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Monotonicity properties of certain expected extreme order statistics
Statistics & Probability Letters 52 (2001) 313 – 319
Monotonicity properties of certain expected extreme order statistics Jes!us De La Cal ∗ , Ana M. Valle Departamento de Matem atica Aplicada y Estad stica e Investigaci on Operativa, Facultad de Ciencias, Universidad del Pa s Vasco, Apartado 644, 48080 Bilbao, Spain Received January 1999; received in revised form September 2000
Abstract If X1 (); : : : ; Xn () are independent random variables having discrete distributions depending on a positive parameter ∈ (0; ) and satisfying appropriate assumptions, the expected extreme order statistics EXn:n () and EX1:n () have nice monotonicity properties. Such properties extend to random extreme order statistics, but not to intermediate order statistics. c 2001 Elsevier Science B.V. All rights reserved MSC: primary 62E10; 62G30 Keywords: Discrete distributions; One-parameter family; Order statistics; Expected value; Monotonicity properties
1. Introduction Let (X1 ; : : : ; Xn ) be a random vector, and denote by X1:n 6 X2:n 6 · · · 6 Xn:n the corresponding order statistics. The expected values of the order statistics have been extensively investigated. Useful accounts of interesting results can be found in David (1981), Arnold et al. (1992), and the references therein. Arnold (1988) surveys bounds on EXn:n under di9erent assumptions on the distributions and=or dependence structure of the Xi ’s. For some recent developments, we refer, for instance, to Balakrishnan (1990), Downey (1990), and Huang (1997). In the present paper, we go in a direction apparently overlooked in the literature. We deal with independent random variables X1 (); : : : ; Xn () having discrete distributions (on the nonnegative integers) depending on a parameter ∈ (0; ), and investigate the behavior of the expected values EXk:n () as functions of . When
Research supported by the University of the Basque Country and by Grant PB98-1577-C02-02 of the Spanish DGESIC.
∗
Corresponding author. E-mail addresses: [email protected] (J. de la Cal), [email protected] (A.M. Valle). 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 0 ) 0 0 2 3 2 - 7
314
J. de la Cal, A.M. Valle / Statistics & Probability Letters 52 (2001) 313 – 319
all the respective families of distributions belong to a certain class P(), we show that the ratios EXn:n () EX1:n () and m() := ; ∈ (0; ) (1) are nonincreasing and nondecreasing, respectively. However, no analogous monotonicity property can be guaranteed for the intermediate order statistics. In the particular case that X1 (); : : : ; Xn () is a random sample of size n from the parent distribution p(), the preceding results provide information about the ratios of the expected order statistics to the population mean (see Remark 4 below). The paper is organized as follows: In the next section, we introduce the classes P(), show that they include the most classical families of discrete distributions, and collect some of their relevant features. Section 3 contains the statements of the monotonicity results. The proof of the main theorem is given in Section 4. Finally, in the last section, we provide a counterexample concerning intermediate order statistics. The results discussed in the present paper can be expected to be useful in statistical problems concerning the mentioned discrete distributions. However, the paper originated in the investigations of the authors on certain best constants for multivariate Bernstein-type operators (cf. de la Cal and Valle, 2000). As it is nicely said in Arnold et al. (1992), problems involving order statistics may arise in the most varied settings. M () :=
2. The classes P() For each ∈ (0; ∞], we denote by P() the class of all families p := {p(): ∈ (0; )} of probability distributions fulJlling the following assumptions: (i) For each ∈ (0; ), p() is a discrete probability distribution on the nonnegative integers. The weight of p() at j = 0; 1; : : : ; will be denoted by pj (). (ii) The weight functions p0 (·); p1 (·); : : : ; are continuously di9erentiable on (0; ) and satisfy the following system of di9erential equations d pj () = jpj () − (j + 1)pj+1 (); 0 ¡ ¡ ; j = 0; 1; : : : : d ∞ (iii) The series j=1 j 2 pj () converges uniformly on each bounded subinterval of (0; ).
Remark 1. Condition (iii) implies that each p() has a Jnite variance. Moreover, it plays the technical role of guaranteeing that series like j pj () and j jpj () can be di9erentiated term by term. Examples. (a) The family p ≡ 0; where p() is the distribution degenerate at 0, belongs to P(), for every ∈ (0; ∞]. (b) For each m = 1; 2; : : : ; we have B(m) ∈ P(1), where B(m) () is the binomial distribution with parameters m and , i.e., m j (1 − )m−j : Bj(m) () := j (c) We have {(): ¿ 0} ∈ P(∞), where () is the Poisson distribution with parameter , i.e., j () := e−
j : j!
(d) For each t ¿ 0, we have b(t) ∈ P(∞), where b(t) () is the negative binomial distribution with parameters t and , i.e., j t+j−1 (t) bj () := : j (1 + )t+j
J. de la Cal, A.M. Valle / Statistics & Probability Letters 52 (2001) 313 – 319
315
(e) Other well-known families of distributions satisfying assumptions (i) – (iii) above can be seen in Johnson et al. (1992, p. 350). The families under consideration share a number of properties. Referring to Sprott (1965), Johnson et al. (1992) mention an important maximum likelihood feature. Other facts are collected in the following proposition. For the notion of the usual stochastic order, we refer to Shaked and Shantikhumar (1994). Proposition 1. Let ∈ (0; ∞] and p ∈ P(). Then; (a) p(·) is nondecreasing for the usual stochastic order. (b) There exists a constant c¿0 such that () = c; 0 ¡ ¡ ; ∞ where () := j=1 jpj () is the mean of p() (we obviously have c = 0 if and only if p ≡ 0). (c) The probability generating function g(; ·) of p() satis
@ @ g(; z) + (1 − z) g(; z) = 0: @ @z
(2)
Proof. (a) From the assumptions on p, we have for all ∈ (0; ) and k = 0; 1; : : : ∞
∞
d d k pj () = pk ()¿0; pj () = d d j=k j=k ∞ showing that j=k pj (·) is a nondecreasing function. (b) It is readily checked that (·) satisJes the di9erential equation d () = () d and the conclusion in part (b) follows. (c) Analogous easy calculations yield Eq. (2).
Remark 2. In the setting of the preceding proposition, we also have that p0 () → 1 as ↓ 0; as it follows from Proposition 1(b), and () = p1 () + o() ( ↓ 0);
(3)
as it is shown in the Appendix at the end of this paper. For some implications of these facts (and Proposition 1(b)), see Remark 4 below. On the other hand, the classes P() have nice closure properties, as those stated in the following proposition. The straightforward proof is omitted. We denote by ‘∗’ the convolution of probability distributions, and the convolution p ∗ q of p; q ∈ P() is deJned by (p ∗ q)() := p() ∗ q();
∈ (0; ):
Proposition 2. Let ∈ (0; ∞]. Then; (a) (Mixtures) For all p; q ∈ P(); and ∈ [0; 1]; we have p + (1 − )q ∈ P(). (b) (Convolutions) For all p; q ∈ P(); we have p ∗ q ∈ P(). Remark 3. Let ∈ (0; ∞], ¿ 0, and p ∈ P(). It is easy to check that p ∈ P( −1 ); where p () := p( );
0 ¡ ¡ −1 :
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J. de la Cal, A.M. Valle / Statistics & Probability Letters 52 (2001) 313 – 319
Moreover, the mapping p → p , from P() into P( −1 ), is one-to-one. In particular, by a linear reparametrization, each class P(), with ¡ ∞, can be identiJed with P(1). On the contrary, P(∞) is substantially di9erent from P(1). Actually, for each m¿1; P(1) contains families p having the Jnite support {0; 1; : : : ; m} (m¿1), i.e., such that pm ≡ 0;
pm+1 ≡ pm+2 ≡ · · · ≡ 0
(4)
(m)
(for instance, the binomial family B in Example (b)), but P(∞) does not, because the assumption (4) implies that the mean function (·) is bounded by m, and this contradicts Proposition 1(b), when = ∞. 3. Monotonicity results Theorem 1. Let ∈ (0; ∞]; p(1) ; : : : ; p(n) ∈ P(); and; for ∈ (0; ); assume that X1 (); X2 (); : : : ; Xn () are independent random variables such that Xi () has the distribution p(i) (). Then; the functions M (·) and m(·) given in (1) are nonincreasing and nondecreasing on (0; ); respectively. Remark 4. Assume that p(1) = · · · = p(n) = p = 0; i.e., (X1 (); : : : ; Xn ()) is a random sample of size n from the parent distribution p(). Then, in view of Proposition 1(b), Theorem 1 gives information about the behavior of the ratios of the expected extreme order statistics to the population mean (). In particular, the ratio of the expected sample range E(Xn:n () − X1:n ()) to the population mean is a nonincreasing function of . Another consequence is that EXn:n () ↑ (n) as ↓ 0; () where (n) is a positive constant depending upon the sample size. We claim that (n) = n. Actually, the inequality (n) 6 n is obvious, and the converse inequality follows from Remark 2 and the fact that EXn:n ()¿P(Xn:n ()¿1)¿n[p0 ()]n−1 p1 (): Analogously, it is easy to see that EX1:n () ↓ 0 as ↓ 0; n¿2: () When dealing with concrete distributions, further results can be obtained. For instance, in the case that p = B(m) (the binomial family in Example (b)), we have that EXn:n () ↑ n as ↓ 0; n¿3; P(Xn:n ()¿1) + m as it is shown in de la Cal and Valle (2000). In the same paper, the reader can also Jnd similar results for the families in Examples (c) and (d). 4. Proof of Theorem 1 We can write, for ∈ (0; ), EXn:n () =
∞
P(Xn:n ()¿k) =
1−
k=1
∞
n ∞
k=1
P(X1:n ()¿k) =
k=1
n
P(Xi () 6 k − 1)
i=1
k=1
and EX1:n () =
∞
i=1
P(Xi ()¿k) :
J. de la Cal, A.M. Valle / Statistics & Probability Letters 52 (2001) 313 – 319
317
Since the series in both right-hand sides can be di9erentiated term by term, we have ∞ n d d − P(Xi () 6 k − 1) P(Xj () 6 k − 1) EXn:n () = d d k=1 i=1
j=i
∞ n k P(Xi () = k) P(Xj () 6 k − 1)
=
k=1
6
∞
i=1
j=i
kP(Xn:n () = k)
k=1
= EXn:n () and
∞ n d k P(Xi () = k) P(Xj ()¿k)¿EX1:n (): EX1:n () = d k=1
i=1
j=i
We therefore conclude that d d 2 M () = EXn:n () − EXn:n () 6 0 d d and d d m() = EX1:n () − EX1:n ()¿0; 2 d d showing the theorem. 5. Intermediate order statistics In this section, we show that the assumptions in Theorem 1 above cannot guarantee the monotonicity of the function EXk:n ()=, when k = 2; : : : ; n − 1. Take, for instance, = 1, and, for i = 1; : : : ; n, let p(i) be the family B(1) in Example (b) above (i.e., the family of Bernoulli distributions). Then, we have for n = 1; 2; : : : ; and k = 1; : : : ; n n EXk:n () = P Xi ()¿n − k + 1 : i=1
Therefore,
n n EX d () k:n Xi () = n − k + 1 − P Xi ()¿n − k + 1 2 = (n − k + 1)P d =
n−k+1
i=1
i=1
(n − k + 1)
n n−k +1
(1 − )
k−1
−
n n r−n+k−1 n−r : (1 − ) r
r=n−k+1
n If n¿3 and k = 2; : : : ; n − 1, the polynomial in brackets takes the value −1 or (n − k)( n−k+1 ) according to = 1 or = 0. By continuity, this entails that the derivative of EXk:n ()= is negative (resp., positive) for close to 1 (resp., for close to 0). Hence, the function EXk:n ()= is not monotone on (0; 1).
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J. de la Cal, A.M. Valle / Statistics & Probability Letters 52 (2001) 313 – 319
Appendix Here, we provide a proof of (3). Let X () be a random variable having distribution p() ( ∈ (0; )). First of all, we observe that d P(X ()¿1) = P(X () = 1) − P(X ()¿1) 6 0; d so that the function P(X ()¿1)= is nonincreasing, and, therefore, it has a limit l as ↓ 0. By Proposition 1(b), we have that l 6 c. Proposition 1(b) also guarantees that we can deJne by continuity P(X (0)¿1) := 0. From the mean value theorem, we conclude that 2
l = lim ↓0
P(X ()¿1) d p1 () = lim P(X ()¿1) = lim ; ↓0 ↓0 d
implying that P(X ()¿1) = p1 () + o() ( ↓ 0); as well as P(X ()¿k) = o() ( ↓ 0);
k = 2; 3; : : : :
(5)
Thus, to achieve (3), we only need to show that h() :=
∞
P(X ()¿k) = o() ( ↓ 0):
(6)
k=2
Let " ¿ 0, and Jx 0 ∈ (0; ) for a while. Since ∞ k=2 P(X (0 )¿k) 6 c; 0 we can Jnd an integer k0 ¿2 such that ∞ k=k0 +1 P(X (0 )¿k) 6 ": 0 ∞ The function k=k0 +1 P(X ()¿k)= is nondecreasing on (0; ), as it can be shown by di9erentiation. Therefore, we can write k0 h() P(X ()¿k) 6 + ";
0 ¡ 6 0
k=2
and, from (5), we conclude that lim sup ↓0
h() 6 ":
This shows (6), because " ¿ 0 is arbitrary, and completes the proof of (3). References Arnold, B.C., 1988. Bounds on the expected maximum. Comm. Statist. Theory Methods 17, 2135–2150. Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1992. A First Course in Order Statistics. Wiley, New York. Balakrishnan, N., 1990. Improving the Hartley–David–Gumbel bound for the mean of extreme order statistics. Statist. Probab. Lett. 9, 291–294.
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319
David, H.A., 1981. Order Statistics, 2nd Edition. Wiley, New York. de la Cal, J., Valle, A.M., 2000. Best constants for tensor products of Bernstein, Sz!asz and Baskakov operators. Bull. Austral. Math. Soc. 62, 211–220. Downey, P.J., 1990. Distribution-free bounds on the expectation of the maximum with scheduling applications. Oper. Res. Lett. 9, 189–201. Huang, J.S., 1997. Sharp bounds for the expected value of order statistics. Statist. Probab. Lett. 33, 105–107. Johnson, N.L., Kotz, S., Kemp, A.W., 1992. Univariate Discrete Distributions, 2nd Edition. Wiley, New York. Shaked, M., Shantikhumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, Boston. Sprott, D.A., 1965. A class of contagious distributions and maximum likelihood estimation. In: Patil, G.P. (Ed.), Classical and Contagious Discrete Distributions. Statistical Publishing Society, Calcutta; Pergamon Press, Oxford, pp. 337–350.
Monotonicity properties of certain expected extreme order statistics Jes!us De La Cal ∗ , Ana M. Valle Departamento de Matem atica Aplicada y Estad stica e Investigaci on Operativa, Facultad de Ciencias, Universidad del Pa s Vasco, Apartado 644, 48080 Bilbao, Spain Received January 1999; received in revised form September 2000
Abstract If X1 (); : : : ; Xn () are independent random variables having discrete distributions depending on a positive parameter ∈ (0; ) and satisfying appropriate assumptions, the expected extreme order statistics EXn:n () and EX1:n () have nice monotonicity properties. Such properties extend to random extreme order statistics, but not to intermediate order statistics. c 2001 Elsevier Science B.V. All rights reserved MSC: primary 62E10; 62G30 Keywords: Discrete distributions; One-parameter family; Order statistics; Expected value; Monotonicity properties
1. Introduction Let (X1 ; : : : ; Xn ) be a random vector, and denote by X1:n 6 X2:n 6 · · · 6 Xn:n the corresponding order statistics. The expected values of the order statistics have been extensively investigated. Useful accounts of interesting results can be found in David (1981), Arnold et al. (1992), and the references therein. Arnold (1988) surveys bounds on EXn:n under di9erent assumptions on the distributions and=or dependence structure of the Xi ’s. For some recent developments, we refer, for instance, to Balakrishnan (1990), Downey (1990), and Huang (1997). In the present paper, we go in a direction apparently overlooked in the literature. We deal with independent random variables X1 (); : : : ; Xn () having discrete distributions (on the nonnegative integers) depending on a parameter ∈ (0; ), and investigate the behavior of the expected values EXk:n () as functions of . When
Research supported by the University of the Basque Country and by Grant PB98-1577-C02-02 of the Spanish DGESIC.
∗
Corresponding author. E-mail addresses: [email protected] (J. de la Cal), [email protected] (A.M. Valle). 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 0 ) 0 0 2 3 2 - 7
314
J. de la Cal, A.M. Valle / Statistics & Probability Letters 52 (2001) 313 – 319
all the respective families of distributions belong to a certain class P(), we show that the ratios EXn:n () EX1:n () and m() := ; ∈ (0; ) (1) are nonincreasing and nondecreasing, respectively. However, no analogous monotonicity property can be guaranteed for the intermediate order statistics. In the particular case that X1 (); : : : ; Xn () is a random sample of size n from the parent distribution p(), the preceding results provide information about the ratios of the expected order statistics to the population mean (see Remark 4 below). The paper is organized as follows: In the next section, we introduce the classes P(), show that they include the most classical families of discrete distributions, and collect some of their relevant features. Section 3 contains the statements of the monotonicity results. The proof of the main theorem is given in Section 4. Finally, in the last section, we provide a counterexample concerning intermediate order statistics. The results discussed in the present paper can be expected to be useful in statistical problems concerning the mentioned discrete distributions. However, the paper originated in the investigations of the authors on certain best constants for multivariate Bernstein-type operators (cf. de la Cal and Valle, 2000). As it is nicely said in Arnold et al. (1992), problems involving order statistics may arise in the most varied settings. M () :=
2. The classes P() For each ∈ (0; ∞], we denote by P() the class of all families p := {p(): ∈ (0; )} of probability distributions fulJlling the following assumptions: (i) For each ∈ (0; ), p() is a discrete probability distribution on the nonnegative integers. The weight of p() at j = 0; 1; : : : ; will be denoted by pj (). (ii) The weight functions p0 (·); p1 (·); : : : ; are continuously di9erentiable on (0; ) and satisfy the following system of di9erential equations d pj () = jpj () − (j + 1)pj+1 (); 0 ¡ ¡ ; j = 0; 1; : : : : d ∞ (iii) The series j=1 j 2 pj () converges uniformly on each bounded subinterval of (0; ).
Remark 1. Condition (iii) implies that each p() has a Jnite variance. Moreover, it plays the technical role of guaranteeing that series like j pj () and j jpj () can be di9erentiated term by term. Examples. (a) The family p ≡ 0; where p() is the distribution degenerate at 0, belongs to P(), for every ∈ (0; ∞]. (b) For each m = 1; 2; : : : ; we have B(m) ∈ P(1), where B(m) () is the binomial distribution with parameters m and , i.e., m j (1 − )m−j : Bj(m) () := j (c) We have {(): ¿ 0} ∈ P(∞), where () is the Poisson distribution with parameter , i.e., j () := e−
j : j!
(d) For each t ¿ 0, we have b(t) ∈ P(∞), where b(t) () is the negative binomial distribution with parameters t and , i.e., j t+j−1 (t) bj () := : j (1 + )t+j
J. de la Cal, A.M. Valle / Statistics & Probability Letters 52 (2001) 313 – 319
315
(e) Other well-known families of distributions satisfying assumptions (i) – (iii) above can be seen in Johnson et al. (1992, p. 350). The families under consideration share a number of properties. Referring to Sprott (1965), Johnson et al. (1992) mention an important maximum likelihood feature. Other facts are collected in the following proposition. For the notion of the usual stochastic order, we refer to Shaked and Shantikhumar (1994). Proposition 1. Let ∈ (0; ∞] and p ∈ P(). Then; (a) p(·) is nondecreasing for the usual stochastic order. (b) There exists a constant c¿0 such that () = c; 0 ¡ ¡ ; ∞ where () := j=1 jpj () is the mean of p() (we obviously have c = 0 if and only if p ≡ 0). (c) The probability generating function g(; ·) of p() satis
@ @ g(; z) + (1 − z) g(; z) = 0: @ @z
(2)
Proof. (a) From the assumptions on p, we have for all ∈ (0; ) and k = 0; 1; : : : ∞
∞
d d k pj () = pk ()¿0; pj () = d d j=k j=k ∞ showing that j=k pj (·) is a nondecreasing function. (b) It is readily checked that (·) satisJes the di9erential equation d () = () d and the conclusion in part (b) follows. (c) Analogous easy calculations yield Eq. (2).
Remark 2. In the setting of the preceding proposition, we also have that p0 () → 1 as ↓ 0; as it follows from Proposition 1(b), and () = p1 () + o() ( ↓ 0);
(3)
as it is shown in the Appendix at the end of this paper. For some implications of these facts (and Proposition 1(b)), see Remark 4 below. On the other hand, the classes P() have nice closure properties, as those stated in the following proposition. The straightforward proof is omitted. We denote by ‘∗’ the convolution of probability distributions, and the convolution p ∗ q of p; q ∈ P() is deJned by (p ∗ q)() := p() ∗ q();
∈ (0; ):
Proposition 2. Let ∈ (0; ∞]. Then; (a) (Mixtures) For all p; q ∈ P(); and ∈ [0; 1]; we have p + (1 − )q ∈ P(). (b) (Convolutions) For all p; q ∈ P(); we have p ∗ q ∈ P(). Remark 3. Let ∈ (0; ∞], ¿ 0, and p ∈ P(). It is easy to check that p ∈ P( −1 ); where p () := p( );
0 ¡ ¡ −1 :
316
J. de la Cal, A.M. Valle / Statistics & Probability Letters 52 (2001) 313 – 319
Moreover, the mapping p → p , from P() into P( −1 ), is one-to-one. In particular, by a linear reparametrization, each class P(), with ¡ ∞, can be identiJed with P(1). On the contrary, P(∞) is substantially di9erent from P(1). Actually, for each m¿1; P(1) contains families p having the Jnite support {0; 1; : : : ; m} (m¿1), i.e., such that pm ≡ 0;
pm+1 ≡ pm+2 ≡ · · · ≡ 0
(4)
(m)
(for instance, the binomial family B in Example (b)), but P(∞) does not, because the assumption (4) implies that the mean function (·) is bounded by m, and this contradicts Proposition 1(b), when = ∞. 3. Monotonicity results Theorem 1. Let ∈ (0; ∞]; p(1) ; : : : ; p(n) ∈ P(); and; for ∈ (0; ); assume that X1 (); X2 (); : : : ; Xn () are independent random variables such that Xi () has the distribution p(i) (). Then; the functions M (·) and m(·) given in (1) are nonincreasing and nondecreasing on (0; ); respectively. Remark 4. Assume that p(1) = · · · = p(n) = p = 0; i.e., (X1 (); : : : ; Xn ()) is a random sample of size n from the parent distribution p(). Then, in view of Proposition 1(b), Theorem 1 gives information about the behavior of the ratios of the expected extreme order statistics to the population mean (). In particular, the ratio of the expected sample range E(Xn:n () − X1:n ()) to the population mean is a nonincreasing function of . Another consequence is that EXn:n () ↑ (n) as ↓ 0; () where (n) is a positive constant depending upon the sample size. We claim that (n) = n. Actually, the inequality (n) 6 n is obvious, and the converse inequality follows from Remark 2 and the fact that EXn:n ()¿P(Xn:n ()¿1)¿n[p0 ()]n−1 p1 (): Analogously, it is easy to see that EX1:n () ↓ 0 as ↓ 0; n¿2: () When dealing with concrete distributions, further results can be obtained. For instance, in the case that p = B(m) (the binomial family in Example (b)), we have that EXn:n () ↑ n as ↓ 0; n¿3; P(Xn:n ()¿1) + m as it is shown in de la Cal and Valle (2000). In the same paper, the reader can also Jnd similar results for the families in Examples (c) and (d). 4. Proof of Theorem 1 We can write, for ∈ (0; ), EXn:n () =
∞
P(Xn:n ()¿k) =
1−
k=1
∞
n ∞
k=1
P(X1:n ()¿k) =
k=1
n
P(Xi () 6 k − 1)
i=1
k=1
and EX1:n () =
∞
i=1
P(Xi ()¿k) :
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317
Since the series in both right-hand sides can be di9erentiated term by term, we have ∞ n d d − P(Xi () 6 k − 1) P(Xj () 6 k − 1) EXn:n () = d d k=1 i=1
j=i
∞ n k P(Xi () = k) P(Xj () 6 k − 1)
=
k=1
6
∞
i=1
j=i
kP(Xn:n () = k)
k=1
= EXn:n () and
∞ n d k P(Xi () = k) P(Xj ()¿k)¿EX1:n (): EX1:n () = d k=1
i=1
j=i
We therefore conclude that d d 2 M () = EXn:n () − EXn:n () 6 0 d d and d d m() = EX1:n () − EX1:n ()¿0; 2 d d showing the theorem. 5. Intermediate order statistics In this section, we show that the assumptions in Theorem 1 above cannot guarantee the monotonicity of the function EXk:n ()=, when k = 2; : : : ; n − 1. Take, for instance, = 1, and, for i = 1; : : : ; n, let p(i) be the family B(1) in Example (b) above (i.e., the family of Bernoulli distributions). Then, we have for n = 1; 2; : : : ; and k = 1; : : : ; n n EXk:n () = P Xi ()¿n − k + 1 : i=1
Therefore,
n n EX d () k:n Xi () = n − k + 1 − P Xi ()¿n − k + 1 2 = (n − k + 1)P d =
n−k+1
i=1
i=1
(n − k + 1)
n n−k +1
(1 − )
k−1
−
n n r−n+k−1 n−r : (1 − ) r
r=n−k+1
n If n¿3 and k = 2; : : : ; n − 1, the polynomial in brackets takes the value −1 or (n − k)( n−k+1 ) according to = 1 or = 0. By continuity, this entails that the derivative of EXk:n ()= is negative (resp., positive) for close to 1 (resp., for close to 0). Hence, the function EXk:n ()= is not monotone on (0; 1).
318
J. de la Cal, A.M. Valle / Statistics & Probability Letters 52 (2001) 313 – 319
Appendix Here, we provide a proof of (3). Let X () be a random variable having distribution p() ( ∈ (0; )). First of all, we observe that d P(X ()¿1) = P(X () = 1) − P(X ()¿1) 6 0; d so that the function P(X ()¿1)= is nonincreasing, and, therefore, it has a limit l as ↓ 0. By Proposition 1(b), we have that l 6 c. Proposition 1(b) also guarantees that we can deJne by continuity P(X (0)¿1) := 0. From the mean value theorem, we conclude that 2
l = lim ↓0
P(X ()¿1) d p1 () = lim P(X ()¿1) = lim ; ↓0 ↓0 d
implying that P(X ()¿1) = p1 () + o() ( ↓ 0); as well as P(X ()¿k) = o() ( ↓ 0);
k = 2; 3; : : : :
(5)
Thus, to achieve (3), we only need to show that h() :=
∞
P(X ()¿k) = o() ( ↓ 0):
(6)
k=2
Let " ¿ 0, and Jx 0 ∈ (0; ) for a while. Since ∞ k=2 P(X (0 )¿k) 6 c; 0 we can Jnd an integer k0 ¿2 such that ∞ k=k0 +1 P(X (0 )¿k) 6 ": 0 ∞ The function k=k0 +1 P(X ()¿k)= is nondecreasing on (0; ), as it can be shown by di9erentiation. Therefore, we can write k0 h() P(X ()¿k) 6 + ";
0 ¡ 6 0
k=2
and, from (5), we conclude that lim sup ↓0
h() 6 ":
This shows (6), because " ¿ 0 is arbitrary, and completes the proof of (3). References Arnold, B.C., 1988. Bounds on the expected maximum. Comm. Statist. Theory Methods 17, 2135–2150. Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1992. A First Course in Order Statistics. Wiley, New York. Balakrishnan, N., 1990. Improving the Hartley–David–Gumbel bound for the mean of extreme order statistics. Statist. Probab. Lett. 9, 291–294.
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David, H.A., 1981. Order Statistics, 2nd Edition. Wiley, New York. de la Cal, J., Valle, A.M., 2000. Best constants for tensor products of Bernstein, Sz!asz and Baskakov operators. Bull. Austral. Math. Soc. 62, 211–220. Downey, P.J., 1990. Distribution-free bounds on the expectation of the maximum with scheduling applications. Oper. Res. Lett. 9, 189–201. Huang, J.S., 1997. Sharp bounds for the expected value of order statistics. Statist. Probab. Lett. 33, 105–107. Johnson, N.L., Kotz, S., Kemp, A.W., 1992. Univariate Discrete Distributions, 2nd Edition. Wiley, New York. Shaked, M., Shantikhumar, J.G., 1994. Stochastic Orders and Their Applications. Academic Press, Boston. Sprott, D.A., 1965. A class of contagious distributions and maximum likelihood estimation. In: Patil, G.P. (Ed.), Classical and Contagious Discrete Distributions. Statistical Publishing Society, Calcutta; Pergamon Press, Oxford, pp. 337–350.