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Two characterizations based on order statistics and records
Journal of Statistical Planning and Inference 124 (2004) 273 – 287
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Two characterizations based on order statistics and records N. Balakrishnana;∗ , A. Stepanovb a Department
of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1 b Department of Mathematics, Kaliningrad State Technical University, Sovietsky Prospect 1, Kaliningrad 236000, Russia Received 27 January 2002; accepted 19 July 2003
Abstract Two characterizations are established in this paper. The 1rst one is a characterization of the uniform distribution based on order statistics. The second characterization makes use of the conditional distribution of the second record value, given the maximum, for characterizing the distributions of exponential type. c 2003 Elsevier B.V. All rights reserved. MSC: 62E10; 62E20 Keywords: Characterizations; Uniform distribution; Order statistics; Records; Conditional distribution; Stirling numbers; Abelian equation
1. Introduction Let X1 ; X2 ; : : : be a sequence of independent random variables with a common distribution function F with support I = [a; b], −∞ 6 a ¡ b 6 ∞, where F will be assumed to be absolutely continuous in Section 2 and to be continuous in Section 3 of this paper. Let X1; n 6 X2; n 6 · · · 6 Xn; n be the order statistics produced by the sample X1 ; X2 ; : : : ; Xn (n ¿ 1), and Mn be the maximum Xn; n (n ¿ 1). The sequences of record times L(n) and record values X (n) are de1ned as follows: L(1) = 1;
X (1) = X1 ;
L(n + 1) = min{j : j ¿ L(n); Xj ¿ XL(n) }; ∗
X (n) = XL(n)
(n ¿ 1):
Corresponding author. Tel.: +1-905-5259140; fax: +1-905-5221676. E-mail addresses: [email protected] (N. Balakrishnan), [email protected] (A. Stepanov).
c 2003 Elsevier B.V. All rights reserved. 0378-3758/$ - see front matter doi:10.1016/j.jspi.2003.07.001
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There are many characterizations of F based on the properties of order statistics and record values. The papers of Ferguson (1967), Kaminsky and Nelson (1975), Huang (1975, 1989), Szekely and Mori (1985), Nagaraja (1984, 1988a, b), and Nagaraja and Nevzorov (1997) contain characterizations based on order statistics. The papers of Ahsanullah (1979, 1981), Ahsanullah and Holland (1984), Kirmani and Beg (1984), Gupta (1984), Nagaraja (1977, 1988b), Stepanov (1990, 1994), Nevzorov (1992), Balakrishnan and Balasubramanian (1995), and Nagaraja and Nevzorov (1997) as well as many others concentrated on characterizations based on records. There are also comprehensive treatments on these two topics in the books of Galambos and Kotz (1978), David (1981), Galambos (1984), Rao and Shanbhag (1993), Ahsanullah (1995), Arnold et al. (1992, 1998), and Nevzorov (2000). In this paper, we 1rst present some of these results as we will refer to them in subsequent sections. Usage of linear regression between two adjacent order statistics or record values is one of the traditional ways of producing characterizations of distributions. The following theorem was proved by Ferguson (1967). Theorem 1.1. The equality E{Xi; n | Xi+1; n = x} = x +
a:s:
(1)
holds for some real and i: the initial continuous F is one of the three following types (except for change of location and scale): F1 (x) = x
(0 ¡ x ¡ 1; 0 ¡ ¡ 1);
F2 (x) = ex
(x ¡ 0; = 1);
F3 (x) = (−x)
(x ¡ − 1; ¿ 1);
where = ={n(1 − )}. As an intermediate result, he had to show that the only possible solution F of the equation (i ¿ 1) x u dF i (u) a = x + ; x ∈ I; (2) F i (x) amongst continuous distribution functions is F1 , F2 or F3 . An obvious dual to the above result (see Ferguson (1967) or Nagaraja (1988b)) is the following result. Theorem 1.2. The equality E{Xi+1; n | Xi; n = x} = x +
a:s:
(3)
holds for some real and i: the initial continuous F is one of the three following types: F4 (x) = 1 − (−x)
(−1 ¡ x ¡ 0; 0 ¡ ¡ 1);
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F5 (x) = 1 − e−x F6 (x) = 1 − x
275
(x ¿ 0; = 1); (x ¿ 1; ¿ 1);
where = ={(n − i)(1 − )}. As an intermediate result, it was also shown that the only possible continuous solution F of the equation (except for change of location and scale) b u d(1 − F(u))n−i = x + ; x ∈ I; (4) − x (1 − F(x))n−i (n − i ¿ 1) is F4 , F5 or F6 . Combining these results, Nagaraja (1988b) produced the following theorem. Theorem 1.3. Equalities (1) and (3) both hold true for some i (1 6 i ¡ n) i: the initial continuous distribution function F is uniform. Remark 1. The functions Fi (1 6 i 6 6) from Theorems 1.1–1.3 are, in fact, in1nitely continuous diJerentiable on their supports, even though they were sought amongst continuous functions. Results similar to Theorem 1.3 hold true for adjacent record values as well; see Nagaraja (1977, 1988b). Theorem 1.4. The equalities E{X (n + 1) | X (n) = x} = x +
a:s:;
E{X (n) | X (n + 1) = x} = x +
a:s:
both hold true for some real , , , and n ¿ 1 i: the initial continuous F is exponential. Recent work of Nagaraja and Nevzorov (1997) showed that the regression function E{X1 | Xi; n } uniquely identi1es the initial continuous F. Theorem 1.5. Let X1 ; X2 ; : : : ; Xn and Y1 ; Y2 ; : : : ; Yn be two sequences of random variables with continuous distribution functions F and G, respectively, concentrated on the same support I = [a; b], −∞ 6 a ¡ b 6 ∞. Assume that EX1 exists and that there exists a point x0 such that for some (0 ¡ ¡ 1) the following conditions are satis=ed: F(x0 ) = G(x0 ) = ;
(5)
E{X1 | X1 6 x0 } = E{Y1 | Y1 6 x0 }:
(6)
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Then, F = G i: the equality b x b x i − 1 a u dF(u) n − i x u dF(u) i − 1 a u dG(u) n − i x u dG(u) = + + n n F(x) n 1 − F(x) n G(x) 1 − G(x) holds for every x ∈ (a; b). They also conjectured that conditions (5) and (6) can possibly be omitted or at least weakened. Another result of Nagaraja and Nevzorov (1997) is as follows. Theorem 1.6. Let the random variables X (2) − M2 and M2 be independent. Then, the initial continuous distribution F belongs to the exponential type of distributions. Motivated by Theorems 1.5 and 1.6, we will prove in this paper the following results: For n ¿ 2, let the equality E{X1 | Xi; n = x} = n (i)x + n (i)
a:s:
hold for some real n (i) ¿ 0 and n (i) at least for two di:erent values of i (1 6 i 6 n). Then, the initial absolutely continuous distribution F is uniform. Random variables min{0; X (2)−Mn } and Mn are independent for some n ¿ 3 i: the initial continuous distribution F coincides (up to the location and scale parameters) with F2 . Random variables max{0; X (2)−Mn } and Mn are independent for some n ¿ 1 i: the initial continuous distribution F coincides (up to the location and scale parameters) with F5 . Random variables X (2) − Mn and Mn are independent for some n = 1; 2; : : : i: the initial continuous distribution F coincides (up to the location and scale parameters) with F5 and n = 1 or n = 2. Some other results regarding the conditional distribution of X (k), given Mn , are established in Section 3 of this paper. 2. Characterization based on order statistics In this section, let F be an absolutely continuous distribution, and let EX1 exist which we shall denote by E. Before we go to the main characterization result of this section, we shall prove the following lemma. Lemma 2.1. If E{X1 | Xi; n = x} = Mi; n (x)
a:s:;
(7)
where Mi; n (x) (n ¿ 2; i ∈ {1; 2; : : : ; n}) is a nondecreasing function on [a; b], then the initial absolutely continuous distribution F is uniquely determined.
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277
Proof. In fact, when i = 1 or n, Lemma 2.1 has been proved by Kotlarski (1972); see also Nagaraja (1977) and Kotz and Shanbhag (1980). Let us now consider other values of i (1 ¡ i ¡ n; n ¿ 3). The conditional distribution of X1 , when Xi; n is 1xed, was given by Nagaraja and Nevzorov (1997) as i − 1 F(u) u ¡ x; n F(x) ; (8) P{X1 6 u | Xi; n = x} = i n − i F(u) − F(x) + ; u ¿ x: n n 1 − F(x) The distribution has the atom 1=n at the point x. In view of (8), one possible choice for E{X1 | Xi; n = x} is x b x i − 1 a u dF(u) n − i x u dF(u) Ti; n (x) = + + : (9) n n F(x) n 1 − F(x) De1ne for i (1 ¡ i ¡ n; n ¿ 3) i−1 ; 0 ¡ pi = 1 − qi ¡ 1; n−1 x n x Ti; n (x) − : u dF(u); Ri; n (x) = I (x) = n−1 n a pi =
Equality (9) can then be rewritten as Ri; n = pi
I (x) E − I (x) + qi : F(x) 1 − F(x)
(10)
The functions I (x)=F(x) and (E − I (x))=(1 − F(x)) have been studied by Ferguson (1967), Kotlarski (1972), and Nagaraja (1975, 1977). If F is absolutely continuous, these two functions are both absolutely continuous and nondecreasing on (a; b). This results in both Ti; n and Ri; n being absolutely continuous and nondecreasing. It is also known that lim
x→a
I (x) = a; F(x)
lim
x→b
E − I (x) = b; 1 − F(x)
E − I (x) I (x) ¡x¡ ; F(x) 1 − F(x) a¡
I (x) ¡ E: F(x)
(11)
The last two inequalities are valid for any x ∈ (a; b). Let every x from the interval [a1 ; b1 ] (a ¡ a1 6 b1 ¡ b) be the pi th quantile of F, i.e. F(x) = pi ;
x ∈ [a1 ; b1 ]:
We 1nd from (10) that Ri; n (x) = E for x ∈ [a1 ; b1 ]. I (x) is constant for this interval I (x) = I (a1 )
(12)
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for any x ∈ [a1 ; b1 ]. Eq. (10) can be changed into a 1rst degree diJerential equation. Indeed, it follows from (10) that E − Ri; n (x) I (x) = Ri; n (x)F(x) + qi (E − Ri; n (x)) + qi pi (13) F(x) − pi for x ∈ (a; a1 ) ∪ (b1 ; b), where E − Ri; n (x) E − Ri; n (x) I (a1 ) − pi E lim = lim = = !: x→a1 −0 F(x) − pi x→b1 +0 F(x) − pi pi q i For [a1 ; b1 ] the integral I (x) is de1ned by (12). The inequality a−E ¡!¡0 qi that presents the bounds for ! is valid due to (11). Upon diJerentiating both sides of (13), we obtain Ri; n (x)F(x)(1 − F(x))(F(x) − pi ) F (x) = (14) (F(x) − pi )2 (Ri; n (x) − x) + qi pi (Ri; n (x) − E) for x ∈ (a; a1 ) ∪ (b1 ; b). For those x that belong to (a1 ; b1 ), F (x) = 0:
(14 )
Eq. (14) is an Abelian equation and the solution can be found directly only in some special cases. In accordance with Cauchy’s theorem, the existence of the solution and its uniqueness is guaranteed everywhere on (a; a1 ) ∪ (b1 ; b). In the interval [a1 ; b1 ], we have one more singular solution F(x) = pi (a ¡ x ¡ b). Taking a particular solution from the general one as a distribution function, we uniquely determine F. It is true that Mi; n (x) = Ti; n (x)
a:s: (n ¿ 3; 1 ¡ i ¡ n);
where Mi; n and Ti; n are both nondecreasing. Following the lines of the proof of Nagaraja and Nevzorov (1997), we conclude that Mi; n (x) = Ti; n (x) for any x ∈ [a; b]. It then follows that Mi; n (x) uniquely determines F since Ti; n (x) (through Ri; n (x)) does. This completes the proof of the lemma. Remark 2. Equality (9) in our approach is viewed as a diJerential equation. Conditions (5) and (6) of Theorem 1.5, in fact, determine the coeMcient E of the equation. Although we additionally assumed that F is absolutely continuous, neither one of the two conditions (5) and (6) can be omitted. Of course, some equivalent substitutions are possible. Remark 3. Since Ti; n (x) in Lemma 2.1 is absolutely continuous, Mi; n (x) is also absolutely continuous. In the following theorem, we use Lemma 2.1 to establish a characterization of the uniform distribution.
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279
Theorem 2.2. For n ¿ 2, let the equality E{X1 | Xi; n = x} = n (i)x + n (i)
a:s:
(15)
hold for some real n (i) ¿ 0 and n (i) for at least for two di:erent values of i (1 6 i 6 n). Then, the initial absolutely continuous distribution F is uniform. Proof. First, with regard to the problem of identi1cation of F and answering the question raised by Nagaraja and Nevzorov (1997) (see Remark 3 of their paper) whether conditions (5) and (6) of Theorem 1.5 can be removed, Lemma 2.1 readily reveals that conditions (5) and (6) cannot be omitted. Next, let Mi; n (x) = n (i)x + n (i), where n (i) ¿ 0 and n (i) (n (i) is positive for Mi; n (x) to be nondecreasing) are some real numbers. The direct solution of the Abelian equation in (14) is complicated. So, let us impose some additional restrictions, namely that equality (14) holds for two diJerent values of i (n ¿ 2; 1 6 i 6 n). In view of Lemma 2.1, it implies for all x ∈ (a; b) pi
E − I (x) I (x) + qi = Ri; n (x); F(x) 1 − F(x)
pj
I (x) E − I (x) + qj = Rj; n (x) F(x) 1 − F(x)
(1 6 i ¡ j 6 n). The functions Ri; n (x) and Rj; n (x) are linear, and the numbers pi ; pj ; qi ; qj are all 1xed. Consequently, I (x)=F(x) and (E − I (x))=(1 − F(x)) are also linear. Now, using the same arguments as those for (2) and (4), we conclude that F is simultaneously of F1 and F4 types. Matching their supports, we 1nd 0 ¡ x ¡ 1 and = = 1 for F. Hence, the theorem.
3. Characterization based on the second record value and maximum In this section, let F be a continuous distribution function. For n ¿ 1, let us denote Sn = X (2) − Mn . Independence of the random variables S2 and M2 characterized the distribution of the exponential type in Theorem 1.6. Now, let us examine this for other values of n. For this purpose, we 1rst establish the following lemma. Lemma 3.1. For any x; y in I, we have P{X (2) 6 y | Mn = x} 0;
l n−1 F(y) 1 n−l ; = n (l − 1)l F(x) l=2 n − 1 1 F(y) − F(x) ; + n n 1 − F(x)
n = 1; 2; y ¡ x;
(A)
n ¿ 3; y ¡ x;
(AA)
n ¿ 1; y ¿ x:
(AAA)
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Proof. Assume 1rst that F(x) is absolutely continuous and p(x) = F (x) for any x ∈ I. Proof of (A). The result is obvious. Indeed, the event {X (2) 6 y; x ¡ Mn 6 x + d x} cannot occur for n = 1; 2 and any y ¡ x. Hence, P{X (2) 6 y | Mn = x} = 0
(n = 1; 2; y ¡ x ∈ I):
Proof of (AA). Let us de1ne the following events: U1 (y) = {X1 ¡ y};
V (i2 ) =
i 2 −1
{Xi 6 X1 };
i=2
U (i2 ; y) = {X1 ¡ Xi2 6 y};
V (i2 ; t; x) =
t−1
{Xi ¡ x};
i=i2 +1
U (t; x) = {Xt ∈ (x; x + d x]};
V (t; x) =
n
{Xi 6 x + d x}:
i=t+1
We can write for y ¡ x when p(x) ¿ 0 P{X (2) 6 y; x ¡ Mn 6 x + d x} =
n t−1
P{U1 (y)V (i2 )U (i2 ; y)V (i2 ; t; x)U (t; x)V (t; x)} + o(d x):
(16)
t=3 i2 =2
Those Xi (1 6 i 6 n) that were registered in (x; x + d x] are to produce the largest value in the sample X1 ; : : : ; Xn . The time when the 1rst such Xi was recorded is denoted as t (3 6 t 6 n). The time when the second record value was registered is denoted as i2 (2 6 i2 ¡ t). Since the second record value does not exceed Mn , it must appear earlier than Xt . The events U1 (y) and U (i2 ; y) reNect the appearance of the 1rst and second record values, respectively. The event U (t; x) describes the appearance of the 1rst Xi that was registered in (x; x +d x]. The event V (i2 ) consists of some trials that do not exceed the 1rst record value (this set could be empty). The event V (i2 ; t; x) formed with the trials Xi2 +1 ; Xi2 +2 ; : : : ; Xt−1 could contain some record values and could be empty. The event V (t; x) consists of the trials Xt+1 ; Xt+2 ; : : : ; Xn and could be empty. The product of the events U1 (y)V (i2 )U (i2 ; y)V (i2 ; t; x)U (t; x)V (t; x) tells us that there are at least two record values belonging to (−∞; y] and Mn ∈ (x; x + d x]. It follows from (16) that for those x for which p(x) ¿ 0 P{X (2) 6 y; x 6 Mn ¡ x + d x} =
n t−1 t=3 i2 =2
y
−∞
(F(y) − F(u))F i2 −2 (u)p(u) du(p(x) d x
N. Balakrishnan, A. Stepanov / Journal of Statistical Planning and Inference 124 (2004) 273 – 287
+ o(d x))F
n−i2 −1
= (p(x) + o(1))F
281
(x + d x)
n−1
(x + d x) d x
n−1 n i2 =2 t=i2 +1
= (p(x) + o(1))F n−1 (x + d x) d x
n−1 i2 =2
1 (i2 − 1)i2
n − i2 (i2 − 1)i2
F(y) F(x + d x)
F(y) F(x + d x)
i2
i2
:
The joint density-distribution function G(y; x) of the vector (X (2); Mn ) can be obtained from the last formula as
l n−1 F(y) n−l G(y; x) = p(x)F n−1 (x) ; (l − 1)l F(x) l=2
where y 6 x ∈ I, which when used with the density of Mn given by nF n−1 (x)p(x), readily yields (AA). Proof of (AAA). First of all, for any x 6 y ∈ I, we have P{X (k) 6 y | Mn = x} n P{N (n) = i; )(x; y) ¿ k − i}; i=1 = k−1 P{N (n) = i; )(x; y) ¿ k − i}; P{N (n) ¿ k} +
(n 6 k)
(B)
(n ¿ k):
(BB)
i=1
Here, N (n) denotes the total number of record values amongst X1 ; X2 ; : : : ; Xn , and )(x; y) denotes the total number of record values in (x; y). It is also known from the works of Shorrock (1972) and Renyi (1976) that P{N (n) = i} =
|Sni | ; n!
where Sni are Stirling numbers de1ned by x(x − 1) · · · (x − (n − 1)) =
∞
Sni xi :
i=0
= 0 and Sn1 = (−1)n−1 (n − 1)! In particular, we have Random variables ), taken over diJerent intervals, reNect the number of record values falling in those intervals. They are independent for diJerent nonoverlapping intervals (see Shorrock, 1972) and that Sn0
P{)(x; y) = i} =
e−* *i i!
(i ¿ 0);
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where
1 − F(y) * = −log 1 − F(x)
:
(17)
Expression (AAA) needs to be considered for two cases, viz. when (a) n ¿ k = 2, and (b) k = 2; n = 1; 2. Case (a): Here, we have P{X (2) 6 y | Mn = x} = P{N (n) ¿ 2} + P{N (n) = 1; )(x; y) ¿ 1} = 1 − P{N (n) = 1} + P{N (n) = 1}P{)(x; y) ¿ 1}
(n ¿ 3);
(18)
where the last product is due to the fact that the variables N and ) are independent. Indeed, the number of record values N (n) amongst X1 ; X2 ; : : : ; Xn is, in fact, the number of record values registered in the interval (−∞; x], because (x ¡ Mn ¡ x+d x). According to Shorrock’s result, the number of records registered in (−∞; x] and the number of records in (x; y] are independent variables. We, therefore, 1nd from (18) that P{X (2) 6 y | Mn = x} = 1 −
1 1 − P{)(x; y) = 0} + : n n
Case (b): For proving this part, we use (B) to write P{X (2) 6 y | Mn = x} =
P{)(x; y) ¿ 1};
(n = 1; k = 2); y ¿ x;
P{N (2) = 1; )(x; y) ¿ 1} + P{N (2) = 2; )(x; y) ¿ 0};
(n = k = 2); y ¿ x:
This readily yields P{X (2) 6 y | Mn = x}
(F(y) − F(x))=(1 − F(x)); = (F(y) − F(x))=(2(1 − F(x))) + 1=2;
(n = 1; k = 2); y ¿ x; (n = k = 2); y ¿ x:
The last equality completes the proof of Lemma 3.1. Remark 4. We assumed at the beginning of the proof of Lemma 3.1 that F(x) is absolutely continuous for x ∈ I. The results holds true even if F is continuous. In the following theorem, we use Lemma 3.1 to establish some additional characterization results. Theorem 3.2. 1. Random variables Sn− = min{0; Sn } and Mn are independent for some n ¿ 3 i: the initial continuous distribution F coincides (up to the location and scale parameters) with F2 . 2. Random variables Sn+ =max{0; Sn } and Mn are independent for some n ¿ 1 i: the initial continuous distribution F coincides (up to the location and scale parameters) with F5 .
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283
3. Random variables Sn and Mn are independent for some n = 1; 2; : : : i: the initial continuous distribution F coincides (up to the location and scale parameters) with F5 and n = 1 or 2. Proof of Part 1 of Theorem 3.2. SuMciency can be veri1ed directly. Let us now prove necessity. It is easily seen that the unconditional distribution of Sn− is concentrated on the interval (−∞; 0]. The conditional distribution of Sn− can be derived from Lemma 3.1 as
l n−1 F(x + y) n−l 1 ; (y ¡ 0) (K) F(x) P{Sn− 6 y | Mn = x} = n l=2 (l − 1)l 1; (y ¿ 0); (KK) n−2 where n ¿ 3, x ∈ I. This conditional distribution has an atom 1=(n−1)+(1=n) l=1 1=l at the point 0. Independence of Sn− and Mn implies that (K) does not depend on x, i.e. l
n−1 F(y + x) cl = H (y); y ¡ 0; x ∈ I; (19) F(x) l=2
where cl = (n − l)={nl(l − 1)} ¿ 0 and H (y) is some continuous function. Actually, H is the continuous part of the conditional distribution function of Sn− . We additionally n−1 de1ne H (0) = l=2 cl = c (0 ¡ c ¡ 1). Now H (x) is continuous on (−∞; 0] with the following properties: H (0) = c ¿ 0, H (−∞) = 0:
(20)
Some point x0 exists such that H (x) ¿ 0
(21)
for all x (x0 ¡ x 6 0). The following proposition is needed for proving the required result. Proposition 1. Let some initial continuous distribution function F satisfy (19) for any y 6 0 and every x ∈ I = [a; b]. Then, a = −∞ and b ¡ + ∞. Proof of Proposition 1. Let us 1rst prove the second statement b ¡ + ∞. Suppose this is not true: F(x) ¡ 1 for any x. Equality (19) can be written with x = −2y ¿ 0 (x ∈ I) as l
n−1 F(−y) cl = H (y): F(−2y) l=2
From here, we get lim H (y) = c ¿ 0:
y→−∞
(22)
This is a direct contradiction to (20). So, our supposition F(x) ¡ 1 (for any x) was not correct. Assuming b ¡ + ∞, we can additionally put (without loss of generality) b = 0.
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Remark 5. Of course, if b ¡ + ∞ and x = −2y ¿ 0 is taken as large for (19), we get the same result (22). However now, we are wrong in applying (19), because x does not belong to I. Let us now prove a = −∞. Suppose the opposite statement is true, viz. a number a exists (−∞ ¡ a ¡ 0) such that F(x) = 0 for any x 6 a and F(x) ¿ 0 for a ¡ x ¡ 0. Taking y = a=m, x = a − a=m (m ¿ 2), we 1nd F(a) F(y + x) = = 0: F(x) F(a − a=m) Equality (19) then gives H (a=m) = 0 for any m ¿ 2. This entails H (y) = 0 for any y ¡ 0, which contradicts (21). This completes the proof of Proposition 1. Let us now return to proving Part 1 of Theorem 3.2. Putting x = 0 and y in equality (19), we obtain n−1
cl F l (y) = H (y);
l=2 n−1
cl
l=2
F(2y) F(y)
l = H (y):
The diJerence of these two equalities gives n−1 l=2
cl
F 2l (y) − F l (2l) =0 F l (y)
(n ¿ 3):
All cl are positive. This entails F 2 (y) = F(2y) for any y 6 0. The last equality can be extended in a similar way to F k (y) = F(ky)
(y 6 0; k ¿ 2):
(23)
The only possible solution of (23) in the class of continuous distribution functions having the support (−∞; 0] is F2 . This completes the proof of Part 1 of Theorem 3.2. Proof of Part 2 of Theorem 3.2. SuMciency of Part 2 of Theorem 3.2 can be checked directly. Let us now establish the necessity. It is obvious that the support of the distribution of Sn+ is [0; ∞). The conditional distribution of Sn+ can be obtained from Lemma 3.1 as (y ¡ 0); 0; + P{Sn 6 y | Mn = x} = n − 1 F(y + x) − F(x) + ; (y ¿ 0); n n(1 − F(x))
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285
where x ∈ I, n ¿ 1. The independence of Sn+ and Mn presumes n − 1 F(x + y) − F(x) + = G(y); (24) n n(1 − F(x)) where G is some continuous function on [0; ∞). Using the substitution G1 (y) = n(G(y) − (n − 1)=n), we can rewrite equality (24) as 1 − F(x + y) (25) = G1 (y) (y ¿ 0; x ∈ I; n ¿ 1): 1 − F(x) This is the Cauchy functional equation. The only possible continuous solution of this equation can be (up to the parameters of location and scale) F5 . Here, we would not like to repeat the known methods of establishing this fact. One of them can be based on the same principle as the method of proving the necessity of Part 1 of Theorem 3.2. We only mention here one important result that we would have to prove if we choose to provide a full-scale proof of Part 2 of Theorem 3.2. Proposition 2. Let some initial continuous distribution function satisfy Eq. (25) for any y ¿ 0 and x ∈ I. Then, its support I = [a; b] is such that a ¿ − ∞ and b = +∞. Note here again that a can be chosen as 0. This completes the proof of Part 2 of Theorem 3.2. Proof of Part 3 of Theorem 3.2. SuMciency can be easily veri1ed for F5 (n = 1; 2). Necessity for n = 1; 2 has already been guaranteed by Theorem 3.2. Indeed, for n = 1; 2 we have Sn+ = Sn a.s. Let us now establish the necessity for n ¿ 3. In fact, we have to prove that there is no distribution that provides the independence of Sn and Mn for any n ¿ 3. Parts (AA) and (AAA) of Lemma 3.1 can be written as (n ¿ 2) P{Sn 6 y | Mn = x} n−1 cl (F(y + x)=F(x))l ; = l=2 (n − 1)=n + (F(x + y) − F(x))=(n(1 − F(x)));
(y ¡ 0)
(AA )
(y ¿ 0):
(AAA )
The independence of Sn and Mn (n ¿ 2) implies that both parts (AA ) and (AAA ) do not depend on x. This, in turn, suggests that Propositions 1 and 2 are valid simultaneously for some continuous distribution function F. This is impossible, because no support of a distribution function can meet the demands of Propositions 1 and 2 at the same time. This then completes the proof of Part 3 of Theorem 3.2. Remark 6. Note here that the conditional distribution function P{X (2) 6 y | Mn = x} n−1 has an atom (1=n) i=1 1=i (n ¿ 2) at the point x. Remark 7. The conditional distribution of X (k), given Mn , can also be derived for arbitrary k and n. However, the corresponding formula is quite cumbersome and therefore may not be suitable for characterizations.
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Acknowledgements We are sincerely grateful to a referee who found an error in Theorem 3.2. The present version of the statement of Theorem 3.2 was oJered by this referee. References Ahsanullah, M., 1979. Characterization of the exponential distribution by record values. SankhyPa, Ser. B 41, 116–121. Ahsanullah, M., 1981. On a characterization of the exponential distribution by weak homoscedasticity of record values. Biometrical J. 23, 715–717. Ahsanullah, M., 1995. Record Statistics. Nova Science Publishers, Commack, New York. Ahsanullah, M., Holland, B., 1984. Record values and the geometric distribution. Statist. Hefte 25, 319–327. Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1992. A First Course in Order Statistics. Wiley, New York. Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1998. Records. Wiley, New York. Balakrishnan, N., Balasubramanian, K., 1995. A characterization of geometric distribution based on record values. J. Appl. Statist. Sci. 2, 233–248. David, H.A., 1981. Order Statistics, 2nd Edition. Wiley, New York. Ferguson, T.S., 1967. On characterizing distributions by properties of order statistics. SankhyPa, Ser. A 29, 265–278. Galambos, J., 1984. The Asymptotic Theory of Extreme Order Statistics, 2nd Edition. Krieger, Malabar, FL. Galambos, J., Kotz, S., 1978. Characterizations of Probability Distribution. In: Lecture Notes in Mathematics, Vol. 657. Springer, Berlin. Gupta, R.C., 1984. Relations between order statistics and record values and some characterization results. J. Appl. Probab. 21, 425–430. Huang, J.S., 1975. Characterization of distributions by the expected values of order statistics. Ann. Inst. Statist. Math. 27, 87–93. Huang, J.S., 1989. Moment problem of order statistics: a review. Internat. Statist. Rev. 57, 57–66. Kaminsky, K.S., Nelson, P.I., 1975. Best linear unbiased predictor of order statistics in location and scale families. J. Amer. Statist. Assoc. 70, 145–150. Kirmani, S.N.U.A., Beg, M.I., 1984. On characterization of distributions by expected records. SankhyPa, Ser. A 46, 463–465. Kotlarski, I., 1972. On a characterization of some distributions by conditional expectation. SankhyPa, Ser. A 34, 461–467. Kotz, S., Shanbhag, D.N., 1980. Some new approaches to probability distributions. Adv. Appl. Probab. 12, 903–921. Nagaraja, H.N., 1977. On a characterization based on record values. Austral. J. Statist. 19, 70–73. Nagaraja, H.N., 1984. Asymptotic linear prediction of extreme order statistics. Ann. Inst. Statist. Math. 36, 289–299. Nagaraja, H.N., 1988a. Record values and related statistics—a review. Comm. Statist.—Theory Methods 17, 2223–2238. Nagaraja, H.N., 1988b. Some characterizations of continuous distributions based on regression of adjacent order statistics and record values. SankhyPa, Ser. A 50, 70–73. Nagaraja, H.N., Nevzorov, V.B., 1997. On characterizations based on record values and order statistics. J. Statist. Plann. Inference 63, 271–284. Nevzorov, V.B., 1992. A characterization of exponential distributions by correlations between records. Math. Methods in Statist. 1, 49–54. Nevzorov, V.B., 2000. Records: Mathematical Theory. In: Translation of Mathematical Monographs, Vol. 194. American Mathematical Society, Providence, RI. Rao, C.R., Shanbhag, D.N., 1993. Choquet–Deny Type Functional Equations with Applications to Stochastic Models. Wiley, Chichester, England.
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287
Renyi, A., 1976. On the extreme elements of observations, In: Selected Papers of Alfred Renyi, Vol. 3. Akademiai Kiado, Budapest, Hungary, pp. 50 – 65. Shorrock, R., 1972. A limit theorem for inter-record times. J. Appl. Probab. 9, 543–555. Stepanov, A., 1990. Characterization of a geometric class of distributions. Theory of Probab. Math. Statist. 41, 133–136. Stepanov, A., 1994. A characterization theorem for weak records. Theory of Probab. Appl. 38, 762–764. Szekely, G.J., Mori, T.F., 1985. An extremal property of rectangular distributions. Statist. Probab. Lett. 3, 107–109.
www.elsevier.com/locate/jspi
Two characterizations based on order statistics and records N. Balakrishnana;∗ , A. Stepanovb a Department
of Mathematics and Statistics, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4K1 b Department of Mathematics, Kaliningrad State Technical University, Sovietsky Prospect 1, Kaliningrad 236000, Russia Received 27 January 2002; accepted 19 July 2003
Abstract Two characterizations are established in this paper. The 1rst one is a characterization of the uniform distribution based on order statistics. The second characterization makes use of the conditional distribution of the second record value, given the maximum, for characterizing the distributions of exponential type. c 2003 Elsevier B.V. All rights reserved. MSC: 62E10; 62E20 Keywords: Characterizations; Uniform distribution; Order statistics; Records; Conditional distribution; Stirling numbers; Abelian equation
1. Introduction Let X1 ; X2 ; : : : be a sequence of independent random variables with a common distribution function F with support I = [a; b], −∞ 6 a ¡ b 6 ∞, where F will be assumed to be absolutely continuous in Section 2 and to be continuous in Section 3 of this paper. Let X1; n 6 X2; n 6 · · · 6 Xn; n be the order statistics produced by the sample X1 ; X2 ; : : : ; Xn (n ¿ 1), and Mn be the maximum Xn; n (n ¿ 1). The sequences of record times L(n) and record values X (n) are de1ned as follows: L(1) = 1;
X (1) = X1 ;
L(n + 1) = min{j : j ¿ L(n); Xj ¿ XL(n) }; ∗
X (n) = XL(n)
(n ¿ 1):
Corresponding author. Tel.: +1-905-5259140; fax: +1-905-5221676. E-mail addresses: [email protected] (N. Balakrishnan), [email protected] (A. Stepanov).
c 2003 Elsevier B.V. All rights reserved. 0378-3758/$ - see front matter doi:10.1016/j.jspi.2003.07.001
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There are many characterizations of F based on the properties of order statistics and record values. The papers of Ferguson (1967), Kaminsky and Nelson (1975), Huang (1975, 1989), Szekely and Mori (1985), Nagaraja (1984, 1988a, b), and Nagaraja and Nevzorov (1997) contain characterizations based on order statistics. The papers of Ahsanullah (1979, 1981), Ahsanullah and Holland (1984), Kirmani and Beg (1984), Gupta (1984), Nagaraja (1977, 1988b), Stepanov (1990, 1994), Nevzorov (1992), Balakrishnan and Balasubramanian (1995), and Nagaraja and Nevzorov (1997) as well as many others concentrated on characterizations based on records. There are also comprehensive treatments on these two topics in the books of Galambos and Kotz (1978), David (1981), Galambos (1984), Rao and Shanbhag (1993), Ahsanullah (1995), Arnold et al. (1992, 1998), and Nevzorov (2000). In this paper, we 1rst present some of these results as we will refer to them in subsequent sections. Usage of linear regression between two adjacent order statistics or record values is one of the traditional ways of producing characterizations of distributions. The following theorem was proved by Ferguson (1967). Theorem 1.1. The equality E{Xi; n | Xi+1; n = x} = x +
a:s:
(1)
holds for some real and i: the initial continuous F is one of the three following types (except for change of location and scale): F1 (x) = x
(0 ¡ x ¡ 1; 0 ¡ ¡ 1);
F2 (x) = ex
(x ¡ 0; = 1);
F3 (x) = (−x)
(x ¡ − 1; ¿ 1);
where = ={n(1 − )}. As an intermediate result, he had to show that the only possible solution F of the equation (i ¿ 1) x u dF i (u) a = x + ; x ∈ I; (2) F i (x) amongst continuous distribution functions is F1 , F2 or F3 . An obvious dual to the above result (see Ferguson (1967) or Nagaraja (1988b)) is the following result. Theorem 1.2. The equality E{Xi+1; n | Xi; n = x} = x +
a:s:
(3)
holds for some real and i: the initial continuous F is one of the three following types: F4 (x) = 1 − (−x)
(−1 ¡ x ¡ 0; 0 ¡ ¡ 1);
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F5 (x) = 1 − e−x F6 (x) = 1 − x
275
(x ¿ 0; = 1); (x ¿ 1; ¿ 1);
where = ={(n − i)(1 − )}. As an intermediate result, it was also shown that the only possible continuous solution F of the equation (except for change of location and scale) b u d(1 − F(u))n−i = x + ; x ∈ I; (4) − x (1 − F(x))n−i (n − i ¿ 1) is F4 , F5 or F6 . Combining these results, Nagaraja (1988b) produced the following theorem. Theorem 1.3. Equalities (1) and (3) both hold true for some i (1 6 i ¡ n) i: the initial continuous distribution function F is uniform. Remark 1. The functions Fi (1 6 i 6 6) from Theorems 1.1–1.3 are, in fact, in1nitely continuous diJerentiable on their supports, even though they were sought amongst continuous functions. Results similar to Theorem 1.3 hold true for adjacent record values as well; see Nagaraja (1977, 1988b). Theorem 1.4. The equalities E{X (n + 1) | X (n) = x} = x +
a:s:;
E{X (n) | X (n + 1) = x} = x +
a:s:
both hold true for some real , , , and n ¿ 1 i: the initial continuous F is exponential. Recent work of Nagaraja and Nevzorov (1997) showed that the regression function E{X1 | Xi; n } uniquely identi1es the initial continuous F. Theorem 1.5. Let X1 ; X2 ; : : : ; Xn and Y1 ; Y2 ; : : : ; Yn be two sequences of random variables with continuous distribution functions F and G, respectively, concentrated on the same support I = [a; b], −∞ 6 a ¡ b 6 ∞. Assume that EX1 exists and that there exists a point x0 such that for some (0 ¡ ¡ 1) the following conditions are satis=ed: F(x0 ) = G(x0 ) = ;
(5)
E{X1 | X1 6 x0 } = E{Y1 | Y1 6 x0 }:
(6)
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Then, F = G i: the equality b x b x i − 1 a u dF(u) n − i x u dF(u) i − 1 a u dG(u) n − i x u dG(u) = + + n n F(x) n 1 − F(x) n G(x) 1 − G(x) holds for every x ∈ (a; b). They also conjectured that conditions (5) and (6) can possibly be omitted or at least weakened. Another result of Nagaraja and Nevzorov (1997) is as follows. Theorem 1.6. Let the random variables X (2) − M2 and M2 be independent. Then, the initial continuous distribution F belongs to the exponential type of distributions. Motivated by Theorems 1.5 and 1.6, we will prove in this paper the following results: For n ¿ 2, let the equality E{X1 | Xi; n = x} = n (i)x + n (i)
a:s:
hold for some real n (i) ¿ 0 and n (i) at least for two di:erent values of i (1 6 i 6 n). Then, the initial absolutely continuous distribution F is uniform. Random variables min{0; X (2)−Mn } and Mn are independent for some n ¿ 3 i: the initial continuous distribution F coincides (up to the location and scale parameters) with F2 . Random variables max{0; X (2)−Mn } and Mn are independent for some n ¿ 1 i: the initial continuous distribution F coincides (up to the location and scale parameters) with F5 . Random variables X (2) − Mn and Mn are independent for some n = 1; 2; : : : i: the initial continuous distribution F coincides (up to the location and scale parameters) with F5 and n = 1 or n = 2. Some other results regarding the conditional distribution of X (k), given Mn , are established in Section 3 of this paper. 2. Characterization based on order statistics In this section, let F be an absolutely continuous distribution, and let EX1 exist which we shall denote by E. Before we go to the main characterization result of this section, we shall prove the following lemma. Lemma 2.1. If E{X1 | Xi; n = x} = Mi; n (x)
a:s:;
(7)
where Mi; n (x) (n ¿ 2; i ∈ {1; 2; : : : ; n}) is a nondecreasing function on [a; b], then the initial absolutely continuous distribution F is uniquely determined.
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277
Proof. In fact, when i = 1 or n, Lemma 2.1 has been proved by Kotlarski (1972); see also Nagaraja (1977) and Kotz and Shanbhag (1980). Let us now consider other values of i (1 ¡ i ¡ n; n ¿ 3). The conditional distribution of X1 , when Xi; n is 1xed, was given by Nagaraja and Nevzorov (1997) as i − 1 F(u) u ¡ x; n F(x) ; (8) P{X1 6 u | Xi; n = x} = i n − i F(u) − F(x) + ; u ¿ x: n n 1 − F(x) The distribution has the atom 1=n at the point x. In view of (8), one possible choice for E{X1 | Xi; n = x} is x b x i − 1 a u dF(u) n − i x u dF(u) Ti; n (x) = + + : (9) n n F(x) n 1 − F(x) De1ne for i (1 ¡ i ¡ n; n ¿ 3) i−1 ; 0 ¡ pi = 1 − qi ¡ 1; n−1 x n x Ti; n (x) − : u dF(u); Ri; n (x) = I (x) = n−1 n a pi =
Equality (9) can then be rewritten as Ri; n = pi
I (x) E − I (x) + qi : F(x) 1 − F(x)
(10)
The functions I (x)=F(x) and (E − I (x))=(1 − F(x)) have been studied by Ferguson (1967), Kotlarski (1972), and Nagaraja (1975, 1977). If F is absolutely continuous, these two functions are both absolutely continuous and nondecreasing on (a; b). This results in both Ti; n and Ri; n being absolutely continuous and nondecreasing. It is also known that lim
x→a
I (x) = a; F(x)
lim
x→b
E − I (x) = b; 1 − F(x)
E − I (x) I (x) ¡x¡ ; F(x) 1 − F(x) a¡
I (x) ¡ E: F(x)
(11)
The last two inequalities are valid for any x ∈ (a; b). Let every x from the interval [a1 ; b1 ] (a ¡ a1 6 b1 ¡ b) be the pi th quantile of F, i.e. F(x) = pi ;
x ∈ [a1 ; b1 ]:
We 1nd from (10) that Ri; n (x) = E for x ∈ [a1 ; b1 ]. I (x) is constant for this interval I (x) = I (a1 )
(12)
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for any x ∈ [a1 ; b1 ]. Eq. (10) can be changed into a 1rst degree diJerential equation. Indeed, it follows from (10) that E − Ri; n (x) I (x) = Ri; n (x)F(x) + qi (E − Ri; n (x)) + qi pi (13) F(x) − pi for x ∈ (a; a1 ) ∪ (b1 ; b), where E − Ri; n (x) E − Ri; n (x) I (a1 ) − pi E lim = lim = = !: x→a1 −0 F(x) − pi x→b1 +0 F(x) − pi pi q i For [a1 ; b1 ] the integral I (x) is de1ned by (12). The inequality a−E ¡!¡0 qi that presents the bounds for ! is valid due to (11). Upon diJerentiating both sides of (13), we obtain Ri; n (x)F(x)(1 − F(x))(F(x) − pi ) F (x) = (14) (F(x) − pi )2 (Ri; n (x) − x) + qi pi (Ri; n (x) − E) for x ∈ (a; a1 ) ∪ (b1 ; b). For those x that belong to (a1 ; b1 ), F (x) = 0:
(14 )
Eq. (14) is an Abelian equation and the solution can be found directly only in some special cases. In accordance with Cauchy’s theorem, the existence of the solution and its uniqueness is guaranteed everywhere on (a; a1 ) ∪ (b1 ; b). In the interval [a1 ; b1 ], we have one more singular solution F(x) = pi (a ¡ x ¡ b). Taking a particular solution from the general one as a distribution function, we uniquely determine F. It is true that Mi; n (x) = Ti; n (x)
a:s: (n ¿ 3; 1 ¡ i ¡ n);
where Mi; n and Ti; n are both nondecreasing. Following the lines of the proof of Nagaraja and Nevzorov (1997), we conclude that Mi; n (x) = Ti; n (x) for any x ∈ [a; b]. It then follows that Mi; n (x) uniquely determines F since Ti; n (x) (through Ri; n (x)) does. This completes the proof of the lemma. Remark 2. Equality (9) in our approach is viewed as a diJerential equation. Conditions (5) and (6) of Theorem 1.5, in fact, determine the coeMcient E of the equation. Although we additionally assumed that F is absolutely continuous, neither one of the two conditions (5) and (6) can be omitted. Of course, some equivalent substitutions are possible. Remark 3. Since Ti; n (x) in Lemma 2.1 is absolutely continuous, Mi; n (x) is also absolutely continuous. In the following theorem, we use Lemma 2.1 to establish a characterization of the uniform distribution.
N. Balakrishnan, A. Stepanov / Journal of Statistical Planning and Inference 124 (2004) 273 – 287
279
Theorem 2.2. For n ¿ 2, let the equality E{X1 | Xi; n = x} = n (i)x + n (i)
a:s:
(15)
hold for some real n (i) ¿ 0 and n (i) for at least for two di:erent values of i (1 6 i 6 n). Then, the initial absolutely continuous distribution F is uniform. Proof. First, with regard to the problem of identi1cation of F and answering the question raised by Nagaraja and Nevzorov (1997) (see Remark 3 of their paper) whether conditions (5) and (6) of Theorem 1.5 can be removed, Lemma 2.1 readily reveals that conditions (5) and (6) cannot be omitted. Next, let Mi; n (x) = n (i)x + n (i), where n (i) ¿ 0 and n (i) (n (i) is positive for Mi; n (x) to be nondecreasing) are some real numbers. The direct solution of the Abelian equation in (14) is complicated. So, let us impose some additional restrictions, namely that equality (14) holds for two diJerent values of i (n ¿ 2; 1 6 i 6 n). In view of Lemma 2.1, it implies for all x ∈ (a; b) pi
E − I (x) I (x) + qi = Ri; n (x); F(x) 1 − F(x)
pj
I (x) E − I (x) + qj = Rj; n (x) F(x) 1 − F(x)
(1 6 i ¡ j 6 n). The functions Ri; n (x) and Rj; n (x) are linear, and the numbers pi ; pj ; qi ; qj are all 1xed. Consequently, I (x)=F(x) and (E − I (x))=(1 − F(x)) are also linear. Now, using the same arguments as those for (2) and (4), we conclude that F is simultaneously of F1 and F4 types. Matching their supports, we 1nd 0 ¡ x ¡ 1 and = = 1 for F. Hence, the theorem.
3. Characterization based on the second record value and maximum In this section, let F be a continuous distribution function. For n ¿ 1, let us denote Sn = X (2) − Mn . Independence of the random variables S2 and M2 characterized the distribution of the exponential type in Theorem 1.6. Now, let us examine this for other values of n. For this purpose, we 1rst establish the following lemma. Lemma 3.1. For any x; y in I, we have P{X (2) 6 y | Mn = x} 0;
l n−1 F(y) 1 n−l ; = n (l − 1)l F(x) l=2 n − 1 1 F(y) − F(x) ; + n n 1 − F(x)
n = 1; 2; y ¡ x;
(A)
n ¿ 3; y ¡ x;
(AA)
n ¿ 1; y ¿ x:
(AAA)
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Proof. Assume 1rst that F(x) is absolutely continuous and p(x) = F (x) for any x ∈ I. Proof of (A). The result is obvious. Indeed, the event {X (2) 6 y; x ¡ Mn 6 x + d x} cannot occur for n = 1; 2 and any y ¡ x. Hence, P{X (2) 6 y | Mn = x} = 0
(n = 1; 2; y ¡ x ∈ I):
Proof of (AA). Let us de1ne the following events: U1 (y) = {X1 ¡ y};
V (i2 ) =
i 2 −1
{Xi 6 X1 };
i=2
U (i2 ; y) = {X1 ¡ Xi2 6 y};
V (i2 ; t; x) =
t−1
{Xi ¡ x};
i=i2 +1
U (t; x) = {Xt ∈ (x; x + d x]};
V (t; x) =
n
{Xi 6 x + d x}:
i=t+1
We can write for y ¡ x when p(x) ¿ 0 P{X (2) 6 y; x ¡ Mn 6 x + d x} =
n t−1
P{U1 (y)V (i2 )U (i2 ; y)V (i2 ; t; x)U (t; x)V (t; x)} + o(d x):
(16)
t=3 i2 =2
Those Xi (1 6 i 6 n) that were registered in (x; x + d x] are to produce the largest value in the sample X1 ; : : : ; Xn . The time when the 1rst such Xi was recorded is denoted as t (3 6 t 6 n). The time when the second record value was registered is denoted as i2 (2 6 i2 ¡ t). Since the second record value does not exceed Mn , it must appear earlier than Xt . The events U1 (y) and U (i2 ; y) reNect the appearance of the 1rst and second record values, respectively. The event U (t; x) describes the appearance of the 1rst Xi that was registered in (x; x +d x]. The event V (i2 ) consists of some trials that do not exceed the 1rst record value (this set could be empty). The event V (i2 ; t; x) formed with the trials Xi2 +1 ; Xi2 +2 ; : : : ; Xt−1 could contain some record values and could be empty. The event V (t; x) consists of the trials Xt+1 ; Xt+2 ; : : : ; Xn and could be empty. The product of the events U1 (y)V (i2 )U (i2 ; y)V (i2 ; t; x)U (t; x)V (t; x) tells us that there are at least two record values belonging to (−∞; y] and Mn ∈ (x; x + d x]. It follows from (16) that for those x for which p(x) ¿ 0 P{X (2) 6 y; x 6 Mn ¡ x + d x} =
n t−1 t=3 i2 =2
y
−∞
(F(y) − F(u))F i2 −2 (u)p(u) du(p(x) d x
N. Balakrishnan, A. Stepanov / Journal of Statistical Planning and Inference 124 (2004) 273 – 287
+ o(d x))F
n−i2 −1
= (p(x) + o(1))F
281
(x + d x)
n−1
(x + d x) d x
n−1 n i2 =2 t=i2 +1
= (p(x) + o(1))F n−1 (x + d x) d x
n−1 i2 =2
1 (i2 − 1)i2
n − i2 (i2 − 1)i2
F(y) F(x + d x)
F(y) F(x + d x)
i2
i2
:
The joint density-distribution function G(y; x) of the vector (X (2); Mn ) can be obtained from the last formula as
l n−1 F(y) n−l G(y; x) = p(x)F n−1 (x) ; (l − 1)l F(x) l=2
where y 6 x ∈ I, which when used with the density of Mn given by nF n−1 (x)p(x), readily yields (AA). Proof of (AAA). First of all, for any x 6 y ∈ I, we have P{X (k) 6 y | Mn = x} n P{N (n) = i; )(x; y) ¿ k − i}; i=1 = k−1 P{N (n) = i; )(x; y) ¿ k − i}; P{N (n) ¿ k} +
(n 6 k)
(B)
(n ¿ k):
(BB)
i=1
Here, N (n) denotes the total number of record values amongst X1 ; X2 ; : : : ; Xn , and )(x; y) denotes the total number of record values in (x; y). It is also known from the works of Shorrock (1972) and Renyi (1976) that P{N (n) = i} =
|Sni | ; n!
where Sni are Stirling numbers de1ned by x(x − 1) · · · (x − (n − 1)) =
∞
Sni xi :
i=0
= 0 and Sn1 = (−1)n−1 (n − 1)! In particular, we have Random variables ), taken over diJerent intervals, reNect the number of record values falling in those intervals. They are independent for diJerent nonoverlapping intervals (see Shorrock, 1972) and that Sn0
P{)(x; y) = i} =
e−* *i i!
(i ¿ 0);
282
N. Balakrishnan, A. Stepanov / Journal of Statistical Planning and Inference 124 (2004) 273 – 287
where
1 − F(y) * = −log 1 − F(x)
:
(17)
Expression (AAA) needs to be considered for two cases, viz. when (a) n ¿ k = 2, and (b) k = 2; n = 1; 2. Case (a): Here, we have P{X (2) 6 y | Mn = x} = P{N (n) ¿ 2} + P{N (n) = 1; )(x; y) ¿ 1} = 1 − P{N (n) = 1} + P{N (n) = 1}P{)(x; y) ¿ 1}
(n ¿ 3);
(18)
where the last product is due to the fact that the variables N and ) are independent. Indeed, the number of record values N (n) amongst X1 ; X2 ; : : : ; Xn is, in fact, the number of record values registered in the interval (−∞; x], because (x ¡ Mn ¡ x+d x). According to Shorrock’s result, the number of records registered in (−∞; x] and the number of records in (x; y] are independent variables. We, therefore, 1nd from (18) that P{X (2) 6 y | Mn = x} = 1 −
1 1 − P{)(x; y) = 0} + : n n
Case (b): For proving this part, we use (B) to write P{X (2) 6 y | Mn = x} =
P{)(x; y) ¿ 1};
(n = 1; k = 2); y ¿ x;
P{N (2) = 1; )(x; y) ¿ 1} + P{N (2) = 2; )(x; y) ¿ 0};
(n = k = 2); y ¿ x:
This readily yields P{X (2) 6 y | Mn = x}
(F(y) − F(x))=(1 − F(x)); = (F(y) − F(x))=(2(1 − F(x))) + 1=2;
(n = 1; k = 2); y ¿ x; (n = k = 2); y ¿ x:
The last equality completes the proof of Lemma 3.1. Remark 4. We assumed at the beginning of the proof of Lemma 3.1 that F(x) is absolutely continuous for x ∈ I. The results holds true even if F is continuous. In the following theorem, we use Lemma 3.1 to establish some additional characterization results. Theorem 3.2. 1. Random variables Sn− = min{0; Sn } and Mn are independent for some n ¿ 3 i: the initial continuous distribution F coincides (up to the location and scale parameters) with F2 . 2. Random variables Sn+ =max{0; Sn } and Mn are independent for some n ¿ 1 i: the initial continuous distribution F coincides (up to the location and scale parameters) with F5 .
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3. Random variables Sn and Mn are independent for some n = 1; 2; : : : i: the initial continuous distribution F coincides (up to the location and scale parameters) with F5 and n = 1 or 2. Proof of Part 1 of Theorem 3.2. SuMciency can be veri1ed directly. Let us now prove necessity. It is easily seen that the unconditional distribution of Sn− is concentrated on the interval (−∞; 0]. The conditional distribution of Sn− can be derived from Lemma 3.1 as
l n−1 F(x + y) n−l 1 ; (y ¡ 0) (K) F(x) P{Sn− 6 y | Mn = x} = n l=2 (l − 1)l 1; (y ¿ 0); (KK) n−2 where n ¿ 3, x ∈ I. This conditional distribution has an atom 1=(n−1)+(1=n) l=1 1=l at the point 0. Independence of Sn− and Mn implies that (K) does not depend on x, i.e. l
n−1 F(y + x) cl = H (y); y ¡ 0; x ∈ I; (19) F(x) l=2
where cl = (n − l)={nl(l − 1)} ¿ 0 and H (y) is some continuous function. Actually, H is the continuous part of the conditional distribution function of Sn− . We additionally n−1 de1ne H (0) = l=2 cl = c (0 ¡ c ¡ 1). Now H (x) is continuous on (−∞; 0] with the following properties: H (0) = c ¿ 0, H (−∞) = 0:
(20)
Some point x0 exists such that H (x) ¿ 0
(21)
for all x (x0 ¡ x 6 0). The following proposition is needed for proving the required result. Proposition 1. Let some initial continuous distribution function F satisfy (19) for any y 6 0 and every x ∈ I = [a; b]. Then, a = −∞ and b ¡ + ∞. Proof of Proposition 1. Let us 1rst prove the second statement b ¡ + ∞. Suppose this is not true: F(x) ¡ 1 for any x. Equality (19) can be written with x = −2y ¿ 0 (x ∈ I) as l
n−1 F(−y) cl = H (y): F(−2y) l=2
From here, we get lim H (y) = c ¿ 0:
y→−∞
(22)
This is a direct contradiction to (20). So, our supposition F(x) ¡ 1 (for any x) was not correct. Assuming b ¡ + ∞, we can additionally put (without loss of generality) b = 0.
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Remark 5. Of course, if b ¡ + ∞ and x = −2y ¿ 0 is taken as large for (19), we get the same result (22). However now, we are wrong in applying (19), because x does not belong to I. Let us now prove a = −∞. Suppose the opposite statement is true, viz. a number a exists (−∞ ¡ a ¡ 0) such that F(x) = 0 for any x 6 a and F(x) ¿ 0 for a ¡ x ¡ 0. Taking y = a=m, x = a − a=m (m ¿ 2), we 1nd F(a) F(y + x) = = 0: F(x) F(a − a=m) Equality (19) then gives H (a=m) = 0 for any m ¿ 2. This entails H (y) = 0 for any y ¡ 0, which contradicts (21). This completes the proof of Proposition 1. Let us now return to proving Part 1 of Theorem 3.2. Putting x = 0 and y in equality (19), we obtain n−1
cl F l (y) = H (y);
l=2 n−1
cl
l=2
F(2y) F(y)
l = H (y):
The diJerence of these two equalities gives n−1 l=2
cl
F 2l (y) − F l (2l) =0 F l (y)
(n ¿ 3):
All cl are positive. This entails F 2 (y) = F(2y) for any y 6 0. The last equality can be extended in a similar way to F k (y) = F(ky)
(y 6 0; k ¿ 2):
(23)
The only possible solution of (23) in the class of continuous distribution functions having the support (−∞; 0] is F2 . This completes the proof of Part 1 of Theorem 3.2. Proof of Part 2 of Theorem 3.2. SuMciency of Part 2 of Theorem 3.2 can be checked directly. Let us now establish the necessity. It is obvious that the support of the distribution of Sn+ is [0; ∞). The conditional distribution of Sn+ can be obtained from Lemma 3.1 as (y ¡ 0); 0; + P{Sn 6 y | Mn = x} = n − 1 F(y + x) − F(x) + ; (y ¿ 0); n n(1 − F(x))
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where x ∈ I, n ¿ 1. The independence of Sn+ and Mn presumes n − 1 F(x + y) − F(x) + = G(y); (24) n n(1 − F(x)) where G is some continuous function on [0; ∞). Using the substitution G1 (y) = n(G(y) − (n − 1)=n), we can rewrite equality (24) as 1 − F(x + y) (25) = G1 (y) (y ¿ 0; x ∈ I; n ¿ 1): 1 − F(x) This is the Cauchy functional equation. The only possible continuous solution of this equation can be (up to the parameters of location and scale) F5 . Here, we would not like to repeat the known methods of establishing this fact. One of them can be based on the same principle as the method of proving the necessity of Part 1 of Theorem 3.2. We only mention here one important result that we would have to prove if we choose to provide a full-scale proof of Part 2 of Theorem 3.2. Proposition 2. Let some initial continuous distribution function satisfy Eq. (25) for any y ¿ 0 and x ∈ I. Then, its support I = [a; b] is such that a ¿ − ∞ and b = +∞. Note here again that a can be chosen as 0. This completes the proof of Part 2 of Theorem 3.2. Proof of Part 3 of Theorem 3.2. SuMciency can be easily veri1ed for F5 (n = 1; 2). Necessity for n = 1; 2 has already been guaranteed by Theorem 3.2. Indeed, for n = 1; 2 we have Sn+ = Sn a.s. Let us now establish the necessity for n ¿ 3. In fact, we have to prove that there is no distribution that provides the independence of Sn and Mn for any n ¿ 3. Parts (AA) and (AAA) of Lemma 3.1 can be written as (n ¿ 2) P{Sn 6 y | Mn = x} n−1 cl (F(y + x)=F(x))l ; = l=2 (n − 1)=n + (F(x + y) − F(x))=(n(1 − F(x)));
(y ¡ 0)
(AA )
(y ¿ 0):
(AAA )
The independence of Sn and Mn (n ¿ 2) implies that both parts (AA ) and (AAA ) do not depend on x. This, in turn, suggests that Propositions 1 and 2 are valid simultaneously for some continuous distribution function F. This is impossible, because no support of a distribution function can meet the demands of Propositions 1 and 2 at the same time. This then completes the proof of Part 3 of Theorem 3.2. Remark 6. Note here that the conditional distribution function P{X (2) 6 y | Mn = x} n−1 has an atom (1=n) i=1 1=i (n ¿ 2) at the point x. Remark 7. The conditional distribution of X (k), given Mn , can also be derived for arbitrary k and n. However, the corresponding formula is quite cumbersome and therefore may not be suitable for characterizations.
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Acknowledgements We are sincerely grateful to a referee who found an error in Theorem 3.2. The present version of the statement of Theorem 3.2 was oJered by this referee. References Ahsanullah, M., 1979. Characterization of the exponential distribution by record values. SankhyPa, Ser. B 41, 116–121. Ahsanullah, M., 1981. On a characterization of the exponential distribution by weak homoscedasticity of record values. Biometrical J. 23, 715–717. Ahsanullah, M., 1995. Record Statistics. Nova Science Publishers, Commack, New York. Ahsanullah, M., Holland, B., 1984. Record values and the geometric distribution. Statist. Hefte 25, 319–327. Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1992. A First Course in Order Statistics. Wiley, New York. Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1998. Records. Wiley, New York. Balakrishnan, N., Balasubramanian, K., 1995. A characterization of geometric distribution based on record values. J. Appl. Statist. Sci. 2, 233–248. David, H.A., 1981. Order Statistics, 2nd Edition. Wiley, New York. Ferguson, T.S., 1967. On characterizing distributions by properties of order statistics. SankhyPa, Ser. A 29, 265–278. Galambos, J., 1984. The Asymptotic Theory of Extreme Order Statistics, 2nd Edition. Krieger, Malabar, FL. Galambos, J., Kotz, S., 1978. Characterizations of Probability Distribution. In: Lecture Notes in Mathematics, Vol. 657. Springer, Berlin. Gupta, R.C., 1984. Relations between order statistics and record values and some characterization results. J. Appl. Probab. 21, 425–430. Huang, J.S., 1975. Characterization of distributions by the expected values of order statistics. Ann. Inst. Statist. Math. 27, 87–93. Huang, J.S., 1989. Moment problem of order statistics: a review. Internat. Statist. Rev. 57, 57–66. Kaminsky, K.S., Nelson, P.I., 1975. Best linear unbiased predictor of order statistics in location and scale families. J. Amer. Statist. Assoc. 70, 145–150. Kirmani, S.N.U.A., Beg, M.I., 1984. On characterization of distributions by expected records. SankhyPa, Ser. A 46, 463–465. Kotlarski, I., 1972. On a characterization of some distributions by conditional expectation. SankhyPa, Ser. A 34, 461–467. Kotz, S., Shanbhag, D.N., 1980. Some new approaches to probability distributions. Adv. Appl. Probab. 12, 903–921. Nagaraja, H.N., 1977. On a characterization based on record values. Austral. J. Statist. 19, 70–73. Nagaraja, H.N., 1984. Asymptotic linear prediction of extreme order statistics. Ann. Inst. Statist. Math. 36, 289–299. Nagaraja, H.N., 1988a. Record values and related statistics—a review. Comm. Statist.—Theory Methods 17, 2223–2238. Nagaraja, H.N., 1988b. Some characterizations of continuous distributions based on regression of adjacent order statistics and record values. SankhyPa, Ser. A 50, 70–73. Nagaraja, H.N., Nevzorov, V.B., 1997. On characterizations based on record values and order statistics. J. Statist. Plann. Inference 63, 271–284. Nevzorov, V.B., 1992. A characterization of exponential distributions by correlations between records. Math. Methods in Statist. 1, 49–54. Nevzorov, V.B., 2000. Records: Mathematical Theory. In: Translation of Mathematical Monographs, Vol. 194. American Mathematical Society, Providence, RI. Rao, C.R., Shanbhag, D.N., 1993. Choquet–Deny Type Functional Equations with Applications to Stochastic Models. Wiley, Chichester, England.
N. Balakrishnan, A. Stepanov / Journal of Statistical Planning and Inference 124 (2004) 273 – 287
287
Renyi, A., 1976. On the extreme elements of observations, In: Selected Papers of Alfred Renyi, Vol. 3. Akademiai Kiado, Budapest, Hungary, pp. 50 – 65. Shorrock, R., 1972. A limit theorem for inter-record times. J. Appl. Probab. 9, 543–555. Stepanov, A., 1990. Characterization of a geometric class of distributions. Theory of Probab. Math. Statist. 41, 133–136. Stepanov, A., 1994. A characterization theorem for weak records. Theory of Probab. Appl. 38, 762–764. Szekely, G.J., Mori, T.F., 1985. An extremal property of rectangular distributions. Statist. Probab. Lett. 3, 107–109.