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On the confidence region for the bivariate co-ordinate-wise quantiles of the continuous bivariate distribution functions
Statistics & Probability Letters 56 (2002) 37 – 43
On the con dence region for the bivariate co-ordinate-wise quantiles of the continuous bivariate distribution functions H.M. Barakat ∗ Department of Mathematics, Faculty of science, Zagazig University, Zagazig, Egypt Received June 2000; received in revised form July 2001
Abstract In this paper, the distribution-free con dence region (rectangular) for the vector of the co-ordinate-wise quantiles of a general continuous bivariate distribution function is explicitly derived. This con dence region, in general, depends on the dependence function. A procedure is suggested which enables one to attach a con dence coe6cient to the estimate of the con dence region even if the dependence function is unknown. Moreover, some approximated distribution-free lower bounds, which are independent of the dependence function, of the con dence coe6cient of this region are derived. c 2002 Finally, some results of this study are extended to the three-dimensional vector of the co-ordinate-wise quantiles. Elsevier Science B.V. All rights reserved Keywords: Distribution-free con dence region; Bivariate order statistics; Bivariate quantiles
1. Introduction If measurements of two characteristics are taken on the same members of the population, then the observed random quantities follow some type of bivariate distribution. Let this distribution be F(x) = F(x1 ; x2 ) = P(X1 ¡ x1 ; X2 ¡ x2 ). Consider a sequence of n independent two-dimensional random variables (r.v.’s) {X j } = {(X1j ; X2j )}; j = 1; 2; : : : ; n with common distribution function (d.f.) F(x). The order statistics of the ith (i = 1; 2) component (characteristic) are Xi; 1:n 6 Xi; 2:n 6 · · · 6 Xi; n:n . Write x 6 y to mean xi 6 yi ; i = 1; 2 and G(x) = P(X ¿ x) to denote the survival function. For the ith (i = 1; 2) component of the vector X = (X1 ; X2 ); let Fi (xi ) and Gi (xi ); be the marginals of F(x) and G(x), respectively. Suppose that the two equations F1 (x1 ) = p1 ; 0 ¡ p1 ¡ 1 and
F2 (x2 ) = p2 ;
0 ¡ p2 ¡ 1
∗ Fax: +55-345452. E-mail address: [email protected] (H.M. Barakat).
c 2002 Elsevier Science B.V. All rights reserved 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 1 4 5 - 6
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H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
have unique solutions, say xt = tpt ; t = 1; 2 which are known as the (population) quantiles of order p1 and p2 of the d.f.’s F1 and F2 , respectively. Clearly, if F() = p; where = (1p1 ; 2p2 ) and 0 ¡ pt ¡ 1; t = 1; 2; then G() = q = 1 − p1 − p2 + p and in this case we may call as the pth quantile vector of the d.f. F(x); where p = (p1 ; p2 ; p): Clearly, the vector of the co-ordinate-wise quantiles of orders p1 and p2 is an extension of the notion of quantiles in the two-dimensions case. It is worth mentioning that there are many diEerent versions of bivariate (and multivariate) quantiles considered in the literature. See, e.g., Chaudhuri (1996) and the review and the references therein. The main object, which distinguishes these versions is the technique which is chosen for ordering the bivariate (and multivariate) random sample, see Barnett (1976). In this paper, we focus our attention only on the co-ordinate-wise version of bivariate quantiles based on what might be meant by M -ordering technique (see, Barnett, 1976), i.e., the ordering of the bivariate random sample X j ; j = 1; 2; : : : ; n takes place within the two marginal samples X1j ; j = 1; 2; : : : ; n and X2j ; j = 1; 2; : : : ; n. The con dence intervals for tpt ; t = 1; 2 (tpt = Ft−1 (pt ) = sup{u : F(u) ¡ pt }) whose endpoints are order statistics and have coverage probabilities free of Ft have been extensively studied, see David (1981, pp. 15 –16), Arnold et al. (1993, pp. 183–185) and references therein. The earliest work in this area seems to be that of Thompson (1936), Namely, the 100tn % con dence interval for tpt is (X1; rt :n ; X2; st :n ); where 1 6 rt ¡ st 6 n; t = 1; 2 and tn = tn (rt ; st ; pt ) =
n−s t i=rt
n pti (1 − pt )n−i ; i
(1.1)
as long as Ft is continuous. The technique developed to obtain this result has been extended to obtain several related results such as con dence intervals for quantile diEerences (see David, 1981, and references therein). While distribution-free con dence intervals for 1p1 ; 0 ¡ p1 ¡ 1; or 2p2 ; 0 ¡ p2 ¡ 1; individually can be obtained by using (1.1), occasions may arise where a joint con dence region for the two quantiles is required. The works for this subject are very limited. However, when p1 = p2 = 12 the nearly only known procedure for constructing a distribution-free con dence region for the median vector is given in Maritz (1995). This procedure mainly depends on a test statistic, by which one can assign an asymptotic critical value (in view of preassigned con dence coe6cient). In order to construct the required con dence region, the observed value of this statistic must be determined for every point in the range of the random vector X = (X1 ; X2 ) and test whether each of these points is a member of the con dence region or not i.e., we construct the region point by point (for the details see Maritz, 1995, Example 7:4). This is of course not an eEective procedure, specially when X is continuous. Moreover, the test statistic of this procedure mainly depends on the symmetries arising from the statistical properties of the median vector, i.e., F1 ( 1 ) = F2 ( 1 ) (p1 = p2 = 12 ) and F( 1 ; 1 ) = G( 1 ; 1 )(p = q): 12
22
12
22
12
22
Therefore, it does not work on the other quantiles. In this paper, a distribution-free con dence region (rectangular) for pth quantile vector of F(x); ∀(0; 0) ¡ (p1 ; p2 ) ¡ (1; 1); is explicity derived. Although this con dence region does not directly depend on the d.f. F(x); (or, of course, on its marginals) it depends clearly on p. Therefore, it will be used only in such cases when the marginal distributions are unknown but its dependence function (for de nition see Galambos, 1978=1987, Chapter 5) is known, e.g., F = F1 F2 (1 − (1 − F1 )(1 − F2 )) (Morgenstern d.f., see Galambos, 1987, Chapter 5), where is known while Ft ; t = 1; 2 are unknown d.f.’s, i.e., in this case we only know the dependence function D(u1 ; u2 ) = u1 u2 (1 − (1 − u1 )(1 − u2 )): However, at the end of Section 2, we suggest a procedure which enables us to attach an approximated con dence coe6cient to our estimate of the con dence region even if the dependence function is unknown. In Section 3, the approximated lower bounds, which are independent of p (or q), of the con dence coe6cient of this region are derived.
H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
39
2. Distribution-free condence region for quantile vector Theorem 2.1. For any given 1 6 rt ¡ st 6 n; t = 1; 2 we have n = n (r1 ; s1 ; r2 ; s2 ; p) = P(X1; r1 :n ¡ 1p1 6 X1; s1 :n ; X2; r2 :n ¡ 2p2 6 X2; s2 :n ) =
n−r 1
i∧j
n−r 2
Ii; j; k:n (q);
(2.1)
i=n−s1 +1 j=n−s2 +1 k=0∨(i+j−n)
where Ii; j; k:n (q) =
n! (q1 − q)i−k qk (q2 − q)j−k (1 − q1 − q2 + q)n−i−j+k ; (i − k)!k!(j − k)!(n − i − j + k)!
q = (q1 ; q2 ; q); qt = 1 − pt ; t = 1; 2 and a ∨ b = max(a; b); a ∧ b = min(a; b) ∀a; b: Proof. Let R = {X1; r1 :n ¡ 1p1 6 X1; s1 :n ; X2; r2 :n ¡ 2p2 6 X2; s2 :n }; 6 Xt; st :n } and Ct = {Xt; st :n ¡ tpt }; where t = 1; 2. Therefore, A1 ∩ A2 = B1 ∩ B2 B1 ∩ C2 C1 ∩ B2 C1 ∩ C2 :
At = {Xt; rt :n ¡ tpt };
Bt = {Xt; rt :n ¡ tpt
In view of the obvious relation Bt ∩ Ct = ; t = 1; 2; we get P(R) = P(B1 ∩ B2 ) = P(A1 ∩ A2 ) − P(B1 ∩ C2 ) − P(C1 ∩ B2 ) − P(C1 ∩ C2 ):
(2.2)
On the other hand, if we write Fk1 ;k2 :n (x) = Fk1 ;k2 :n (x1 ; x2 ) = P(X1; k1 :n ¡ x1 ; X2; k2 :n ¡ x2 ); we get P(A1 ∩ A2 ) = P(X1; r1 :n ¡ 1p1 ; X2; r2 :n ¡ 2p2 ) = Fr1 ;r2 :n (1p1 ; 2p2 ); P(B1 ∩ C2 ) = P(X1; r1 :n ¡ 1p1 6 X1; s1 :n ; X2; s2 :n ¡ 2p2 ) = Fr1 ;s2 :n (1p1 ; 2p2 )−Fs1 ;s2 :n (1p1 ; 2p2 ); P(C1 ∩B2 ) = P(X2; r2 :n ¡ 2p2 6 X2; s2 :n ; X1; s1 :n ¡ 1p1 ) = Fs1 ;r2 :n (1p1 ; 2p2 ) − Fs1 ;s2 :n (1p1 ; 2p2 ) and P(C1 ∩ C2 ) = P(X1; s1 :n ¡ 1p1 ; X2; s2 :n ¡ 2p2 ) = Fs1 ;s2 :n (1p1 ; 2p2 ). Hence, P(R) = Fr1 ;r2 :n (1p1 ; 2p2 ) − Fr1 ;s2 :n (1p1 ; 2p2 ) − Fs1 ;r2 :n (1p1 ; 2p2 ) + Fs1 ; s2 :n (1p1 ; 2p2 ):
(2.3)
It can be shown that (see Barakat, 1990, 1998, 1999)
Fn−k+1; n−k +1:n (x) =
k−1 k −1
i∧j
i=0 j=0 r=0∨(i+j−n)
n! (G1 (x1 ) − G(x))i−r (i − r)!r!(j − r)!(n − i − j + r)!
×G r (x)(G2 (x2 ) − G(x))j−r (1 − G1 (x1 ) − G2 (x2 ) + G(x))n−i−j+r ; (2.4) where, in introducing (2.4), we use the summation i=0 ai ; ∀a ¿ 0 to denote the summation 1 + i=0 ai . This, combined with (2.3), yields n−r 1 n−r 2 n−r 1 n−s 2 n−s 1 n−r 2 n−s 1 n−s 2 Ii;∗j; k:n (q); P(R) = − − + i=0 j=0
i=0 j=0
where Ii;∗j; k:n (q) =
i∧j
Ii; j; k:n (q):
k=0∨(i+j−n)
i=0 j=0
i=0 j=0
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H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
It follows that
P(R) =
n−r 1
i=0
=
n−r 2 j=0
n−r 2
n−r 1
+
j=0
n−s 1 i=0
n−s 2
−
j=0
n−r 2
Ii;∗j; k:n (q)
j=0
n−r 1 n−s 1 Ii;∗j; k:n (q) −
j=n−s2 +1
=
−
n−s 2
i=0
n−r 2
i=0
Ii;∗j; k:n (q) =
i=n−s1 +1 j=n−s2 +1
n−r 1
n−r 2
i∧j
Ii; j; k:n (q):
i=n−s1 +1 j=n−s2 +1 k=0∨(i+j−n)
The proof is completed. General case (unknown dependence function): In order to estimate a con dence region for the quantile vector for some speci ed values of p1 and p2 when the dependence function is unknown, i.e., p (or q) is not preassigned, we must seek a procedure which rst enables us to estimate p (or q) and then by using (2.1), we can construct the desired con dence region for the given p1 and p2 . However, to obtain a suitable point estimate of q (or p); we rst estimate = (1p1 ; 2p2 ). It is known that (see Gibbons, 1993) the lt th-order statistics is a consistent estimator of the pt th quantile of the d.f. Ft (t = 1; 2); where lt =n = pt remains xed. A de nition of the pt th sample quantile, which provides a unique number xt; lt :n (xt; lt :n is the observed value of the lt th-order statistic Xt; lt :n ), is to choose if npt is an integer; npt ; lt = [npt + 1]; if npt is not an integer: ˆ Therefore, a logical point estimate of = (1p1 ; 2p2 ) would be the sample quantile vector n = (x1; l1 :n ; x2; l2 :n ). Consequently, we can estimate q by the estimator qˆ = Mn (q1 ; q2 )=n; where Mn (q1 ; q2 ) = i=1 Yi:n (x1; l1 :n ; x2; l2 :n ) and 1 if X1i ¿ a; X2i ¿ b; Yi:n (a; b) = 0 otherwise:
3. Dependence function (i.e., p)-free lower bound of P(R) Of course, there exists a natural lower bound of P(R) = n (r1 ; s1 ; r2 ; s2 ; p); which does not depend on p; i.e., depends only on (p1 ; p2 ). This lower bound is the FrPechet lower bound (see Galambos, 1978, 1987, Chapter 5) n = 0 ∨ (1n + 2n − 1). In this section, we establish a dependence function-free lower bound of P(R) which, in general, is larger than n : Theorem 3.1. Let n 6 s1 + s2 − 2 and p1 + p2 ¿ 1 (see Remark 3:1). Then P(R) ¿ Qn =
n−r 1
n−r 2
i=n−s1 +1j=n−s2 +1 i+j6n
n! qi q j (1 − q1 − q2 )n−i−j : i!j!(n − i − j)! 1 2
(3.1)
H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
41
Let n ¿ r1 + r2 and p1 + p2 ¡ 1 (see Remark 3:1). Then P(R) ¿ ˆn =
n−r 1
n−r 2
i=n−s1 +1j=n−s2 +1 i+j¿n
n! pn−j p2n−i (1 − p1 − p2 )i+j−n : i!j!(i + j − n)! 1
(3.2)
Proof. Let us look at the RHS of (2.1) as an algebraic polynomial of q; for which 0 6 q 6 q1 ∧ q2 . Clearly, all individual terms of this polynomial are nonnegative. Therefore, this polynomial and L1 (q) = (q1 − q)i−k qk (q2 − q) j−k (1 − q1 − q2 + q)n−i−j+k ¿ 0; ∀n − s1 + 1 6 i 6 n − r1 ; n − s2 + 1 6 j 6 n − r2 ; 0 ∧ (i + j − n) ¡ k 6 i ∧ j; attain their minimum at the same value of q. On the other hand L1 (0) = 0. Therefore, if there exist i and j such that n − s1 + 1 6 i 6 n − r1 ; n − s2 + 1 6 j 6 n − r2 and i + j − n 6 0; then (3.1) holds provided that 1 − q1 − q2 ¿ 0. Clearly, if minn−s1 +16i6n−r1 i + minn−s2 +16j6n−r2 j − n 6 0; i.e., n 6 s1 + s2 − 2; then there exist two values of i and j such that i + j − n 6 0. In order to prove (3.2), we look at the RHS of (2.1) as a polynomial of p((q1 − q) = (p2 − p); (q2 − q) = (p1 − p) and q = 1 − p1 − p2 + p) for which 0 6 p 6 p1 ∧ p2 . This polynomial and L2 (p) = (p1 − p) j−k (1 − p1 − p2 + p)k (p2 − p)i−k pn−i−j+k ¿ 0; ∀n − s1 + 1 6 i 6 n − r1 ; n − s2 + 1 6 j 6 n − r2 ; 0 ∧ (i + j − n) ¡ k 6 i ∧ j (note that L2 (p) = L1 (q)) attain their minimum at the same value of p. On the other hand, L2 (0) = 0. Therefore, if there exist i and j such that n − s1 + 1 6 i 6 n − r1 ; n − s2 + 1 6 j 6 n − r2 and i + j − n ¿ 0; then (3.2) holds provided that 1 − p1 − p2 ¿ 0. Clearly, if maxn−s1 +16i6n−r1 i + maxn−s2 +16j6n−r2 j − n ¿ 0; i.e., n ¿ r1 + r2 ; then there exist two values of i and j such that i + j − n ¿ 0. This concludes the proof. Remark 3.1. Although, in most of the applications p1 (q1 ) and p2 (q2 ) are preassigned, in using (3.1) and (3.2) it is necessarily to take, respectively, p1 ∨ p2 large (closed to one) and p1 ∧ p2 small (closed to zero). This is because ∀x we must have one and only one of the two relations 0 ¡ G(x) 6 G1 (x1 ) ∧ G2 (x2 ) and 0 = G(x) = G1 (x1 ) ∧ G2 (x2 ) and also one and only one of the two relations 0 ¡ F(x) 6 F1 (x1 ) ∧ F2 (x2 ) and 0 = F(x) = F1 (x1 ) ∧ F2 (x2 ): Example 3.1. Let p1 = p2 = 14 and n = 8. Let, further, r1 = r2 = 6 and s1 = s2 = 8. Then, we have to use (3.1) to get Qn = 0:3701; while 1n = 2n = 0:5783. Therefore, n ¿ Qn = 0:3701 ¿ n = 1n + 2n − 1 = 0:1567:
4. The extension to the three-dimensions case Consider a sequence of n independent three-dimensional random vectors X j = (X1j ; X2j ; X3j ); j = 1; 2; : : : ; n with common d.f. F(x) = P(X1j ¡ x1 ; X2j ¡ x2 ; X3j ¡ x3 ). Let Fi (xi ) be the ith marginal of F(x); i = 1; 2; 3 and for 1 6 i1 = i2 = i3 6 3; let Fi1 i2 (xi1 ; xi2 ) = limxi3 →∞ F(x). Let also G(x) = P(X ¿ x) = P(X1j ¿ x1 ; X2j ¿ x2 ; X3j ¿ x3 ); Gi1 i2 (xi1 ; xi2 ) = limxi3 →−∞ G(x); for any 1 6 i1 = i2 = i3 6 3; and Gi (xi ) be the ith marginal of G(x); i = 1; 2; 3. Finally, let Xi; 1:n 6 Xi; 2:n 6 · · · 6 Xi; n:n be the order statistics of the ith marginal random sample Xi1 ; Xi2 ; : : : ; Xin ; i = 1; 2; 3. Keeping the notations of Section 1, we can write ipi = Fi−1 (pi ); i = 1; 2; 3; 3 Fij (ipi ; jpj ) = pij ; 1 6 i = j 6 3; F() = p. Also, it is not di6cult to show that G() = q = 1 − i−1 pi + 3 i−1 qi + 16i¡j63 pij −p and p = 1− 16i¡j63 qij −q; where qi = 1−pi ; i = 1; 2; 3; and qij = 1−p1 − p2 + pij ; 1 6 i = j 6 3. The following theorem, which is the three-dimensional version of Theorem 2.1, gives the con dence region of the pth quantile vector = (1p1 ; 2p2 ; 3p3 ) of the d.f. F(x); where p = (p1 ; p2 ; p3 ; p12 ; p13 ; p23 ; p):
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H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
Theorem 2.1. For any given 1 6 rt ¡ st 6 n; t = 1; 2; 3 we have
3
n = n (r1 ; s1 ; r2 ; s2 ; r3 ; s3 ; p) = P {Xi; ri :n ¡ ipi 6 Xi; si :n } i=1 n−r 1
=
n−r 2
n−r 3
Ii∗1 ;i2 ;i3 :n (q);
(4.1)
i1 =n−s1 +1 i2 =n−s2 +1 i3 =n−s3 +1
where q = (q1 ; q2 ; q3 ; q12 ; q13 ; q23 ; q); Ii∗1 ;i2 ;i3 :n (q) =
i 2 +j3 −i1 −i2 −i3 +n) 2 ∧i3 i 1 ∧i3 i 1 ∧i2 j1 ∧j2 ∧j3 ∧( j1 +j j1 =0 j2 =0 j3 =0
Ii1 ;i2 ;i3 ;j1 ;j2 ;j3 ;k:n (q);
k=k0
k0 = max(0; j1 + j2 − i3 ; j1 + j3 − i2 ; j2 + j3 − i1 ) and 3 n−i1 −i2 −i3 +j1 +j2 +j3 −k n!qk (1 − i=1 qi + 16i¡j63 qij − q) Ii1 ;i2 ;i3 ;j1 ;j2 ;j3 ;k:n (q) = ; k!(n − i1 − i2 − i3 + j1 + j2 + j3 − k)! :
3 t=1 16t1 ¡t2 63 t1 ; t2 =t
(qt1 t2 − q)jt −k (qt − qtt1 − qtt2 + q)it −jt1 −jt2 +k : (jt − k)!(it − jt1 − jt2 + k)!
Proof. The key ingredient of the proof consists of two main steps, the rst is to derive the d.f. of the vector (X1; n−k1 +1:n ; X2; n−k2 +1:n ; X3; n−k3 +1:n ). This d.f. is given by Fn−k1 +1;n−k2 +1;n−k3 +1:n (x) =
k 3 −1 1 −1 k 2 −1 k
Ii∗1 ;i2 ;i3 :n (G(x));
(4.2)
i1 =0 i2 =0 i3 =0
where G(x) = (G1 (x1 ); G2 (x2 ); G3 (x3 ); G12 (x1 ; x2 ); G13 (x1 ; x3 ); G23 (x2 ; x3 ); G(x)). The proof of (4.2) follows closely as the the proof of (2.4), except that it requires more routine cumbersome calculations. We thus omit the details. In order to accomplish the second 3 step. We de ne At ; Bt and Ct ; for t = 1; 2; 3; as in the proof of Theorem 2.1. Clearly A1 ∩ A2 ∩ A3 = t=1 (Bt ∪ Ct ). Moreover, n = P(B1 ∩ B2 ∩ B3 ). Therefore, in view of the relation Bt ∩ Ct = ; t = 1; 2; 3; we get n = P(A1 ∩ A2 ∩ A3 ) −
3
(P(Ct ∩ Bt1 ∩ Bt2 ) + P(Bt ∩ Ct1 ∩ Ct2 ))
t=1 16t1 ¡t2 63 t1 ; t2 =t
− P(C1 ∩ C2 ∩ C3 ): This leads, after some routine calculations, to the equation n = Fr1 ;r2 ;r3 :n () − Fr1 ;r2 ;s3 :n () − Fr1 ;s2 ;r3 :n () − Fs1 ;r2 ;r3 :n () + Fs1 ;s2 ;r3 :n () + Fs1 ;r2 ;s3 :n () + Fr1 ;s2 ;s3 :n () − Fs1 ;s2 ;s3 :n ();
(4.3)
H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
43
where, for any integers 1 6 k1 ; k2 ; k3 6 n; Fk1 ;k2 ;k3 :n () =
n−k 1 n−k 2 n−k 3
Ii∗1 ;i2 ;i3 :n (q):
i1 =0 i2 =0 i3 =0
Finally (4.3), after simple calculations, yields the claimed result. Theorem 4:1 reveals the fact that Theorem 2.1 and its proof may be extended for multivariate co-ordinate-wise quantiles. Moreover, the suggested procedure, at the end of Section 2, may also be applied to attach a con dence coe6cient to the estimate of the three-dimensional (and the multi-dimensional) con dence region. Namely, by keeping the de nition of lt ; for t = 1; 2; 3; the point estimate of = (1p1 ; 2p2 ; 3p3 ) would be the sample quantile vector ˆ = (x1; l1 :n ; x2; l2 :n ; x3; l3 :n ). Consequently, q and qij ; 1 6 i = j 6 3 may be estimated, respectively, by the estimators qˆ = (1=n)Mn (q1 ; q2 ; q3 ); and qˆij = (1=n)Mnij (q1 ; q2 ; q3 ); 1 6 i = j 6 3; n n ij where Mn (q1 ; q2 ; q3 ) = t=1 Yt:n (x1; l1 :n ; x2; l2 :n ; x3; l3 :n ); Mnij (q1 ; q2 ; q3 ) = t=1 Yt:n (xi; li :n ; xj; lj :n ); 1 if X1t ¿ a; X2t ¿ b; X3t ¿ c; Yt:n (a; b; c) = 0 otherwise: and
ij Yt:n (a; b) =
1 if Xit ¿ a; 0 otherwise:
Xjt ¿ b;
Acknowledgements The author is grateful to the anonymous referee for his valuable comments and helpful suggestions that improved the presentation of the paper. References Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1993. A First Course in Order Statistics. Wiley, New York. Barakat, H.M., 1990. Limit theorems for lower–upper extreme values from two-dimensional distribution function. J. Statist. Plann. Inference 24, 69–79. Barakat, H.M., 1998. Asymptotic properties of bivariate random extremes. J. Statist. Plann. Inference 61, 203–217. Barakat, H.M., 1999. On the limit behaviour of bivariate extremes. Statistica (Bologna-Italy) 2, 351–358. Barnett, V., 1976. The ordering of mulivariate data (with comments). J. Roy. Statist. Soc. Ser. A 139, 318–354. Chaudhuri, P., 1996. On a Geometric notion of quantiles for multivariate data. JASA Theory Methods 91 (434), 862–872. David, H.A., 1981. Order Statistics, 2nd Edition. Wiley, New York. Galambos, J., 1978=1987. The Asymptotic Theory of Extreme Order Statistics, Wiley, New York, (1st Edition) Krieger, New York (2nd Edition). Gibbons, J.D., 1993. Nonparametric Statistics an Introduction. Wiley, New York. Maritz, J.S., 1995. Distribution-free Statistical Method, 2nd Edition. Chapman & Hall, London. Thompson, W.R., 1936. On con dence ranges for the median and other expectation distribution for populations of unknown distribution form. Ann. Math. Statist. 7, 122–128.
On the con dence region for the bivariate co-ordinate-wise quantiles of the continuous bivariate distribution functions H.M. Barakat ∗ Department of Mathematics, Faculty of science, Zagazig University, Zagazig, Egypt Received June 2000; received in revised form July 2001
Abstract In this paper, the distribution-free con dence region (rectangular) for the vector of the co-ordinate-wise quantiles of a general continuous bivariate distribution function is explicitly derived. This con dence region, in general, depends on the dependence function. A procedure is suggested which enables one to attach a con dence coe6cient to the estimate of the con dence region even if the dependence function is unknown. Moreover, some approximated distribution-free lower bounds, which are independent of the dependence function, of the con dence coe6cient of this region are derived. c 2002 Finally, some results of this study are extended to the three-dimensional vector of the co-ordinate-wise quantiles. Elsevier Science B.V. All rights reserved Keywords: Distribution-free con dence region; Bivariate order statistics; Bivariate quantiles
1. Introduction If measurements of two characteristics are taken on the same members of the population, then the observed random quantities follow some type of bivariate distribution. Let this distribution be F(x) = F(x1 ; x2 ) = P(X1 ¡ x1 ; X2 ¡ x2 ). Consider a sequence of n independent two-dimensional random variables (r.v.’s) {X j } = {(X1j ; X2j )}; j = 1; 2; : : : ; n with common distribution function (d.f.) F(x). The order statistics of the ith (i = 1; 2) component (characteristic) are Xi; 1:n 6 Xi; 2:n 6 · · · 6 Xi; n:n . Write x 6 y to mean xi 6 yi ; i = 1; 2 and G(x) = P(X ¿ x) to denote the survival function. For the ith (i = 1; 2) component of the vector X = (X1 ; X2 ); let Fi (xi ) and Gi (xi ); be the marginals of F(x) and G(x), respectively. Suppose that the two equations F1 (x1 ) = p1 ; 0 ¡ p1 ¡ 1 and
F2 (x2 ) = p2 ;
0 ¡ p2 ¡ 1
∗ Fax: +55-345452. E-mail address: [email protected] (H.M. Barakat).
c 2002 Elsevier Science B.V. All rights reserved 0167-7152/02/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 1 4 5 - 6
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H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
have unique solutions, say xt = tpt ; t = 1; 2 which are known as the (population) quantiles of order p1 and p2 of the d.f.’s F1 and F2 , respectively. Clearly, if F() = p; where = (1p1 ; 2p2 ) and 0 ¡ pt ¡ 1; t = 1; 2; then G() = q = 1 − p1 − p2 + p and in this case we may call as the pth quantile vector of the d.f. F(x); where p = (p1 ; p2 ; p): Clearly, the vector of the co-ordinate-wise quantiles of orders p1 and p2 is an extension of the notion of quantiles in the two-dimensions case. It is worth mentioning that there are many diEerent versions of bivariate (and multivariate) quantiles considered in the literature. See, e.g., Chaudhuri (1996) and the review and the references therein. The main object, which distinguishes these versions is the technique which is chosen for ordering the bivariate (and multivariate) random sample, see Barnett (1976). In this paper, we focus our attention only on the co-ordinate-wise version of bivariate quantiles based on what might be meant by M -ordering technique (see, Barnett, 1976), i.e., the ordering of the bivariate random sample X j ; j = 1; 2; : : : ; n takes place within the two marginal samples X1j ; j = 1; 2; : : : ; n and X2j ; j = 1; 2; : : : ; n. The con dence intervals for tpt ; t = 1; 2 (tpt = Ft−1 (pt ) = sup{u : F(u) ¡ pt }) whose endpoints are order statistics and have coverage probabilities free of Ft have been extensively studied, see David (1981, pp. 15 –16), Arnold et al. (1993, pp. 183–185) and references therein. The earliest work in this area seems to be that of Thompson (1936), Namely, the 100tn % con dence interval for tpt is (X1; rt :n ; X2; st :n ); where 1 6 rt ¡ st 6 n; t = 1; 2 and tn = tn (rt ; st ; pt ) =
n−s t i=rt
n pti (1 − pt )n−i ; i
(1.1)
as long as Ft is continuous. The technique developed to obtain this result has been extended to obtain several related results such as con dence intervals for quantile diEerences (see David, 1981, and references therein). While distribution-free con dence intervals for 1p1 ; 0 ¡ p1 ¡ 1; or 2p2 ; 0 ¡ p2 ¡ 1; individually can be obtained by using (1.1), occasions may arise where a joint con dence region for the two quantiles is required. The works for this subject are very limited. However, when p1 = p2 = 12 the nearly only known procedure for constructing a distribution-free con dence region for the median vector is given in Maritz (1995). This procedure mainly depends on a test statistic, by which one can assign an asymptotic critical value (in view of preassigned con dence coe6cient). In order to construct the required con dence region, the observed value of this statistic must be determined for every point in the range of the random vector X = (X1 ; X2 ) and test whether each of these points is a member of the con dence region or not i.e., we construct the region point by point (for the details see Maritz, 1995, Example 7:4). This is of course not an eEective procedure, specially when X is continuous. Moreover, the test statistic of this procedure mainly depends on the symmetries arising from the statistical properties of the median vector, i.e., F1 ( 1 ) = F2 ( 1 ) (p1 = p2 = 12 ) and F( 1 ; 1 ) = G( 1 ; 1 )(p = q): 12
22
12
22
12
22
Therefore, it does not work on the other quantiles. In this paper, a distribution-free con dence region (rectangular) for pth quantile vector of F(x); ∀(0; 0) ¡ (p1 ; p2 ) ¡ (1; 1); is explicity derived. Although this con dence region does not directly depend on the d.f. F(x); (or, of course, on its marginals) it depends clearly on p. Therefore, it will be used only in such cases when the marginal distributions are unknown but its dependence function (for de nition see Galambos, 1978=1987, Chapter 5) is known, e.g., F = F1 F2 (1 − (1 − F1 )(1 − F2 )) (Morgenstern d.f., see Galambos, 1987, Chapter 5), where is known while Ft ; t = 1; 2 are unknown d.f.’s, i.e., in this case we only know the dependence function D(u1 ; u2 ) = u1 u2 (1 − (1 − u1 )(1 − u2 )): However, at the end of Section 2, we suggest a procedure which enables us to attach an approximated con dence coe6cient to our estimate of the con dence region even if the dependence function is unknown. In Section 3, the approximated lower bounds, which are independent of p (or q), of the con dence coe6cient of this region are derived.
H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
39
2. Distribution-free condence region for quantile vector Theorem 2.1. For any given 1 6 rt ¡ st 6 n; t = 1; 2 we have n = n (r1 ; s1 ; r2 ; s2 ; p) = P(X1; r1 :n ¡ 1p1 6 X1; s1 :n ; X2; r2 :n ¡ 2p2 6 X2; s2 :n ) =
n−r 1
i∧j
n−r 2
Ii; j; k:n (q);
(2.1)
i=n−s1 +1 j=n−s2 +1 k=0∨(i+j−n)
where Ii; j; k:n (q) =
n! (q1 − q)i−k qk (q2 − q)j−k (1 − q1 − q2 + q)n−i−j+k ; (i − k)!k!(j − k)!(n − i − j + k)!
q = (q1 ; q2 ; q); qt = 1 − pt ; t = 1; 2 and a ∨ b = max(a; b); a ∧ b = min(a; b) ∀a; b: Proof. Let R = {X1; r1 :n ¡ 1p1 6 X1; s1 :n ; X2; r2 :n ¡ 2p2 6 X2; s2 :n }; 6 Xt; st :n } and Ct = {Xt; st :n ¡ tpt }; where t = 1; 2. Therefore, A1 ∩ A2 = B1 ∩ B2 B1 ∩ C2 C1 ∩ B2 C1 ∩ C2 :
At = {Xt; rt :n ¡ tpt };
Bt = {Xt; rt :n ¡ tpt
In view of the obvious relation Bt ∩ Ct = ; t = 1; 2; we get P(R) = P(B1 ∩ B2 ) = P(A1 ∩ A2 ) − P(B1 ∩ C2 ) − P(C1 ∩ B2 ) − P(C1 ∩ C2 ):
(2.2)
On the other hand, if we write Fk1 ;k2 :n (x) = Fk1 ;k2 :n (x1 ; x2 ) = P(X1; k1 :n ¡ x1 ; X2; k2 :n ¡ x2 ); we get P(A1 ∩ A2 ) = P(X1; r1 :n ¡ 1p1 ; X2; r2 :n ¡ 2p2 ) = Fr1 ;r2 :n (1p1 ; 2p2 ); P(B1 ∩ C2 ) = P(X1; r1 :n ¡ 1p1 6 X1; s1 :n ; X2; s2 :n ¡ 2p2 ) = Fr1 ;s2 :n (1p1 ; 2p2 )−Fs1 ;s2 :n (1p1 ; 2p2 ); P(C1 ∩B2 ) = P(X2; r2 :n ¡ 2p2 6 X2; s2 :n ; X1; s1 :n ¡ 1p1 ) = Fs1 ;r2 :n (1p1 ; 2p2 ) − Fs1 ;s2 :n (1p1 ; 2p2 ) and P(C1 ∩ C2 ) = P(X1; s1 :n ¡ 1p1 ; X2; s2 :n ¡ 2p2 ) = Fs1 ;s2 :n (1p1 ; 2p2 ). Hence, P(R) = Fr1 ;r2 :n (1p1 ; 2p2 ) − Fr1 ;s2 :n (1p1 ; 2p2 ) − Fs1 ;r2 :n (1p1 ; 2p2 ) + Fs1 ; s2 :n (1p1 ; 2p2 ):
(2.3)
It can be shown that (see Barakat, 1990, 1998, 1999)
Fn−k+1; n−k +1:n (x) =
k−1 k −1
i∧j
i=0 j=0 r=0∨(i+j−n)
n! (G1 (x1 ) − G(x))i−r (i − r)!r!(j − r)!(n − i − j + r)!
×G r (x)(G2 (x2 ) − G(x))j−r (1 − G1 (x1 ) − G2 (x2 ) + G(x))n−i−j+r ; (2.4) where, in introducing (2.4), we use the summation i=0 ai ; ∀a ¿ 0 to denote the summation 1 + i=0 ai . This, combined with (2.3), yields n−r 1 n−r 2 n−r 1 n−s 2 n−s 1 n−r 2 n−s 1 n−s 2 Ii;∗j; k:n (q); P(R) = − − + i=0 j=0
i=0 j=0
where Ii;∗j; k:n (q) =
i∧j
Ii; j; k:n (q):
k=0∨(i+j−n)
i=0 j=0
i=0 j=0
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H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
It follows that
P(R) =
n−r 1
i=0
=
n−r 2 j=0
n−r 2
n−r 1
+
j=0
n−s 1 i=0
n−s 2
−
j=0
n−r 2
Ii;∗j; k:n (q)
j=0
n−r 1 n−s 1 Ii;∗j; k:n (q) −
j=n−s2 +1
=
−
n−s 2
i=0
n−r 2
i=0
Ii;∗j; k:n (q) =
i=n−s1 +1 j=n−s2 +1
n−r 1
n−r 2
i∧j
Ii; j; k:n (q):
i=n−s1 +1 j=n−s2 +1 k=0∨(i+j−n)
The proof is completed. General case (unknown dependence function): In order to estimate a con dence region for the quantile vector for some speci ed values of p1 and p2 when the dependence function is unknown, i.e., p (or q) is not preassigned, we must seek a procedure which rst enables us to estimate p (or q) and then by using (2.1), we can construct the desired con dence region for the given p1 and p2 . However, to obtain a suitable point estimate of q (or p); we rst estimate = (1p1 ; 2p2 ). It is known that (see Gibbons, 1993) the lt th-order statistics is a consistent estimator of the pt th quantile of the d.f. Ft (t = 1; 2); where lt =n = pt remains xed. A de nition of the pt th sample quantile, which provides a unique number xt; lt :n (xt; lt :n is the observed value of the lt th-order statistic Xt; lt :n ), is to choose if npt is an integer; npt ; lt = [npt + 1]; if npt is not an integer: ˆ Therefore, a logical point estimate of = (1p1 ; 2p2 ) would be the sample quantile vector n = (x1; l1 :n ; x2; l2 :n ). Consequently, we can estimate q by the estimator qˆ = Mn (q1 ; q2 )=n; where Mn (q1 ; q2 ) = i=1 Yi:n (x1; l1 :n ; x2; l2 :n ) and 1 if X1i ¿ a; X2i ¿ b; Yi:n (a; b) = 0 otherwise:
3. Dependence function (i.e., p)-free lower bound of P(R) Of course, there exists a natural lower bound of P(R) = n (r1 ; s1 ; r2 ; s2 ; p); which does not depend on p; i.e., depends only on (p1 ; p2 ). This lower bound is the FrPechet lower bound (see Galambos, 1978, 1987, Chapter 5) n = 0 ∨ (1n + 2n − 1). In this section, we establish a dependence function-free lower bound of P(R) which, in general, is larger than n : Theorem 3.1. Let n 6 s1 + s2 − 2 and p1 + p2 ¿ 1 (see Remark 3:1). Then P(R) ¿ Qn =
n−r 1
n−r 2
i=n−s1 +1j=n−s2 +1 i+j6n
n! qi q j (1 − q1 − q2 )n−i−j : i!j!(n − i − j)! 1 2
(3.1)
H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
41
Let n ¿ r1 + r2 and p1 + p2 ¡ 1 (see Remark 3:1). Then P(R) ¿ ˆn =
n−r 1
n−r 2
i=n−s1 +1j=n−s2 +1 i+j¿n
n! pn−j p2n−i (1 − p1 − p2 )i+j−n : i!j!(i + j − n)! 1
(3.2)
Proof. Let us look at the RHS of (2.1) as an algebraic polynomial of q; for which 0 6 q 6 q1 ∧ q2 . Clearly, all individual terms of this polynomial are nonnegative. Therefore, this polynomial and L1 (q) = (q1 − q)i−k qk (q2 − q) j−k (1 − q1 − q2 + q)n−i−j+k ¿ 0; ∀n − s1 + 1 6 i 6 n − r1 ; n − s2 + 1 6 j 6 n − r2 ; 0 ∧ (i + j − n) ¡ k 6 i ∧ j; attain their minimum at the same value of q. On the other hand L1 (0) = 0. Therefore, if there exist i and j such that n − s1 + 1 6 i 6 n − r1 ; n − s2 + 1 6 j 6 n − r2 and i + j − n 6 0; then (3.1) holds provided that 1 − q1 − q2 ¿ 0. Clearly, if minn−s1 +16i6n−r1 i + minn−s2 +16j6n−r2 j − n 6 0; i.e., n 6 s1 + s2 − 2; then there exist two values of i and j such that i + j − n 6 0. In order to prove (3.2), we look at the RHS of (2.1) as a polynomial of p((q1 − q) = (p2 − p); (q2 − q) = (p1 − p) and q = 1 − p1 − p2 + p) for which 0 6 p 6 p1 ∧ p2 . This polynomial and L2 (p) = (p1 − p) j−k (1 − p1 − p2 + p)k (p2 − p)i−k pn−i−j+k ¿ 0; ∀n − s1 + 1 6 i 6 n − r1 ; n − s2 + 1 6 j 6 n − r2 ; 0 ∧ (i + j − n) ¡ k 6 i ∧ j (note that L2 (p) = L1 (q)) attain their minimum at the same value of p. On the other hand, L2 (0) = 0. Therefore, if there exist i and j such that n − s1 + 1 6 i 6 n − r1 ; n − s2 + 1 6 j 6 n − r2 and i + j − n ¿ 0; then (3.2) holds provided that 1 − p1 − p2 ¿ 0. Clearly, if maxn−s1 +16i6n−r1 i + maxn−s2 +16j6n−r2 j − n ¿ 0; i.e., n ¿ r1 + r2 ; then there exist two values of i and j such that i + j − n ¿ 0. This concludes the proof. Remark 3.1. Although, in most of the applications p1 (q1 ) and p2 (q2 ) are preassigned, in using (3.1) and (3.2) it is necessarily to take, respectively, p1 ∨ p2 large (closed to one) and p1 ∧ p2 small (closed to zero). This is because ∀x we must have one and only one of the two relations 0 ¡ G(x) 6 G1 (x1 ) ∧ G2 (x2 ) and 0 = G(x) = G1 (x1 ) ∧ G2 (x2 ) and also one and only one of the two relations 0 ¡ F(x) 6 F1 (x1 ) ∧ F2 (x2 ) and 0 = F(x) = F1 (x1 ) ∧ F2 (x2 ): Example 3.1. Let p1 = p2 = 14 and n = 8. Let, further, r1 = r2 = 6 and s1 = s2 = 8. Then, we have to use (3.1) to get Qn = 0:3701; while 1n = 2n = 0:5783. Therefore, n ¿ Qn = 0:3701 ¿ n = 1n + 2n − 1 = 0:1567:
4. The extension to the three-dimensions case Consider a sequence of n independent three-dimensional random vectors X j = (X1j ; X2j ; X3j ); j = 1; 2; : : : ; n with common d.f. F(x) = P(X1j ¡ x1 ; X2j ¡ x2 ; X3j ¡ x3 ). Let Fi (xi ) be the ith marginal of F(x); i = 1; 2; 3 and for 1 6 i1 = i2 = i3 6 3; let Fi1 i2 (xi1 ; xi2 ) = limxi3 →∞ F(x). Let also G(x) = P(X ¿ x) = P(X1j ¿ x1 ; X2j ¿ x2 ; X3j ¿ x3 ); Gi1 i2 (xi1 ; xi2 ) = limxi3 →−∞ G(x); for any 1 6 i1 = i2 = i3 6 3; and Gi (xi ) be the ith marginal of G(x); i = 1; 2; 3. Finally, let Xi; 1:n 6 Xi; 2:n 6 · · · 6 Xi; n:n be the order statistics of the ith marginal random sample Xi1 ; Xi2 ; : : : ; Xin ; i = 1; 2; 3. Keeping the notations of Section 1, we can write ipi = Fi−1 (pi ); i = 1; 2; 3; 3 Fij (ipi ; jpj ) = pij ; 1 6 i = j 6 3; F() = p. Also, it is not di6cult to show that G() = q = 1 − i−1 pi + 3 i−1 qi + 16i¡j63 pij −p and p = 1− 16i¡j63 qij −q; where qi = 1−pi ; i = 1; 2; 3; and qij = 1−p1 − p2 + pij ; 1 6 i = j 6 3. The following theorem, which is the three-dimensional version of Theorem 2.1, gives the con dence region of the pth quantile vector = (1p1 ; 2p2 ; 3p3 ) of the d.f. F(x); where p = (p1 ; p2 ; p3 ; p12 ; p13 ; p23 ; p):
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H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
Theorem 2.1. For any given 1 6 rt ¡ st 6 n; t = 1; 2; 3 we have
3
n = n (r1 ; s1 ; r2 ; s2 ; r3 ; s3 ; p) = P {Xi; ri :n ¡ ipi 6 Xi; si :n } i=1 n−r 1
=
n−r 2
n−r 3
Ii∗1 ;i2 ;i3 :n (q);
(4.1)
i1 =n−s1 +1 i2 =n−s2 +1 i3 =n−s3 +1
where q = (q1 ; q2 ; q3 ; q12 ; q13 ; q23 ; q); Ii∗1 ;i2 ;i3 :n (q) =
i 2 +j3 −i1 −i2 −i3 +n) 2 ∧i3 i 1 ∧i3 i 1 ∧i2 j1 ∧j2 ∧j3 ∧( j1 +j j1 =0 j2 =0 j3 =0
Ii1 ;i2 ;i3 ;j1 ;j2 ;j3 ;k:n (q);
k=k0
k0 = max(0; j1 + j2 − i3 ; j1 + j3 − i2 ; j2 + j3 − i1 ) and 3 n−i1 −i2 −i3 +j1 +j2 +j3 −k n!qk (1 − i=1 qi + 16i¡j63 qij − q) Ii1 ;i2 ;i3 ;j1 ;j2 ;j3 ;k:n (q) = ; k!(n − i1 − i2 − i3 + j1 + j2 + j3 − k)! :
3 t=1 16t1 ¡t2 63 t1 ; t2 =t
(qt1 t2 − q)jt −k (qt − qtt1 − qtt2 + q)it −jt1 −jt2 +k : (jt − k)!(it − jt1 − jt2 + k)!
Proof. The key ingredient of the proof consists of two main steps, the rst is to derive the d.f. of the vector (X1; n−k1 +1:n ; X2; n−k2 +1:n ; X3; n−k3 +1:n ). This d.f. is given by Fn−k1 +1;n−k2 +1;n−k3 +1:n (x) =
k 3 −1 1 −1 k 2 −1 k
Ii∗1 ;i2 ;i3 :n (G(x));
(4.2)
i1 =0 i2 =0 i3 =0
where G(x) = (G1 (x1 ); G2 (x2 ); G3 (x3 ); G12 (x1 ; x2 ); G13 (x1 ; x3 ); G23 (x2 ; x3 ); G(x)). The proof of (4.2) follows closely as the the proof of (2.4), except that it requires more routine cumbersome calculations. We thus omit the details. In order to accomplish the second 3 step. We de ne At ; Bt and Ct ; for t = 1; 2; 3; as in the proof of Theorem 2.1. Clearly A1 ∩ A2 ∩ A3 = t=1 (Bt ∪ Ct ). Moreover, n = P(B1 ∩ B2 ∩ B3 ). Therefore, in view of the relation Bt ∩ Ct = ; t = 1; 2; 3; we get n = P(A1 ∩ A2 ∩ A3 ) −
3
(P(Ct ∩ Bt1 ∩ Bt2 ) + P(Bt ∩ Ct1 ∩ Ct2 ))
t=1 16t1 ¡t2 63 t1 ; t2 =t
− P(C1 ∩ C2 ∩ C3 ): This leads, after some routine calculations, to the equation n = Fr1 ;r2 ;r3 :n () − Fr1 ;r2 ;s3 :n () − Fr1 ;s2 ;r3 :n () − Fs1 ;r2 ;r3 :n () + Fs1 ;s2 ;r3 :n () + Fs1 ;r2 ;s3 :n () + Fr1 ;s2 ;s3 :n () − Fs1 ;s2 ;s3 :n ();
(4.3)
H.M. Barakat / Statistics & Probability Letters 56 (2002) 37 – 43
43
where, for any integers 1 6 k1 ; k2 ; k3 6 n; Fk1 ;k2 ;k3 :n () =
n−k 1 n−k 2 n−k 3
Ii∗1 ;i2 ;i3 :n (q):
i1 =0 i2 =0 i3 =0
Finally (4.3), after simple calculations, yields the claimed result. Theorem 4:1 reveals the fact that Theorem 2.1 and its proof may be extended for multivariate co-ordinate-wise quantiles. Moreover, the suggested procedure, at the end of Section 2, may also be applied to attach a con dence coe6cient to the estimate of the three-dimensional (and the multi-dimensional) con dence region. Namely, by keeping the de nition of lt ; for t = 1; 2; 3; the point estimate of = (1p1 ; 2p2 ; 3p3 ) would be the sample quantile vector ˆ = (x1; l1 :n ; x2; l2 :n ; x3; l3 :n ). Consequently, q and qij ; 1 6 i = j 6 3 may be estimated, respectively, by the estimators qˆ = (1=n)Mn (q1 ; q2 ; q3 ); and qˆij = (1=n)Mnij (q1 ; q2 ; q3 ); 1 6 i = j 6 3; n n ij where Mn (q1 ; q2 ; q3 ) = t=1 Yt:n (x1; l1 :n ; x2; l2 :n ; x3; l3 :n ); Mnij (q1 ; q2 ; q3 ) = t=1 Yt:n (xi; li :n ; xj; lj :n ); 1 if X1t ¿ a; X2t ¿ b; X3t ¿ c; Yt:n (a; b; c) = 0 otherwise: and
ij Yt:n (a; b) =
1 if Xit ¿ a; 0 otherwise:
Xjt ¿ b;
Acknowledgements The author is grateful to the anonymous referee for his valuable comments and helpful suggestions that improved the presentation of the paper. References Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1993. A First Course in Order Statistics. Wiley, New York. Barakat, H.M., 1990. Limit theorems for lower–upper extreme values from two-dimensional distribution function. J. Statist. Plann. Inference 24, 69–79. Barakat, H.M., 1998. Asymptotic properties of bivariate random extremes. J. Statist. Plann. Inference 61, 203–217. Barakat, H.M., 1999. On the limit behaviour of bivariate extremes. Statistica (Bologna-Italy) 2, 351–358. Barnett, V., 1976. The ordering of mulivariate data (with comments). J. Roy. Statist. Soc. Ser. A 139, 318–354. Chaudhuri, P., 1996. On a Geometric notion of quantiles for multivariate data. JASA Theory Methods 91 (434), 862–872. David, H.A., 1981. Order Statistics, 2nd Edition. Wiley, New York. Galambos, J., 1978=1987. The Asymptotic Theory of Extreme Order Statistics, Wiley, New York, (1st Edition) Krieger, New York (2nd Edition). Gibbons, J.D., 1993. Nonparametric Statistics an Introduction. Wiley, New York. Maritz, J.S., 1995. Distribution-free Statistical Method, 2nd Edition. Chapman & Hall, London. Thompson, W.R., 1936. On con dence ranges for the median and other expectation distribution for populations of unknown distribution form. Ann. Math. Statist. 7, 122–128.