`,s`:νn r`:νn s`:νn −n→ r,s
∞ (m,k)
= Ω (zκ ,zκ )dH(z). (iii)
r,s 1 2
Z0
Then any two of the above conditions imply the remaining one, where x = x = a x +
n 1n n 1
b ,y = x = a x +b ,a > 0,b are suitable normalizing constants, x = (x ,x ) =
n n 2n n 2 n n n 1 2
(x,y),r`= n r+1< n s+1 = s`,Z(n) = Xr`i:n−bn,i = 1,2,(r`,r`) = (r`,s`),Φˆ(m,k)(x,y)
− − r`i:n an 1 2 r,s
3
is a non-degenerate df, H(z) is a df with H(+0) = 0,
1 Γ (κm+1), x y,
(m,k) − Rs 2 ≥
Ω (κ ,κ ) =
r,s 1 2  1−ΓRr(κm1 +1)− Γ(R1r) κ∞m1+1I κm2+1 (Rs,Rr −Rs)uRr−1e−udu, x ≤ y,
(cid:18) u (cid:19)
R

Γ (x) = 1 xθr 1e θdθ denotestheincompletegammaratiofunction,κ = (x ),i =
r Γ(r) 0 − − i Uj;α i
1,2,j ∈ {1,2R,3},U1;α(xi) = x−i α,xi ≤ 0;U2;α(xi) = (−xi)α,xi > 0,α > 0 and U3(xi) =
3;α(xi) = e−xi, xi.
U ∀
Remark 2.1. The continuity of the limit df Φˆ(m,k)(x,y) in (i) implies the continuity of
r,s
the limit Ψˆ(m,k)(x,y). Hence the convergence in (iii) is uniform with respect to x and y.
r,s
Remark 2.2. It is natural to look for the limitations on ν , under which we get the
n
relation Φˆ(m,k)(x,y) Ψˆ(m,k)(x,y) x,y. In view of Theorem 2.1, the last equation is
r,s r,s
≡ ∀
satisfied if and only if the df H(z) is degenerate at one, which means the asymptotically
almostrandomlessnessofν .Weassume,duetoRemark2.2,thatH(z)isanon-degenerate
n
df and H(+0) = 0, i.e., continuous at zero.
(m,k)
Proof of the implication (i)+(ii) (iii): First, we note that Φ (x ,y ) can be
⇒ r`,s`:n n n
written in the form (see, Theorem 2.3 in Barakat et al., 2014b)
1 N
Φ(m,k)(x ,y )= 1 Γ (NL (x )) I (R ,R R )uRr 1e udu,
r`,s`:n n n − Rr m n − Γ(R ) NLm(yn) s r− s − −
r ZNLm(xn) u
(2.1)
where L (.) = 1 L (.). Now by using the total probability rule we get,
m m
−
(m,k) ∞ (m,k)
Φ (x ,y ) = Φ (x ,y )P(ν =t). (2.2)
r`,s`:νn n n r`,s`:t n n n
t=r
X
Assume that H (z) = P(ν = t) = P(ν z) and z = [t], where [θ] denotes the
n t z n n ≤ n
≤
greatest integer part ofPθ. Thus, the relation (2.1) show that the sum term in (2.2) is a
Riemann sum of the integral
(m,k) ∞ (m,k)
Φ (x ,y )= Φ (x ,y ,z)dH (nz), (2.3)
r`,s`:νn n n r`,s`:n n n n
Z0
where, for sufficiently large n, we have
1 zN`
Φ(m,k)(x ,y ,z) = 1 Γ (zN`L (x )) I (R ,R R )uRr 1e udu,
r`,s`:n n n − Rr m n −Γ(Rr)ZzN`Lm(xn) zN`Lmu(yn) s r− s − −
where N` = (ℓ 1 +n) n. Appealing to the condition (i), Theorem 2.3 in Barakat et al.
−z ∼
(2014b) and Remark 2.2, we get
(m,k) w (m,k)
Φ (x ,y ,z) Ω (zκ ,zκ ), (2.4)
r`,s`:n n n −n→ r,s 1 2
4
where the convergence is uniform with respect to x and y, over any finite interval of z.
Now, let ξ be a continuity point of H(z) such that 1 H(ξ) < ǫ, (ǫ is an aribtary small
−
value). Then, we have
∞ (m,k) ∞
Ω (zκ ,zκ )dH(z) dH(z) = 1 H(ξ) < ǫ. (2.5)
r,s 1 2
≤ −
Zξ Zξ
In view of the condition (ii) we get, for sufficiently large n, that
∞ (m,k)
Φ (x ,y ,z)dH (nz) 1 H (nξ) 2(1 H(ξ)) < 2ǫ. (2.6)
r`,s`:n n n n ≤ − n ≤ −
Zξ
On the other hand, by the triangle inequality, we get
ξ ξ
(m,k) (m,k)
Φ (x ,y ,z)dH (nz) Ω (zκ ,zκ )dH(z)
r`,s`:n n n n − r,s 1 2
(cid:12)Z0 Z0 (cid:12)
(cid:12) (cid:12)
(cid:12) ξ ξ (cid:12)
(cid:12) (m,k) (m,k) (cid:12)
Φ (x ,y ,z)dH (nz) Ω (zκ ,zκ )dH (nz)
≤ r`,s`:n n n n − r,s 1 2 n
(cid:12)Z0 Z0 (cid:12)
(cid:12) (cid:12)
(cid:12) ξ (m,k) ξ (m,k) (cid:12)
+(cid:12) Ω (zκ ,zκ )dH (nz) Ω (zκ ,zκ )dH(z) ,(cid:12) (2.7)
r,s 1 2 n r,s 1 2
−
(cid:12)Z0 Z0 (cid:12)
(cid:12) (cid:12)
where the conver(cid:12)gence in (2.4) is uniform over the finite interval 0 z ξ(cid:12). Therefore, for
(cid:12) ≤ ≤ (cid:12)
arbitrary ǫ > 0 and for sufficiently large n, we have
ξ
(m,k) (m,k)
Φ (x ,y ,z) Ω (zκ ,zκ ) dH (nz) ǫH (nξ) ǫ. (2.8)
r`,s`:n n n − r,s 1 2 n ≤ n ≤
(cid:12)(cid:12)Z0 h i (cid:12)(cid:12)
In order to(cid:12)estimate the second difference on the right hand(cid:12)side of (2.7), we construct
(cid:12) (cid:12)
Riemann sums which are close to the integral there. Let T be a fixed number and 0 =
ξ < ξ < ... < ξ = ξ be continuity points of H(z). Furthermore, let T and ξ be such
0 1 T i
that
ξ T
(m,k) (m,k)
Ω (zκ ,zκ )dH (nz) Ω (ξ κ ,ξ κ )(H (nξ ) H (nξ )) < ǫ
r,s 1 2 n r,s i 1 i 2 n i n i 1
(cid:12)(cid:12)Z0 −Xi=1 − − (cid:12)(cid:12)
(cid:12) (cid:12)
and(cid:12) (cid:12)
(cid:12) (cid:12)
ξ T
(m,k) (m,k)
Ω (zκ ,zκ )dH(z) Ω (ξ κ ,ξ κ )(H(ξ ) H(ξ )) < ǫ.
r,s 1 2 r,s i 1 i 2 i i 1
(cid:12)(cid:12)Z0 −Xi=1 − − (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) w (cid:12)
Since,b(cid:12)ytheassumption Hn(nξi) H(ξi), 0 i T,thetwoRiemannsu(cid:12)msarecloser
−n→ ≤ ≤
to each other than ǫ forall nsufficiently large. Thus,once again bythetriangle inequality,
the absolute value of the difference of the integrals is smaller than 3ǫ. Combining this fact
with (2.8), the left hand side of (2.7) becomes smaller than 4ǫ for all large n. Therefore,
in view of (2.5), (2.6) and (2.4), we have
ξ ξ
Φ(m,k)(x ,y ) Ψˆ(m,k)(x,y) < Φ(m,k)(x ,y ,z)dH (nz) Ω(m,k)(zκ ,zκ )dH(z)
r`,s`:νn n n − r,s r`,s`:n n n n − r,s 1 2
(cid:12) (cid:12) (cid:12)Z0 Z0 (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) 5 (cid:12)
∞ (m,k) ∞ (m,k)
+ Φ (x ,y ,z)dH (nz)+ Ω (zκ ,zκ )dH(z) < 7ǫ.
r`,s`:n n n n r,s 1 2
Zξ Zξ
This completes the proof of the first part of the theorem.
Proof of the implication (i) + (iii) (ii): Starting with the relation (2.3), we
⇒
select a subsequence n′ of n for which Hn′(n′z) converges weakly to an extended
{ } { }
df H (z) (i.e., H ( ) H (0) 1 and such a subsequence exists by the compactness
′ ′ ′
∞ − ≤
of df’s). Then, by repeating the first part of the theorem for the subsequence n ,
′
{ }
with the exception that we choose ξ so that H ( ) H (ξ) < ǫ, we get Ψˆ(m,k)(x,y) =
′ ′ r,s
∞ −
0∞Ωr(m,s,k)(zκ1,zκ2)dH′(z). Since the function Ψˆr(m,s,k)(x,y) is a df, we get Ψˆr(m,s,k)(∞,∞) =
R1 = 0∞dH′(z) = H′(∞) − H′(0), which implies that H′(z) is a df. Now, if Hn(nz)
did nRot converge weakly, then we can select two subsequences n and n such that
′ ′′
{ } { }
w w
Hn′(n′z) −n→′ H′(z) and Hn′′(n′′z) −n→′′ H′′(z), where H′(z) and H′′(z) are df’s. In this
case, we get
Ψˆ(m,k)(x,y) = ∞Ω(m,k)(zκ ,zκ )dH (z)= ∞Ω(m,k)(zκ ,zκ )dH (z).
r,s r,s 1 2 ′ r,s 1 2 ′′
Z0 Z0
Thus, let (y ), we get
→ ∞
∞Γ (zκm+1)dH (z) = ∞Γ (zκm+1)dH (z). (2.9)
Rr 1 ′ Rr 1 ′′
Z0 Z0
Appealing to equation (2.9) and by using the same argument which is applied in the proof
of the second part of Theorem 2.1 in Barakat, 1997, we can easily prove H (z) = H (z).
′ ′′
This complete the proof of the second part.
Proof of the implication (ii)+(iii) (i): For proving this part, we need first present
⇒
the following lemma.
Lemma 2.1. For all x ,i = 1,2, we have
i
(n)
[1 Γ (NL (x ))] σ P(Z < x ) [1 Γ (NL (x ))]+ρ (2.10)
− Rri m in − i,N ≤ r`i:n i ≤ − Rri m in i,N
and
(n)
Γ (NL (x )) ρ P(Z x ) Γ (NL (x ))+σ , (2.11)
Rri m in − i,N ≤ r`i:n ≥ i ≤ Rri m in i,N
where 0 < ρ ,σ 0 (or equivalently, as n ),(R ,R ) = (R ,R ) and
i,N i,N −N→ −→ ∞ r1 r2 r s
(r ,r ) = (r,s).
1 2
Proof. Since 0 Γ (x) 1 x, the proof of the lemma will immediately follow from
≤ Rri ≤ ∀
the result of Smirnov (1952) (Theorem 3, p. 133, or Lemma 2.1 in Barakat, 1997).
WenowturntotheproofofthelastpartofTheorem2.1. InviewofRemark2.1,wecan
assume, without any loss of generality, that the df Ψˆ(m,k)(x,y) is continuous. Therefore,
r,s
6
the condition (iii) willbesatisfied for all univariate marginals of Ψˆ(m,k)(x,y), i.e., wehave
r,s
Φ(m,k)(x ) w Ψˆ(m,k)(x ),i = 1,2, (2.12)
r`i:νn in −n→ ri i
whereΨˆr(mi ,k)(xi)= 0∞[1−ΓRri(zκmi +1)]dH(z),i = 1,2,isthemarginalsdf’sofΨˆr(m,s,k)(x,y).
We shall now proveR
Φ(m,k)(x ) w Φˆ(m,k)(x )= 1 Γ (κm+1),i = 1,2. (2.13)
r`i:n in −n→ ri i − Rri i
(n)
In view of Lemma 2.1, we first show that the sequence Z , i = 1,2, is stochastically
{ r`i:n}n
bounded (see, Feller, 1979). If we assume the contrary, we would find ε ,ε > 0 such
i,1 i,2
that at least one of the two following relations
n
(a) nlim P(Zrh`i:in ≥ xi) ≥ εi,1 > 0, ∀ xi > 0, i= 1,2,
→∞ n
(b) nlim P(Zrh`i:in < xi) ≥ εi,2 > 0, ∀ xi < 0, i = 1,2
→∞ (n)
is satisfied. The assertions (a) and (b) mean that the sequence Z , is not stochasti-
{ r`i:n}n
cally bounded at the left ( ) and at the right (+ ), respectively. Let the assumption
−∞ ∞
(a) be true. Since H(z) is non-degenerate df, we find ε > 0 and β > 0 such that
0
ν
n
P β ε , for sufficiently large n. (2.14)
0
n ≥ ≥
(cid:16) (cid:17)
Using the following well known inequality, for i = 1,2,
(n) n
P Zr`i:n1 ≥ xi ≥ P Zrh`i:in2 ≥ xi , ∀ n1 ≥ n2. (2.15)
(cid:16) (cid:17) (cid:16) (cid:17)
We thus get the following inequalities, for sufficiently large n,
(n) (n)
P Z x P Z x P(ν = t)
r`i:νn ≥ i ≥ r`i:t ≥ i n
(cid:16) (cid:17) t≥X[nβ] (cid:16) (cid:17)
n (n)
≥ P Zrh`i:i[nβ] ≥ xi P (νn ≥ [nβ]) ≥ ε0P Zr`i:[nβ] ≥ xi , i =1,2,
(cid:16) (cid:17) (cid:16) (cid:17)
(note that P (ν [nβ]) P (ν nβ)). Therefore,
n n
≥ ≥ ≥
n (n)
nlim P(Zrh`i:iνn ≥ xi)≥ ε0 nlim P(Zr`i:[nβ] ≥ xi).
→∞ →∞
n n
Now,ifwefindε′i,1 > 0suchthat nlim P(Zrh`i:i[nβ] ≥ xi) ≥ ε′i,1 > 0,weget nlim P(Zrh`i:iνn ≥
→∞ →∞
x ) ǫ ε > 0, which contradicts the right stochastic boundedness of the sequence
i ≥ 0 ′i,1
(n)
Z and consequently contradicts the relation (2.12). However, if such an ε > 0
{ r`i:νn}n ′i,1
n
does not exist we have nlim P(Zrh`i:i[nβ] ≥ xi) = 0, which in view of Lemma 2.1 (relation
→∞ n
(2.10)) leads to the following chain of implications (∀ xi > 0) P(Zrh`i:i[nβ] ≥ xi) → 0 ⇒
Γ ([Nβ]L (x )) 0 [Nβ]L (x ) 0 NL (x ) 0 (since NL (x ) 0)
Rri m in → ⇒ m in → ⇒ m in → m in →
7
(n)
Γ (NL (x )) 0 P(Z x ) 0, which contradicts the assumption (a).
⇒ Rri m in → ⇒ r`i:n ≥ i →
Consider the assumption (b). Since H(z) is a df we can find a positive integer δ and real
number α > 0 such that
ν
n
P( δ) α, for sufficiently large n. (2.16)
n ≤ ≥
Therefore, in view of (2.16) and the inequality (2.15), we have
δn
ν
(n) (n) (n) n
P(Z < x ) P(Z < x )P(ν = t) P(Z < x )P( δ)
r`i:νn i ≥ r`i:t i n ≥ r`i:δn i n ≤
t=r
X
(n)
αP(Z < x ), i = 1,2.
≥ r`i:δn i
n n
Hence, we get nlim P(Zrh`i:iνn < xi) ≥ α nlim P(Zrh`i:iδn < xi). By using Lemma 2.1
→∞ →∞
(relation (2.11)) and applyingthesame argumentas in thecase (a), it is easy to show that
the last inequality leads to a contradiction (the last inequality, in view of the assumption
(n)
(b)), which yields that the sequences Z ,i = 1,2, is not stochastically bounded at
{ r`i:n}n
(n)
theleft. Thiscompletes theproofthatthesequences Z , i= 1,2, arestochastically
{ r`i:n}n
(m,k)
bounded. Now, if Φ (x ),i = 1,2, did not converge weakly, then we could select two
r`i:n in
subsequences {n′} and {n′′} such that Φ(r`mi:n,k′)(xin′) would converge weakly to Φˆr′(im,k)(xi)
and Φ(r`mi:n,k′′)(xin′′) to another limit df Φˆr′′i(m,k)(xi). In this case we get (by repeating the
′ ′′
first part of Theorem 2.1 for the univariate case and for the two subsequences n , n )
{ } { }
(m,k) ∞ ′(m+1) ∞ ′′(m+1)
Ω (x ) = 1 Γ (zκ ) dH(z) = 1 Γ (zκ ) dH(z).
ri i − Rri i − Rri i
Z0 h i Z0 h i
However, Lemma 3.2 in Barakat (1997) shows that the last equalities, cannot hold unless
′ ′′
κ κ ,i = 1,2. Hence the relation (2.13) is proved. Hence, the proof of Theorem 2.1 is
i ≡ i
completed.
Let and be the classes of all possible limit df’s in (i) and (iii), respectively.
ν
G G
The class is fully determined by Barakat et al. (2014b). Furthermore, let S and S
ν
G
be the necessary and sufficient conditions for the validity of the relations (i) and (iii),
respectively. The following corollary characterizes the class .
ν
G
Corollary 2.1. For every df Ψˆ(m,k)(x,y) in there exists a unique df Φˆ(m,k)(x,y) in
ν
G
, such that Ψˆ(m,k)(x,y) is uniquely determined by the representation (iii). Moreover,
G
S = S .
ν
Proof of corollary 2.1. Let us first prove the implication Φˆ′(m,k)(x,y) = Φˆ′′(m,k)(x,y)
r,s r,s
{ 6 }
= Ψˆ′(m,k)(x,y) = Ψˆ′′(m,k)(x,y) . If we assume the contrary, we get Ψˆ′(m,k)(x,y) =
r,s r,s r,s
⇒ { 6 }
Ψˆ′′(m,k)(x,y), while Φˆ′(m,k)(x,y) = Φˆ′′(m,k)(x,y). Appealing to the first part of Theorem
r,s r,s r,s
6
8
′(m+1) ′′(m+1)
2.1 , we get 0∞ 1−ΓRri(zκi ) dH(z) = 0∞ 1−ΓRri(zκi ) dH(z),i = 1,2.
The last equaRlitiehs, as we have seenibefore, fromR Lhemma 3.2 in Barakait (1997), cannot
hold unless κ = κ ,i = 1,2. Therefore, Corollary 2.1 is followed as a consequence of
′i ′i′
Theorem 2.1 and the last implication. This completes the proof of Corollary 2.1.
Theorem 2.2. Consider the following three conditions :
Φ(m,k)(x ,y ) =P(Z(n) < x) = P(Z(n) < x,Z(n) < y) w Φ(m,k)(x,y),x y, (i)
r,s:n n n r,s:n r:n s:n −n→ r,s ≤
νn w
H (nz) = P( < z) H(z), (ii)
n
n −n→
Φ(m,k)(x ,y )= P(Z(n) < x) = P(Z(n) < x,Z(n) < y) w Ψ(m,k)(x,y)
r,s:νn n n r,s:νn r:νn s:νn −n→ r,s
= ∞Ω(m,k)(zρ ,zρ )dH(z). (iii)
r,s 1 2
Z0
Then any two of the above conditions imply the remainingone, wherex = c x+d ,y =
n n n n
cny+dn,cn > 0,dn aresuitablenormalizingconstants, 1 ≤r < s ≤ n,Zr(in:)n = Xri:cnn−dn,i =
(m,k)
1,2,Φ (x,y) is a non-degenerate df, H(z) is a df with H(+0) = 0,
r,s
Γ (ρ ), x y,
Ω(m,k)(ρ ,ρ ) = s 2 ≥
r,s 1 2  1 ρ1Γ (ρ u)ur 1e udu, x y,
 (r−1)! 0 s−r 2− − − ≤
R
ρ = (x ),i = 1,2,j 1,2,3 , (x ) = ( x ) β,x 0; (x ) = xβ,x > 0,β > 0
i Vj;β i ∈ { } V1;β i − i − i ≤ V2;β i i i
and 3(xi)= 3;β(xi)= exi, xi.
V V ∀
Theorem 2.3. Consider the following three conditions :
Φ(m,k)(x ,y ) = P(Z(n) < x) = P(Z(n) < x,Z(n) < y) w Φ(m,k)(x)Φˆ(m,k)(y)
r,s`:n n n r,s`:n r:n s`:n −n→ r s
= Γ (ρ ))[1 Γ (κm+1),1 r,s n, (i)
r 1 − Rs 2 ≤ ≤
νn w
H (nz) = P( < z) H(z), (ii)
n
n −n→
Φ(m,k)(x ,y )= P(Z(n) < x) = P(Z(n) < x,Z(n) < y) w Ω(m,k)(zρ )Ω(m,k)(zκ ).
r,s`:νn n n r,s`:νn r:νn s`:νn −n→ r 1 s 2
(iii)
Thenanytwo oftheaboveconditions implytheremainingone, where x = c x+d ,y =
n n n n
any + bn,an,cn > 0,bn,dn are suitable normalizing constants, Zr(n:n) = Xr:cnn−dn,Zs`(:nn) =
Xs`:ann−bn,Φr(m,k)(x),Φˆs(m,k)(y) are non-degenerate df’s, H(z) is a df with H(+0) = 0,
Ω(m,k)(zρ ) = ∞Γ (zρ )dH(z) and Ω(m,k)(zκ ) = ∞ 1 Γ (zκm+1) dH(z).
r 1 r 1 s 2 − Rs 2
Z0 Z0
(cid:2) (cid:3)
Proof of Theorems 2.2 and 2.3. Without significant modifications, the method of the
proof of Theorems 2.2 and 2.3 are the same as that Theorem 2.1, except only the obvious
changes. Hence, for brevity the details of the proof are omitted.
9
2.2 The interrelation of ν and the basic rv’s is not restricted
n
When theinterrelation between therandomindex andthebasicvariables is notrestricted,
paralleltheoremofTheorem2.1maybeprovedbyreplacingthecondition(ii)byastronger
one. Namely, the weak convergence of the df H (nz) must bereplaced by the convergence
n
in probability of the rv νn to a positive rv . However, the key ingredient of the proof
n T
of this parallel result is to prove the mixing property, due to R´enyi (see, Barakat and
Nigm, 1996) of the sequence of order statistics under consideration. In the sense of R´enyi
a sequence u of rv’s is called mixing if for any event of positive probability, the
n
{ } E
conditional df of u , under the condition , converges weakly to a non-degenerate df,
n
{ } E
which does not depend on , as n . The following lemma proves the mixing property
E → ∞
(n)
for the sequence Z .
{ r`,s`:n}n
(n)
Lemma 2.2. Under the condition (i) in Theorem 2.1 the sequence Z is mixing.
{ r`,s`:n}n
(n) (l)
Proof. The lemma will be proved if one shows the relation P(Z < x Z <
r`,s`:n | r`,s`:l
x) w Φˆ(m,k)(x,y), for all integers l = r,r +1,.... The sufficiency of the above relation
r,s
−n→
can easily be proved as a direct multivariate extension of Lemma 6.2.1, of Galambos
(1987). However, this relation is equivalent to
(n) l w (m,k)
P(Zr`,s`:n ≥x |Zr`h,si`:l ≥ x) −n→ Φr,s (x,y), (2.17)
where Φ(m,k)(x,y) is the survival function of the limit df Φˆ(m,k)(x,y), i.e.,
r,s r,s
Φ(m,k)(x,y) = 1 Φˆ(m,k)(x) Φˆ(m,k)(y)+Φˆ(m,k)(x,y).
r,s r s r,s
− −
Therefore, our lemma will be established if one proves the relation (2.17). Now, we can
write
(n) (l) (n) (n) (l)
P(Z x Z x) = P(Z x,Z < x Z x)
r`,s`:n ≥ | r`,s`:l ≥ r`,s`:n ≥ r`,s`:l | r`,s`:l ≥
(n) (n) (l)
+P(Z x,Z x Z x). (2.18)
r`,s`:n ≥ r`,s`:l ≥ | r`,s`:l ≥
Bearing in mind that all X ,r = 1,2,...,n, are i.i.d rv’s, the first term in (2.18) can be
r:n
written in the form
(n) (n) (l) (n) (n) (l)
P(Z x,Z < x Z x) = P(Z∗ x,Z < x Z x)
r`,s`:n ≥ r`,s`:l | r`,s`:l ≥ r`,s`:(n l) ≥ r`,s`:l | r`,s`:l ≥
−
(n) (n) (n) (l)
= P(Z∗ x) P(Z∗ x,Z x Z x),
r`,s`:(n l) ≥ − r`,s`:(n l) ≥ r`,s`:l ≥ | r`,s`:l ≥
− −
where
(n) (n) (n)
Z∗ = (Z∗ ,Z∗ ),
r`,s`:(n l) r`:(n l) s`:(n l)
− − −
10