- Email: [email protected]
An extension of the Erdős–Neveu–Rényi theorem with applications to order statistics
Statistics & Probability Letters 55 (2001) 181 – 186
An extension of the Erd˝os–Neveu–R%enyi theorem with applications to order statistics M. Kaluszka ∗ , A. Okolewski Institute of Mathematics, Technical University of Lodz, ul. Zwirki 36, 90-924 Lodz, Poland Received January 2001; received in revised form May 2001
Abstract The necessary and su3cient conditions are given for some stochastic process to be an empirical distribution function from some exchangeable random variables. The result is applied to establish sharp lower and upper bounds for order c 2001 Elsevier Science B.V. All rights reserved statistics based on possibly dependent random variables. MSC: 62G30; 60G09 Keywords: Dependent random variables; Exchangeable random variables; Empirical distribution function; Order statistics; Bonferroni-type inequalities
1. Introduction The following theorem, proved by Erd˝os et al. (1963), shows that if the assumption of independence of n Bernoulli trials is omitted, the number of successes can be arbitrarily distributed on {0; 1; : : : ; n}: Before rewriting it, let us denote by IA the indicator of A and recall that the events A1 ; A2 ; : : : ; An are said to be exchangeable if P(Ai1 ∩ · · · ∩ Air ) = P(A1 ∩ · · · ∩ Ar ) for every choice of the subscripts 1 6 i1 ¡ · · · ¡ ir 6 n: Theorem 1. Let {pk }nk=0 be a distribution on {0; 1; : : : ; n}: Then there is a probability space ( ; A; P) and a sequence of exchangeable events A1 ; A2 ; : : : ; An in A such that P( n = k) = pk for each k; where n = IA1 + · · · + IAn : The result of Theorem 1 has some interesting applications (see Galambos and Simonelli, 1996). The present version, however, appears to be not strong enough in many situations and consequently there is a need for
∗
Corresponding author. E-mail address: [email protected] (M. Kaluszka).
c 2001 Elsevier Science B.V. All rights reserved 0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 1 1 4 - 6
182
M. Kaluszka, A. Okolewski / Statistics & Probability Letters 55 (2001) 181 – 186
its generalization. The aim of this paper is to extend the result of Theorem 1 to collections of monotonically increasing events A1 (t); : : : ; An (t); t ∈ R: Such extension not only leads to natural generalizations of some Bonferroni-type inequalities as well as Rychlik’s (1993, 1995) bounds for order statistics (see also Rychlik, 1992; Gascuel and Caraux, 1992a, b), but also indicates that both the mentioned types of inequalities can be treated as bounds for some spaces of moments of n : In addition, our theorem provides a tool which can be used to obtain easily new inequalities for other functionals of order statistics. The mentioned generalizations as well as some examples of new bounds for order statistics are presented in Section 3. 2. The result d
We will use the symbol X =Y to denote that the random variables X and Y are identically distributed. For a given distribution function F, we will denote by F −1 the quantile function, i.e., F −1 (t) = sup{s ∈ R: F(s) 6 t}; t ∈ [0; 1): Theorem 2. Let { n (t): t ∈ R} be a stochastic process on some probability space ( ; A; P) with values in {0; 1; : : : ; n}: The following conditions are equivalent: ˜ A;˜ P) ˜ and a sequence X1 ; X2 ; : : : ; Xn of exchangeable random (i) There exists a probability space ( ; ˜ variables de9ned on such that n d I{Xi 6t} ; (1) n (t) = i=1
for every t ∈ R: (ii) R t → P( n (t) ¿ k) is a distribution function on R for every k = 1; 2; : : : ; n: Proof. The implication (i) ⇒ (ii) is immediate. The proof of the reverse implication consists in the con˜ A;˜ P) ˜ and a sequence X1 ; X2 ; : : : ; Xn of exchangeable random variables struction of a probability space ( ; ˜ deLned on : Let us Lrst deal with the probability space. Set ˜ = [0; 1) × ; where is the collection of all permutations of {1; : : : ; n}: DeLne A˜ to be the -Leld generated by the class of all sets of the form B × A; where A ∈ and B is a Borel subset of [0; 1): Let P˜ be the product measure of the Lebesgue measure on [0; 1) and the uniform distribution on : We can now deLne the sequence of random variables. Set Fk (t) = P( n (t) ¿ k); k = 1; 2; : : : ; n; t ∈ R: By assumption, Fk ’s are distribution functions. Write Yk (s) = Fk−1 (s);
s ∈ [0; 1)
and X1 (s; ) = Y(1) (s); : : : ; Xn (s; ) = Y(n) (s);
˜ (s; ) ∈ [0; 1) × = ;
where = ((1); : : : ; (n)) is a permutation of the set {1; : : : ; n}: We next show that X1 ; X2 ; : : : ; Xn are exchangeable. Consider arbitrary k = 1; 2; : : : ; n; 1 6 i1 ¡ i2 ¡ · · · ¡ ik 6 n and t1 ; : : : ; tk ∈ R: By the deLnition of Xi ’s, ˜ i1 6 t1 ; : : : ; Xik 6 tk ) P(X = =
˜ (i1 ) 6 t1 ; : : : ; Y(ik ) 6 tk | (i1 ) = l1 ; : : : ; (ik ) = lk ) (n − k)! P(Y n!
˜ l1 6 t1 ; : : : ; Ylk 6 tk ) (n − k)! P(Y n!
M. Kaluszka, A. Okolewski / Statistics & Probability Letters 55 (2001) 181 – 186
=
183
˜ (1) 6 t1 ; : : : ; Y(k) 6 tk | (1) = l1 ; : : : ; (k) = lk ) (n − k)! P(Y n!
˜ 1 6 t1 ; : : : ; Xk 6 tk ); = P(X where the sums are over all possible l1 ; : : : ; lk : It remains to prove that (1) is satisLed. Observe that Y1 6 Y2 6 ; : : : ; 6 Yn P˜ a.s. Combining this with the deLnition of X1 ; X2 ; : : : ; Xn gives n n ˜ k 6 t; Yk+1 ¿ t) P˜ I{Xi 6t} = k = P˜ I{Yi 6t} = k = P(Y i=1
i=1
˜ k 6 t; Yk+1 6 t) ˜ k 6 t) − P(Y = P(Y ˜ k 6 t) − P(Y ˜ k+1 6 t) = Fk (t) − Fk+1 (t) = P(Y = P( n (t) ¿ k) − P( n (t) ¿ k + 1) = P( n (t) = k); and the proof is complete. Remarks. (1) The original proof of Erd˝os–Neveu–R%enyi cannot be adapted here because the construction of the measure P˜ depends on t: Another proof of Theorem 1 due to Sibuya (1991) is known. Since the method presented therein may not provide monotone collections of sets, Sibuya’s reasoning cannot be used as well. (2) Theorem 2 states the necessary and su3cient conditions for the stochastic process n (t)=n to be an empirical distribution function from some exchangeable random variables X1 ; : : : ; Xn : (3) If we take a Lxed t ∈ R; we recover Theorem 1 of Erd˝os–Neveu–R%enyi.
3. Applications 3.1. Rychlik-type bounds on L-estimates We extend Rychlik’s (1993) bounds on expectation of linear combinations of order statistics to nonidentical distributions. Let X1 ; X2 ; : : : ; Xn be possibly dependent but not necessarily identically distributed random variables. Let Xk:n denote the kth order statistic from the sample X1 ; : : : ; Xn ; and let c1 ; : : : ; cn be arbitrary real numbers. Then 1 1 n −1 N F (t) d C(t) 6 ck EXk:n 6 F −1 (t) dC(t); (2) 0
k=1
0
i n where F(t) = (1=n) j=1 P(Xj 6 t); C is the greatest convex function such that C(0) = 0 and C(i=n) 6 k=1 ck ; i N N i = 0; 1; : : : ; n; and CN is the smallest concave function such that C(0) = 0 and C(i=n) ¿ k=1 ck ; i = 0; 1; : : : ; n 0 (see Rychlik, 1993). Here and subsequently, we adopt the convention k=1 ck = 0: Furthermore, if X1 ; X2 ; : : : ; Xn are identically distributed, there exists a sequence of exchangeable random variables such that the lower bound is attainable (the same conclusion can be drawn for the upper bound). To deduce (2) from Theorem 2, we use the result due to M%ori and Sz%ekely (1985) (see also Galambos and Simonelli, 1996, Lemma III.1). The result states that the range of the moments (Ef1 ( n ); : : : ; Efl ( n )); where f1 ; : : : ; fl are arbitrary real functions on {0; 1; : : : ; n}; is the conv{(f1 (i); : : : ; fl (i)): i = 0; 1; : : : ; n} if the distribution of n varies over all distributions on {0; 1; : : : ; n}:
184
M. Kaluszka, A. Okolewski / Statistics & Probability Letters 55 (2001) 181 – 186
Let n (t) =
n
i=1 I{Xi 6t} ; n
1 Ef1 ( n (t)) = n
f1 (t) = t=n and f2 (t) =
P(Xi 6 t);
n
Ef2 ( n (t)) =
i=1
k=1 ck I[k; ∞) (t):
n
It is clear that
ck P(Xk:n 6 t):
k=1
By M%ori and Sz%ekely’s characterization, n n i 1 i ; P(Xi 6 t); ck P(Xk:n 6 t) ∈ conv ck : i = 0; 1; : : : ; n : n n i=1
k=1
k=1
Hence for every t ∈ R n N C(F(t)) 6 ck P(Xk:n 6 t) 6 C(F(t)): k=1
From what has already been proved, it follows easily that (2) is satisLed. What is left is to show that the N Let n (t); t ∈ R; be a lower bound is sharp. Let n1 ; : : : ; nm denote the endpoints of the linear pieces of C: stochastic process such that P( n (t) = ni ) = p(t) = 1 − P( n (t) = ni+1 ) for F(t) ∈ [ni =n; ni+1 =n]; i = 1; : : : ; m − 1; where p(t) is chosen so that Ef1 ( n (t)) = F(t): We check at once that p(t) = (ni+1 − nF(t))=(ni+1 − ni ) and Ef2 ( n (t)) =
ni
ck +
k=1
ni+1
ck
(1 − p(t))
k=ni +1
N i+1 ) − C(n N i ))(1 − p(t)) N i ) + (C(n = C(n n n i i+1 N ; : = C(F(t)); F(t) ∈ n n Since R t → P( n (t) ¿ k) is a distribution function for every k = 1; 2; : : : ; n; the attainability conditions for identical distributions are an immediate consequence of Theorem 2. The corresponding problem of attainability for nonidentical distributions seems to be rather di3cult (see Rychlik, 1995). 3.2. New bounds for order statistics Similar arguments applied to moments diQerent from the ones considered above lead to new bounds for order statistics. For example, let f1 (t) = t and f2 (t) = I[k; l) (t); where 1 6 k ¡ l 6 n: Then n Ef1 ( n (t)) = P(Xi 6 t); i=1
Ef2 ( n (t)) = P(k 6 n (t) ¡ l) = P(Xk:n 6 t; Xl:n ¿ t) and (Ef1 ( n (t)); Ef2 ( n (t))) ∈ conv{(i; I[k; l) (i)): i = 0; 1; : : : ; n}: It is easily seen that 0 6 Ef2 ( n (t)) 6 min
n − Ef1 ( n (t)) Ef1 ( n (t)) ; 1; k n−l+1
and so 0 6 P(Xk:n 6 t; Xl:n ¿ t) 6 min
n(1 − F(t)) n F(t); 1; k n−l+1
; ;
(3)
M. Kaluszka, A. Okolewski / Statistics & Probability Letters 55 (2001) 181 – 186
185
n where F(t) = (1=n) i=1 P(Xi 6 t): The lower and upper bounds are sharp, i.e. they are attained for some exchangeable random variables. This follows by the same method as in Section 3.1. Since similar constructions concerning attainability apply to Sections 3.3–3.5, we will omit them. 3.3. Bounds for empirical distribution function with common distribution function F and let Fn Let X1 ; X2 ; : : : ; Xn be possibly dependent random variables n denote the empirical distribution function, i.e., Fn (t) = (1=n) i=1 I{Xi 6t} ; t ∈ R: Applying (3) with l = k + 1 yields
n n(1 − F(t)) k 6 min F(t); ∀t∈R P Fn (t) = n k n−k for k = 1; 2; : : : ; n − 1: Moreover, analysis similar to that in Section 3.2 shows that ∀t∈R
P(Fn (t) = 0) 6 1 − F(t)
and
P(Fn (t) = 1) 6 F(t):
By Theorem 2, the bounds are attained for some exchangeable Xi ’s. In thereminder of this section, we assume that Xi ’s are similar to those in Section 3.1 and we write n S1 (t) = i=1 P(Xi 6 t); S2 (t) = 16i¡j6n P(Xi 6 t; Xj 6 t): 3.4. Bonferroni-type inequalities Choose f1 (t) = t and f2 (t) = t(t − 1)=2: It leads to so-called binomial moments (see e.g. Sibuya, 1991, Lemma 2:2) Ef1 ( n (t)) = S1 (t)
and
Ef2 ( n (t)) = S2 (t);
which are closely related with the Bonferroni inequalities (see Galambos and Simonelli, 1996). From the relation (S1 (t); S2 (t)) ∈ conv{(i; i(i − 1)=2): i = 0; 1; : : : ; n}; it follows that for every t ∈ R max{iS1 (t) − i(i + 1)=2: 0 6 i 6 n − 1} 6 S2 (t) 6 (n − 1)S1 (t)=2: The bounds are sharp. 3.5. Bounds for sample median Given m = 1; : : : ; n; consider f1 (t) = I[m; ∞) (t); f2 (t) = t; f3 (t) = t(t − 1)=2: Then Ef1 ( n (t)) = P(Xm:n 6 t); Ef2 ( n (t)) = S1 (t); Ef3 ( n (t)) = S2 (t): We have (P(Xm:n 6 t); S1 (t); S2 (t)) ∈ conv{(I[m; ∞) (i); i; i(i − 1)=2): i = 0; 1; : : : ; n}: Put n = 3; m = 2 and note that the sample median Med(X1 ; X2 ; X3 ) = X2:3 : Consequently, for every t ∈ R max{S2 (t)=3; 2S1 (t) − S2 (t) − 2} 6 P(Med(X1 ; X2 ; X3 ) 6 t) 6 min{S2 (t); (2S1 (t) − S2 (t))=3}: The bounds are attained for some exchangeable X1 ; X2 ; X3 :
186
M. Kaluszka, A. Okolewski / Statistics & Probability Letters 55 (2001) 181 – 186
Acknowledgements The authors would like to thank the referee for his valuable comments which led to improvements in the paper. References Erd˝os, P., Neveu, J., R%enyi, A., 1963. An elementary inequality between the probabilities of events. Math. Scand. 13, 99–104. Galambos, J., Simonelli, I., 1996. Bonferroni-type Inequalities with Applications. Springer, New York. Gascuel, O., Caraux, G., 1992a. Bounds on distribution functions of order statistics for dependent variates. Statist. Probab. Lett. 14, 103–105. Gascuel, O., Caraux, G., 1992b. Bounds on expectations of order statistics via extremal dependencies. Statist. Probab. Lett. 15, 143–148. M%ori, T.F., Sz%ekely, G.J., 1985. A note on the background of several Bonferroni–Galambos-type inequalities. J. Appl. Probab. 22, 836–843. Rychlik, T., 1992. Stochastically extremal distributions of order statistics for dependent samples. Statist. Probab. Lett. 13, 337–341. Rychlik, T., 1993. Bounds for expectation of L-estimates for dependent samples. Statistics 24, 1–7. Rychlik, T., 1995. Bounds for order statistics based on dependent variables with given nonidentical distributions. Statist. Probab. Lett. 23, 351–358. Sibuya, M., 1991. Bonferroni-type inequalities; Chebyshev-type inequalities for the distributions on [0; n]. Ann. Inst. Statist. Math. 43, 261–285.
An extension of the Erd˝os–Neveu–R%enyi theorem with applications to order statistics M. Kaluszka ∗ , A. Okolewski Institute of Mathematics, Technical University of Lodz, ul. Zwirki 36, 90-924 Lodz, Poland Received January 2001; received in revised form May 2001
Abstract The necessary and su3cient conditions are given for some stochastic process to be an empirical distribution function from some exchangeable random variables. The result is applied to establish sharp lower and upper bounds for order c 2001 Elsevier Science B.V. All rights reserved statistics based on possibly dependent random variables. MSC: 62G30; 60G09 Keywords: Dependent random variables; Exchangeable random variables; Empirical distribution function; Order statistics; Bonferroni-type inequalities
1. Introduction The following theorem, proved by Erd˝os et al. (1963), shows that if the assumption of independence of n Bernoulli trials is omitted, the number of successes can be arbitrarily distributed on {0; 1; : : : ; n}: Before rewriting it, let us denote by IA the indicator of A and recall that the events A1 ; A2 ; : : : ; An are said to be exchangeable if P(Ai1 ∩ · · · ∩ Air ) = P(A1 ∩ · · · ∩ Ar ) for every choice of the subscripts 1 6 i1 ¡ · · · ¡ ir 6 n: Theorem 1. Let {pk }nk=0 be a distribution on {0; 1; : : : ; n}: Then there is a probability space ( ; A; P) and a sequence of exchangeable events A1 ; A2 ; : : : ; An in A such that P( n = k) = pk for each k; where n = IA1 + · · · + IAn : The result of Theorem 1 has some interesting applications (see Galambos and Simonelli, 1996). The present version, however, appears to be not strong enough in many situations and consequently there is a need for
∗
Corresponding author. E-mail address: [email protected] (M. Kaluszka).
c 2001 Elsevier Science B.V. All rights reserved 0167-7152/01/$ - see front matter PII: S 0 1 6 7 - 7 1 5 2 ( 0 1 ) 0 0 1 1 4 - 6
182
M. Kaluszka, A. Okolewski / Statistics & Probability Letters 55 (2001) 181 – 186
its generalization. The aim of this paper is to extend the result of Theorem 1 to collections of monotonically increasing events A1 (t); : : : ; An (t); t ∈ R: Such extension not only leads to natural generalizations of some Bonferroni-type inequalities as well as Rychlik’s (1993, 1995) bounds for order statistics (see also Rychlik, 1992; Gascuel and Caraux, 1992a, b), but also indicates that both the mentioned types of inequalities can be treated as bounds for some spaces of moments of n : In addition, our theorem provides a tool which can be used to obtain easily new inequalities for other functionals of order statistics. The mentioned generalizations as well as some examples of new bounds for order statistics are presented in Section 3. 2. The result d
We will use the symbol X =Y to denote that the random variables X and Y are identically distributed. For a given distribution function F, we will denote by F −1 the quantile function, i.e., F −1 (t) = sup{s ∈ R: F(s) 6 t}; t ∈ [0; 1): Theorem 2. Let { n (t): t ∈ R} be a stochastic process on some probability space ( ; A; P) with values in {0; 1; : : : ; n}: The following conditions are equivalent: ˜ A;˜ P) ˜ and a sequence X1 ; X2 ; : : : ; Xn of exchangeable random (i) There exists a probability space ( ; ˜ variables de9ned on such that n d I{Xi 6t} ; (1) n (t) = i=1
for every t ∈ R: (ii) R t → P( n (t) ¿ k) is a distribution function on R for every k = 1; 2; : : : ; n: Proof. The implication (i) ⇒ (ii) is immediate. The proof of the reverse implication consists in the con˜ A;˜ P) ˜ and a sequence X1 ; X2 ; : : : ; Xn of exchangeable random variables struction of a probability space ( ; ˜ deLned on : Let us Lrst deal with the probability space. Set ˜ = [0; 1) × ; where is the collection of all permutations of {1; : : : ; n}: DeLne A˜ to be the -Leld generated by the class of all sets of the form B × A; where A ∈ and B is a Borel subset of [0; 1): Let P˜ be the product measure of the Lebesgue measure on [0; 1) and the uniform distribution on : We can now deLne the sequence of random variables. Set Fk (t) = P( n (t) ¿ k); k = 1; 2; : : : ; n; t ∈ R: By assumption, Fk ’s are distribution functions. Write Yk (s) = Fk−1 (s);
s ∈ [0; 1)
and X1 (s; ) = Y(1) (s); : : : ; Xn (s; ) = Y(n) (s);
˜ (s; ) ∈ [0; 1) × = ;
where = ((1); : : : ; (n)) is a permutation of the set {1; : : : ; n}: We next show that X1 ; X2 ; : : : ; Xn are exchangeable. Consider arbitrary k = 1; 2; : : : ; n; 1 6 i1 ¡ i2 ¡ · · · ¡ ik 6 n and t1 ; : : : ; tk ∈ R: By the deLnition of Xi ’s, ˜ i1 6 t1 ; : : : ; Xik 6 tk ) P(X = =
˜ (i1 ) 6 t1 ; : : : ; Y(ik ) 6 tk | (i1 ) = l1 ; : : : ; (ik ) = lk ) (n − k)! P(Y n!
˜ l1 6 t1 ; : : : ; Ylk 6 tk ) (n − k)! P(Y n!
M. Kaluszka, A. Okolewski / Statistics & Probability Letters 55 (2001) 181 – 186
=
183
˜ (1) 6 t1 ; : : : ; Y(k) 6 tk | (1) = l1 ; : : : ; (k) = lk ) (n − k)! P(Y n!
˜ 1 6 t1 ; : : : ; Xk 6 tk ); = P(X where the sums are over all possible l1 ; : : : ; lk : It remains to prove that (1) is satisLed. Observe that Y1 6 Y2 6 ; : : : ; 6 Yn P˜ a.s. Combining this with the deLnition of X1 ; X2 ; : : : ; Xn gives n n ˜ k 6 t; Yk+1 ¿ t) P˜ I{Xi 6t} = k = P˜ I{Yi 6t} = k = P(Y i=1
i=1
˜ k 6 t; Yk+1 6 t) ˜ k 6 t) − P(Y = P(Y ˜ k 6 t) − P(Y ˜ k+1 6 t) = Fk (t) − Fk+1 (t) = P(Y = P( n (t) ¿ k) − P( n (t) ¿ k + 1) = P( n (t) = k); and the proof is complete. Remarks. (1) The original proof of Erd˝os–Neveu–R%enyi cannot be adapted here because the construction of the measure P˜ depends on t: Another proof of Theorem 1 due to Sibuya (1991) is known. Since the method presented therein may not provide monotone collections of sets, Sibuya’s reasoning cannot be used as well. (2) Theorem 2 states the necessary and su3cient conditions for the stochastic process n (t)=n to be an empirical distribution function from some exchangeable random variables X1 ; : : : ; Xn : (3) If we take a Lxed t ∈ R; we recover Theorem 1 of Erd˝os–Neveu–R%enyi.
3. Applications 3.1. Rychlik-type bounds on L-estimates We extend Rychlik’s (1993) bounds on expectation of linear combinations of order statistics to nonidentical distributions. Let X1 ; X2 ; : : : ; Xn be possibly dependent but not necessarily identically distributed random variables. Let Xk:n denote the kth order statistic from the sample X1 ; : : : ; Xn ; and let c1 ; : : : ; cn be arbitrary real numbers. Then 1 1 n −1 N F (t) d C(t) 6 ck EXk:n 6 F −1 (t) dC(t); (2) 0
k=1
0
i n where F(t) = (1=n) j=1 P(Xj 6 t); C is the greatest convex function such that C(0) = 0 and C(i=n) 6 k=1 ck ; i N N i = 0; 1; : : : ; n; and CN is the smallest concave function such that C(0) = 0 and C(i=n) ¿ k=1 ck ; i = 0; 1; : : : ; n 0 (see Rychlik, 1993). Here and subsequently, we adopt the convention k=1 ck = 0: Furthermore, if X1 ; X2 ; : : : ; Xn are identically distributed, there exists a sequence of exchangeable random variables such that the lower bound is attainable (the same conclusion can be drawn for the upper bound). To deduce (2) from Theorem 2, we use the result due to M%ori and Sz%ekely (1985) (see also Galambos and Simonelli, 1996, Lemma III.1). The result states that the range of the moments (Ef1 ( n ); : : : ; Efl ( n )); where f1 ; : : : ; fl are arbitrary real functions on {0; 1; : : : ; n}; is the conv{(f1 (i); : : : ; fl (i)): i = 0; 1; : : : ; n} if the distribution of n varies over all distributions on {0; 1; : : : ; n}:
184
M. Kaluszka, A. Okolewski / Statistics & Probability Letters 55 (2001) 181 – 186
Let n (t) =
n
i=1 I{Xi 6t} ; n
1 Ef1 ( n (t)) = n
f1 (t) = t=n and f2 (t) =
P(Xi 6 t);
n
Ef2 ( n (t)) =
i=1
k=1 ck I[k; ∞) (t):
n
It is clear that
ck P(Xk:n 6 t):
k=1
By M%ori and Sz%ekely’s characterization, n n i 1 i ; P(Xi 6 t); ck P(Xk:n 6 t) ∈ conv ck : i = 0; 1; : : : ; n : n n i=1
k=1
k=1
Hence for every t ∈ R n N C(F(t)) 6 ck P(Xk:n 6 t) 6 C(F(t)): k=1
From what has already been proved, it follows easily that (2) is satisLed. What is left is to show that the N Let n (t); t ∈ R; be a lower bound is sharp. Let n1 ; : : : ; nm denote the endpoints of the linear pieces of C: stochastic process such that P( n (t) = ni ) = p(t) = 1 − P( n (t) = ni+1 ) for F(t) ∈ [ni =n; ni+1 =n]; i = 1; : : : ; m − 1; where p(t) is chosen so that Ef1 ( n (t)) = F(t): We check at once that p(t) = (ni+1 − nF(t))=(ni+1 − ni ) and Ef2 ( n (t)) =
ni
ck +
k=1
ni+1
ck
(1 − p(t))
k=ni +1
N i+1 ) − C(n N i ))(1 − p(t)) N i ) + (C(n = C(n n n i i+1 N ; : = C(F(t)); F(t) ∈ n n Since R t → P( n (t) ¿ k) is a distribution function for every k = 1; 2; : : : ; n; the attainability conditions for identical distributions are an immediate consequence of Theorem 2. The corresponding problem of attainability for nonidentical distributions seems to be rather di3cult (see Rychlik, 1995). 3.2. New bounds for order statistics Similar arguments applied to moments diQerent from the ones considered above lead to new bounds for order statistics. For example, let f1 (t) = t and f2 (t) = I[k; l) (t); where 1 6 k ¡ l 6 n: Then n Ef1 ( n (t)) = P(Xi 6 t); i=1
Ef2 ( n (t)) = P(k 6 n (t) ¡ l) = P(Xk:n 6 t; Xl:n ¿ t) and (Ef1 ( n (t)); Ef2 ( n (t))) ∈ conv{(i; I[k; l) (i)): i = 0; 1; : : : ; n}: It is easily seen that 0 6 Ef2 ( n (t)) 6 min
n − Ef1 ( n (t)) Ef1 ( n (t)) ; 1; k n−l+1
and so 0 6 P(Xk:n 6 t; Xl:n ¿ t) 6 min
n(1 − F(t)) n F(t); 1; k n−l+1
; ;
(3)
M. Kaluszka, A. Okolewski / Statistics & Probability Letters 55 (2001) 181 – 186
185
n where F(t) = (1=n) i=1 P(Xi 6 t): The lower and upper bounds are sharp, i.e. they are attained for some exchangeable random variables. This follows by the same method as in Section 3.1. Since similar constructions concerning attainability apply to Sections 3.3–3.5, we will omit them. 3.3. Bounds for empirical distribution function with common distribution function F and let Fn Let X1 ; X2 ; : : : ; Xn be possibly dependent random variables n denote the empirical distribution function, i.e., Fn (t) = (1=n) i=1 I{Xi 6t} ; t ∈ R: Applying (3) with l = k + 1 yields
n n(1 − F(t)) k 6 min F(t); ∀t∈R P Fn (t) = n k n−k for k = 1; 2; : : : ; n − 1: Moreover, analysis similar to that in Section 3.2 shows that ∀t∈R
P(Fn (t) = 0) 6 1 − F(t)
and
P(Fn (t) = 1) 6 F(t):
By Theorem 2, the bounds are attained for some exchangeable Xi ’s. In thereminder of this section, we assume that Xi ’s are similar to those in Section 3.1 and we write n S1 (t) = i=1 P(Xi 6 t); S2 (t) = 16i¡j6n P(Xi 6 t; Xj 6 t): 3.4. Bonferroni-type inequalities Choose f1 (t) = t and f2 (t) = t(t − 1)=2: It leads to so-called binomial moments (see e.g. Sibuya, 1991, Lemma 2:2) Ef1 ( n (t)) = S1 (t)
and
Ef2 ( n (t)) = S2 (t);
which are closely related with the Bonferroni inequalities (see Galambos and Simonelli, 1996). From the relation (S1 (t); S2 (t)) ∈ conv{(i; i(i − 1)=2): i = 0; 1; : : : ; n}; it follows that for every t ∈ R max{iS1 (t) − i(i + 1)=2: 0 6 i 6 n − 1} 6 S2 (t) 6 (n − 1)S1 (t)=2: The bounds are sharp. 3.5. Bounds for sample median Given m = 1; : : : ; n; consider f1 (t) = I[m; ∞) (t); f2 (t) = t; f3 (t) = t(t − 1)=2: Then Ef1 ( n (t)) = P(Xm:n 6 t); Ef2 ( n (t)) = S1 (t); Ef3 ( n (t)) = S2 (t): We have (P(Xm:n 6 t); S1 (t); S2 (t)) ∈ conv{(I[m; ∞) (i); i; i(i − 1)=2): i = 0; 1; : : : ; n}: Put n = 3; m = 2 and note that the sample median Med(X1 ; X2 ; X3 ) = X2:3 : Consequently, for every t ∈ R max{S2 (t)=3; 2S1 (t) − S2 (t) − 2} 6 P(Med(X1 ; X2 ; X3 ) 6 t) 6 min{S2 (t); (2S1 (t) − S2 (t))=3}: The bounds are attained for some exchangeable X1 ; X2 ; X3 :
186
M. Kaluszka, A. Okolewski / Statistics & Probability Letters 55 (2001) 181 – 186
Acknowledgements The authors would like to thank the referee for his valuable comments which led to improvements in the paper. References Erd˝os, P., Neveu, J., R%enyi, A., 1963. An elementary inequality between the probabilities of events. Math. Scand. 13, 99–104. Galambos, J., Simonelli, I., 1996. Bonferroni-type Inequalities with Applications. Springer, New York. Gascuel, O., Caraux, G., 1992a. Bounds on distribution functions of order statistics for dependent variates. Statist. Probab. Lett. 14, 103–105. Gascuel, O., Caraux, G., 1992b. Bounds on expectations of order statistics via extremal dependencies. Statist. Probab. Lett. 15, 143–148. M%ori, T.F., Sz%ekely, G.J., 1985. A note on the background of several Bonferroni–Galambos-type inequalities. J. Appl. Probab. 22, 836–843. Rychlik, T., 1992. Stochastically extremal distributions of order statistics for dependent samples. Statist. Probab. Lett. 13, 337–341. Rychlik, T., 1993. Bounds for expectation of L-estimates for dependent samples. Statistics 24, 1–7. Rychlik, T., 1995. Bounds for order statistics based on dependent variables with given nonidentical distributions. Statist. Probab. Lett. 23, 351–358. Sibuya, M., 1991. Bonferroni-type inequalities; Chebyshev-type inequalities for the distributions on [0; n]. Ann. Inst. Statist. Math. 43, 261–285.