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On characterizations of the logistic distribution
Journal of Statistical Planning and Inference 138 (2008) 1147 – 1156 www.elsevier.com/locate/jspi
On characterizations of the logistic distribution Gwo Dong Lina,∗ , Chin-Yuan Hub a Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, ROC b Department of Business Education, National Changhua University of Education, Changhua 50058, Taiwan, ROC
Received 31 October 2004; received in revised form 2 September 2006; accepted 17 April 2007 Available online 3 July 2007
Abstract We modify and extend George and Mudholkar’s [1981. A characterization of the logistic distribution by a sample median. Ann. Inst. Statist. Math. 33, 125–129] characterization result about the logistic distribution, which is in terms of the sample median and Laplace distribution. Moreover, we give some new characterization results in terms of the smallest order statistics and the exponential distribution. © 2007 Elsevier B.V. All rights reserved. MSC: primary 62E10; 62G30; 60E10 Keywords: Characterization; Order statistics; Logistic distribution; Exponential distribution; Laplace distribution
1. Introduction Almost 170 years ago, Verhulst (1838, 1845) applied the logistic distribution in his pioneering work on demography. Gumbel (1944) might be the first one who gave a genesis of the distribution: the logistic distribution arises in a purely statistical manner as the limiting distribution of standardized midranges of random samples from a symmetric distribution of exponential-type, where the midrange of a random sample means the average of the largest and the smallest sample values. It has found several important applications in many fields including survival analysis, growth model and public health (see Berkson, 1944; Johnson et al., 1995). The logistic distribution is very similar in shape to the normal distribution because its density function is symmetric and has a bell shape. Besides, the maximum difference between two distribution functions can be less than 0.01 (Mudholkar and George, 1978). So the logistic distribution has a close approximation to the normal distribution. However, Kotz in 1974 made the following comment: while the normal distribution with a more complex distribution function is well characterized, the characteristic property of the logistic distribution has not been thoroughly developed. This comment still remains valid nowadays, although some characterization results are recently available in the literature. In the next section we first review basic properties of the logistic distribution and some existing characterization results. In Section 3, we modify and extend George and Mudholkar’s (1981) characterization result about the logistic distribution, which is in terms of the sample median and Laplace distribution. Finally, in Section 4 we give some new characterization results in terms of the smallest order statistics and the exponential distribution. ∗ Corresponding author.
E-mail address: [email protected] (G.D. Lin). 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2007.04.030
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2. Basic properties We need some notations in the sequel. Let X be a random variable with distribution function F and characteristic function (cf) . Let {Xk }nk=1 be a random sample from F, and denote the corresponding order statistics by X1,n X2,n · · · Xn,n . The distribution function and cf of Xk,n are denoted by Fk,n and k,n , respectively. Especially, for odd sample size n = 2m − 1, denote the sample median Xm,2m−1 by X(m) and its distribution function and cf by F(m) and (m) , respectively. As usual, define the quantile function of F by F −1 (t) = inf{x: F (x) t}, t ∈ (0, 1). d
Moreover, let R be the real line and denote equality in distribution by =. We now review some basic properties of the logistic distribution. Most of the following results are available in the literature (see, e.g., Balakrishnan, 1992; Balakrishnan and Nevzorov, 2003, Chapter 22) except the general form 1,n in (h) which can be proved directly by using beta function. Proposition 1. Let X have the standard logistic distribution function F (x) = (1 + e−x )−1 , x ∈ R, and let U obey the uniform distribution on [0, 1]. Then the following properties hold. (a) The quantile function of F is F −1 (t) = log[t/(1 − t)], t ∈ (0, 1), namely, the logit transformation of t, and hence d
X = log[U/(1 − U )]. (b) The moment generating function of X is M(s) = (1 + s)(1 − s), s ∈ (−1, 1). 2 −1 = it/ sin(it) = t/ sinh(t), t ∈ R. (c) The cf (t) = M(it) = (1 + it)(1 − it) = ∞ j =1 [1 + (t/j ) ] d
(d) X = V1 − V2 , where V1 and V2 are independent random variables having the standard extreme-value distribution function H (x) = exp(−e−x ), x ∈ R. 2k ) = 2(2k), k = 1, 2, . . . , where is the Riemann zeta function defined by (e) The moments ∞ E(X −x (x) = j =1 j for x > 1. In particular, E(X 2 ) = 2 /3. (f) The distribution function F is equal to its hazard rate function, namely, F (x) = f (x)/(1 − F (x)), x ∈ R, where f is the density function of F. 2 2 (g) For n = 2m − 13, the cf of the sample median X(m) is (m) (t) = m−1 j =1 (1 + t /j )(t), t ∈ R. In particular, (2) (t) = (1 + t 2 )(t), t ∈ R. (h) The cf of the smallest order statistic X1,n is 1,n (t) = (1 + it)(n − it)/(n) = n−1 j =1 (1 − it/j )(t), t ∈ R. In particular, 1,2 (t) = (1 − it)(t) and 1,3 (t) = (1 − it/2)(1 − it)(t), t ∈ R. d
(i) For n = 3, the sample median is distributed as the midrange, namely, X2,3 =(X1,3 + X3,3 )/2. (j) The product moment is of the form: E(Xi,n Xj,n ) = (j ) + [(i) − (n + 1)][(j ) − (n − j + 1)] +
∞ B(n − i + 1, i + k) k=1
kB(i, n − i + 1)
[(j + k) − (n − j + 1)],
where the function (x) = (x)/(x) for x > 0 and B(a, b) = (a)(b)/(a + b). (k) Another representation of the product moment is 2 E(Xi,n Xj,n ) = E(Xj,n )+
j −1 k−1
(−1)
k=i =1
× E(Xj +−k,n+−k ) +
k+i
k − 1 n j − k + − 1 B(, n − k + 1) i−1 k
n j −i−1 n−i i {− (n − j + 1) (−1)k k i+k i k=0
+ [(n − j + 1) − (n − i − k + 1)][(j − i − k) − (n − j + 1)]}. () The moments of order statistics satisfy the recurrence relation k E(Xj,n ) = E(Xjk−1,n ) +
kn E(Xjk−1 −1,n−1 ). (j − 1)(n − j + 1)
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Propositions 1(j) and 1(k) are essentially due to Shah (1966), but here we have modified the arguments of the digamma and trigamma functions ( and ) (see Gupta and Balakrishnan, 1992). Moreover, in Shah’s paper, the numerator i of the fraction i/(i + k) in (k) was missing. Applying the Müntz–Szász Theorem, we are able to characterize the logistic distribution by using an infinite set of recurrence relations () (see Lin, 1988). d The logistic distribution is infinitely divisible, because, by Proposition 1(c), we have the identity: X = ∞ j =1 Lj /j , where the Lj are independent and standard Laplace random variables. This fact also follows from Proposition 1(d) and the infinite divisibility of the extreme-value distribution (see, e.g., Steutel and van Harn, 2004, p. 209). The question whether the median-midrange identity in Proposition 1(i) characterizes the logistic distribution remains open (see Arnold et al., 1992, p. 150; Kotz et al., 2001, p. 127; Nevzorov et al., 2003). We next review some existing characterizations of the logistic distribution. Theorem A given by Galambos and Kotz (1978, p. 27) applies the conditional distribution. Theorem B applying a monotone transformation of the underlying distribution is due to Galambos (1992). Theorem C follows from a well-known characterization for general distributions (see also Galambos, 1992, p. 176). The results by properties of sample medians and of minimal order statistics are investigated in the next two sections. For other characterization results, see the survey works by Galambos (1992), Gather et al. (1998) and Kamps (1998).
Theorem A. Let X be a random variable with continuous distribution function F symmetric at zero. Then the following statements are equivalent: (a) F (x) = (1 + e−x )−1 , x ∈ R, where > 0 is a constant. (b) P (−x X x|X x) = 1 − e−x for all x > 0, where > 0 is a constant. (x+y) F (x+y) (c) [1−F1−F (x)][1−F (y)] = F (x)F (y) for all x, y 0. Theorem B. Let X be a random variable and define T = 1 + e−X . If P (T uv|T u) = P (T v) for all u 1 and v 1, then X has the standard logistic distribution. Theorem C. Let X be a random variable with E(|X(k0 ) |) < ∞ for some integer k0 1. If EX(m) =0 and EXm,2m =− m1 for all mk0 , then X has the standard logistic distribution. Remark. Theorem C is a consequence of the following facts: (a) The set of expected order statistics {EX (m) , EXm,2m }∞ m=k0 uniquely determines the larger set {EX k,n : k0 k n − k0 + 1, n2k0 − 1}, which in turn characterizes the distribution of X. [In fact, we can delete finitely many elements from {EX(m) , EXm,2m }∞ m=k0 .] (b) If X and Y obey the standard logistic and exponential distributions, respectively, then EX (m) = 0 and EXm,2m = 2m 1 1 1 EY m,2m − EY m+1,2m = 2m i=m+1 i − i=m i = − m . 3. Characterizations by properties of sample medians George and Mudholkar (1981) claimed that the identity for (2) in Proposition 1(g) is a characteristic property of the logistic distribution under the assumptions: F (0) = 21 ,
f (−∞) = f (−∞) = 0
and the function t(t) is integrable,
where f is the density function of F. Their proof however requires implicitly a stronger assumption that the function t 2 (t) is integrable over R. On the other hand, the boundary conditions on the density f and its derivative are actually consequences of the stronger condition. We now modify their result as follows. Theorem 1. Let X be a random variable with distribution function F satisfying F (0) = 21 and with cf . Assume further that the function t 2 (t) is absolutely integrable over R. Then a necessary and sufficient condition for F to be the standard logistic distribution function is that the cf (2) of the sample median X2,3 satisfies the equation (2) (t) = (1 + t 2 )(t)
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for all t ∈ R, or equivalently, d
X = X2,3 + L, where X2,3 and L are independent and L has the standard Laplace distribution with cf L
(1) (t) = 1/(1 + t 2 ),
t ∈ R.
We shall give an alternative proof of Theorem 1. For convenience, denote the left and the right extremities of F by F and rF , respectively. Namely, for any > 0, we have that F (F − ) = 0, F (F + ) > 0, F (rF − ) < 1 while F (rF ) = 1. It is well known that Proposition 1(f) is a characteristic property of the logistic distribution up to location parameter. This fact will be used in the sequel and we restate it as follows. Lemma 1. Let F be an absolutely continuous distribution function with density f satisfying f (x) = F (x)(1 − F (x)), x ∈ (F , rF ). Then for some ∈ R, F (x) = (1 + e−(x− ) )−1 , x ∈ R. ∞ Proof of Theorem 1. From the absolute integrability of the function t 2 (t) it follows that −∞ |(t)| dt < ∞ and ∞ 2 −∞ |t(t)| dt < ∞ because is continuous on R and |(t)| |t(t)| |t (t)| for |t| 1. By the inversion formula for cfs, F has a continuous density function
∞ 1 e−itx (t) dt, x ∈ R, (2) f (x) = 2 −∞ and ∞ f (∞) ≡ limx→∞ f (x) = 0, f (−∞) ≡ limx→−∞ f (x) = 0 due to the Riemann–Lebesgue lemma. Since −∞ |t(t)| dt < ∞, the function f in (2) is differentiable by Lebesgue’s Dominated Convergence Theorem, and we can take differentiation under integral to obtain
−i ∞ −itx e t(t) dt, x ∈ R, f (∞) = f (−∞) = 0. f (x) = 2 −∞ ∞ Similarly, by the assumption −∞ |t 2 (t)| dt < ∞, we have
−1 ∞ −itx 2 f (x) = e t (t) dt, x ∈ R, (3) 2 −∞ which is uniformly continuous on R. The necessity part of the theorem is exactly Proposition 1(g). To prove the sufficiency part, suppose that equality (1) holds. Then the cf (2) is absolutely integrable and we have by the inversion formula for cfs
∞ 1 (x) = e−itx (2) (t) dt, x ∈ R. (4) F(2) 2 −∞ = f − f , or equivalently, F = F − F , by the boundary condition f (∞) = 0. Combining (1) through (4) yields F(2) (2) On the other hand,
F (x) t (1 − t) dt = 3F 2 (x) − 2F 3 (x), x ∈ R, F(2) (x) = 6 0
and hence it remains to solve the equation 2F 3 − 3F 2 + F − F = 0.
(5)
Setting G = F (1 − F ), we have G = (1 − 2F )F and F = G(1 − 2F ) by (5). These together imply that F F = GG , or equivalently, (F )2 = G2 by the boundary condition f (∞) = 0. Therefore, F (x) = G(x) = F (x)(1 − F (x)) > 0 for x ∈ (F , rF ) because f (x)0. Lemma 1 together with the condition F (0) = 21 completes the proof. We now extend Theorem 1 to the following. Theorem 2. Let X be a random variable with distribution function F satisfying F (0) = 21 and for some fixed integer m 2, let (m−1) be the cf of the sample median X(m−1) . Assume further that the function t 2 (m−1) (t) is absolutely
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integrable over R. Then a necessary and sufficient condition for F to be the standard logistic distribution function is 2 that (m) (t) = [1 + t 2 ](m−1) (t) for all t ∈ R holds, or equivalently, (m−1)
d
X(m−1) = X(m) + L/(m − 1),
(6)
where X(m) and L are independent and L has the standard Laplace distribution with cf L (t) = 1/(1 + t 2 ), t ∈ R. To prove Theorem 2, we need the following two lemmas, in which the first one follows from the remarkable formula for distribution functions of order statistics (see, e.g., Hwang and Lin, 1984, pp. 179, 184; Nevzorov, 2001, p. 6). Lemma 2. Let n = 2m − 1. Then the distribution function of the sample median X(m) is 2F (x)−1 2m − 1 m F(m) (x) = 2m−1 (1 − v 2 )m−1 dv for all x ∈ R. m 2 −1 Lemma 3. For m 2, the distribution functions of sample medians satisfy the relation 1 2m − 2 F(m) (x) = F(m−1) (x) + [F (x)(1 − F (x))]m−1 (2F (x) − 1) for all x ∈ R. 2 m−1 Proof. Using integration by parts, we have
u 1 2(m − 1) u (1 − v 2 )m−1 dv = (1 − v 2 )m−2 dv, u(1 − u2 )m−1 + 2m − 1 2m − 1 −1 −1 The required recurrence relation follows from Lemma 2 and the identity above.
u ∈ [−1, 1].
Proof of Theorem 2. The necessity part follows immediately from Proposition 1(g). To prove the sufficiency part, suppose that equality (6) holds. Then by the inversion formula for cfs, we have F(m) (x) = F(m−1) (x) −
1 (m − 1)2
F(m−1) (x),
x ∈ R,
(7)
and f(m−1) (∞) = F(m−1) (∞) = 0, f(m−1) (∞) = 0. It follows from (7) and Lemma 3 that 1 2m − 2 1 F(m−1) (x) = [F (x)(1 − F (x))]m−1 (2F (x) − 1) for all x ∈ R. − 2 m−1 (m − 1)2
On the other hand, Lemma 2 implies d m − 1 2m − 3 [1 − (2F (x) − 1)2 ]m−2 (2F (x) − 1), F(m−1) (x) = 2m−3 m−1 2 dx
x ∈ R.
(8)
(9)
Multiplying (8) by (9) and simplifying the result leads to 1 2m − 3 m − 1 2m − 2 − [1 − (2F − 1)2 ]2m−3 (2F − 1)(2F − 1) . F F = (m−1) (m−1) m−1 24m−4 m − 1 (m − 1)2 Therefore, we have, by the boundary condition f(m−1) (∞) = 0, 2 1 2m − 3 2 1 = 2m−2 [1 − (2F − 1)2 ]2m−2 . F(m−1) m−1 m−1 2 This implies 1 1 = 2m−2 F(m−1) m−1 2
2m − 3 [1 − (2F − 1)2 ]m−1 . m−1
(10)
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Combining (9) and (10) yields that d 1 (2F (x) − 1) = [1 − (2F (x) − 1)2 ], x ∈ (F , rF ), dx 2 or equivalently, f (x) = F (x)(1 − F (x)), x ∈ (F , rF ). Lemma 1 together with the condition F (0) = 21 completes the proof. In view of Proposition 1(g) and the inversion formula for cfs, the logistic density function f satisfies the equality: for m2, m−1
1−
j =1
D2 j2
f (x) =
(2m − 1)! {F (x)(1 − F (x))}m−1 f (x) ≡ (x), [(m − 1)!]2
x ∈ R,
(11)
where the operator symbol D stands for differentiation. In order to extend Theorem 1, George and Mudholkar (1980, Theorem 8) tried to solve Eq. (11). However, their proof was incomplete because it was ignored that the function in the RHS of (11) should be independent of the unknown f when applying Hirschman and Widder’s (1955) Theorem 9.3 on the uniqueness of the solution of a differential equation. (For example, each differentiable function on R satisfies the equation Df (x) = f (x), x ∈ R; namely, the solution of this equation is not unique.) We have instead the following corollary to Theorem 2, in which the logistic distribution is characterized by two identities given in Proposition 1(g). It is not clear whether one such identity with m 3 is enough to characterize the distribution. Corollary 1. Let X be a random variable with distribution function F satisfying F (0) = 21 and with cf . For some integer k 4, let the function t 2k−2 (t) integrable over R. Assume further that for m = k − 1, k, the be absolutely 2 /j 2 )(t), t ∈ R, or equivalently, following equality holds: (m) (t) = m−1 (1 + t j =1 d
X = X(m) +
m−1
Lj /j ,
j =1
where X(m) and Lj , j = 1, . . . , m − 1, are independent random variables and each Lj obeys the standard Laplace distribution. Then F is the standard logistic distribution function. 4. Characterizations by properties of minimal order statistics The next theorem, also due to George and Mudholkar (1982), gives another characterization of the logistic distribution in terms of the minimal order statistic of a random sample from the underlying distribution. Theorem 3. Let X be a random variable with distribution function F satisfying F (0) = 21 and with cf . Assume further that the function t(t) is absolutely integrable over R. Then a necessary and sufficient condition for F to be the standard logistic distribution function is that the cf 1,2 of the smallest order statistic X1,2 satisfies the equation d
1,2 (t) = (1 − it)(t), t ∈ R, or equivalently, X = X1,2 + Y , where X1,2 and Y are independent and Y has the standard exponential distribution with cf Y (t) = 1/(1 − it), t ∈ R. We give another interesting result below. Theorem 4. Let X have a symmetric distribution function F satisfying the conditions of Theorem 1. Assume further that Y1 and Y2 , independent of the random sample {Xk }3k=1 from F, are two independent random variables having the standard exponential distribution. Then a necessary and sufficient condition for F to be the standard logistic distribution function is that the smallest order statistic X1,3 satisfies the distribution equation d
X1,3 + 21 Y2 + Y1 = X.
(12)
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Proof. The necessity part follows from Proposition 1(h). To prove the sufficiency part, suppose that the relation (12 ) holds, namely, 1,3 (t) = (1 − 21 it)(1 − it)(t),
t ∈ R.
(13)
On the other hand, by the symmetry property of F, the cf of the largest order statistic X3,3 satisfies the equation 3,3 (t) = 1,3 (t) = (1 + 21 it)(1 + it)(t),
t ∈ R,
(14)
where w denotes the conjugate of a complex number w. Recall further the identity 1,3 + 2,3 + 3,3 = 3
(15)
which follows from the fact F1,3 + F2,3 + F3,3 = 3F. Equalities (13)–(15) together imply that 2,3 for t ∈ R. Theorem 1 finally completes the proof.
(t) = (1 + t 2 )(t)
In the next result, a variant of Theorem 3 above, we assume the differentiation of the distribution function instead of the integrability of the cf. Theorem 5. Let X be a random variable with distribution function F satisfying F (0) = 21 . Assume further that F is differentiable on R. Then a necessary and sufficient condition for F to be the standard logistic distribution function is d that X = X1,2 + Y , where X1,2 and Y are independent and Y has the standard exponential distribution. d
Proof. It is enough to prove the sufficiency part. Suppose X = X1,2 + Y , where X1,2 and Y are independent and Y has the standard exponential distribution. Then we want to prove that X obeys the standard logistic distribution. Let V = eX d
and let U obey the uniform distribution on [0, 1]. Then by the assumption and the fact that Y = − log U , we have d
V = V1,2 /U ,
(16)
where V1,2 and U are independent and V1,2 is the smallest order statistic of a random sample (of size 2) from the distribution of V. Let G be the distribution of V and G = 1 − G the survival function of V. Then it follows from (16) that
1 2 G(x) = G (ux) du, x > 0, 0
namely,
x
xG(x) =
2
G (y) dy,
x > 0.
0
2
Differentiating yields xG (x) + G(x) = G (x), x > 0, or equivalently, xH (x) = H (x), x ∈ (G , rG ), where H (x) = 1/G(x) − 1. Therefore, H (x) = x, x ∈ (G , rG ), for some constant > 0. This in turn implies that G(x) = 1/(1 + x), x > 0, or equivalently, F (x) = 1 − (1 + ex )−1 , x ∈ R. The assumption F (0) = 21 finally completes the proof. To give a general result, we impose some smoothness conditions on the exponential transformation of the underlying distribution. Theorem 6. Let n > 2 be an integer and let X be a random variable with continuous distribution function F satisfying F (0) = 21 . Assume further that the distribution function G∗ of eX is real analytic and strictly increasing in [0, ∞) and (k) that for each k 1, its kth derivative G∗ is strictly monotone in some interval [0, k ). Then a necessary and sufficient condition for F to be the standard logistic distribution function is that d
X = X1,n +
n−1
Yj /j ,
j =1
where X1,n and Yj , j = 1, . . . , n − 1, are independent and each Yj has the standard exponential distribution.
(17)
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To prove Theorem 6, we shall apply the method of intensively monotone operator introduced by Kakosyan et al. (1984). Definition 1. Let C=C[0, ∞) be a space of all real-valued functions defined and continuous in the interval [0, ∞). The notation f g for f, g ∈ C means that f (t)g(t) for all t ∈ [0, ∞). Let A be an operator mapping some set E ⊂ C into C. We say that the operator A is intensively monotone, if for any f1 and f2 , belonging to E, the condition f1 ( ) f2 ( ) for all ∈ (0, t) implies that (Af 1 )( ) (Af 2 )( ) for ∈ (0, t) and, in addition, the condition f1 ( ) > f2 ( ) for all ∈ (0, t) implies that (Af 1 )(t) > (Af 2 )(t). Definition 2. Let E ⊂ C and {f }∈ be a family of elements of E. We say that the family {f }∈ is strongly E-positive if the following conditions hold: (1) for any f ∈ E there are t0 > 0 and 0 ∈ such that f (t0 ) = f0 (t0 ); (2) for any f ∈ E and any ∈ either f (t) = f (t) for all t ∈ [0, ∞), or there is > 0 such that the difference f (t) − f (t) does not vanish (preserves its sign) in the interval (0, ]. Lemma 4. Let A be an intensively monotone operator on E ⊂ C and let {f }∈ be a strongly E-positive family. Assume that Af = f for all ∈ . Then the condition Af = f , where f ∈ E, implies that there is ∈ such that f = f . In other words, all solutions of the equation Af = f , belonging to E, coincide with elements of the family {f }∈ . Proof. See Kakosyan et al. (1984, pp. 2–3).
Lemma 5. Let f and g be two functions real analytic and strictly monotone in [0, ∞). Assume that for each n 1, the nth derivatives f (n) and g (n) are strictly monotone in some interval [0, n ). Let {xn }∞ n=1 be a sequence of positive real numbers converging to zero. If f (xn ) = g(xn ), n = 1, 2, . . . , then f = g. Proof. Note first that f (0) = g(0) by the assumption f (xn ) = g(xn ) and the continuity of f and g. Next, since f and g are real analytic in [0, ∞), we have the Taylor’s expansions: f (x) =
∞ f (n) (0) n=0
n!
xn
and
g(x) =
∞ g (n) (0) n=0
n!
xn,
x 0.
We shall prove that f (n) (0) = g (n) (0), n = 1, 2, . . . , under the assumptions. By the Generalized Mean-Value Theorem, there exists yn ∈ (0, xn ) such that f (yn )(g(xn ) − g(0)) = g (yn )(f (xn ) − f (0)), n = 1, 2, . . . . Therefore, f (yn ) = g (yn ), n = 1, 2, . . . , and yn converges to zero as n goes to infinity. This in turn implies that f (0) = g (0). Again, there exists zn ∈ (0, yn ) such that f (zn )(g (yn ) − g (0)) = g (zn )(f (yn ) − f (0)), n = 1, 2, . . . . This implies that f (zn ) = g (zn ) for sufficiently large n, due to the monotone property of f and g in the interval [0, 1 ). Moreover, zn converges to zero as n goes to infinity and f (0) = g (0). By induction, we conclude that f (n) (0) = g (n) (0), n1, which completes the proof. Lemma 6. Let Y1 , Y2 , . . . , Yn be independent random variables having the standard exponential distribution. Then d Yn,n = nj=1 Yj /j . Proof. See, e.g., David and Nagaraja (2003, pp. 17–18). Proof of Theorem 6. The necessity part is exactly Proposition 1(h). To prove the sufficiency part, suppose that the condition (17) holds. Then we want to prove that X obeys the standard logistic distribution. Let V = eX and Uj = e−Yj , j = 1, . . . , n − 1, each Uj obeying the uniform distribution on [0, 1]. It follows from condition (17) and Lemma 6 that d
V = V1,n /U1,n−1 ,
(18)
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where V1,n is the smallest order statistic of a random sample (of size n) from the distribution of V, and is independent of U1,n−1 = min{U1 , . . . , Un−1 }. By identity (18), the survival function G∗ of V satisfies
1 n G∗ (x) = (n − 1) G∗ (ux)(1 − u)n−2 du, x > 0. (19) 0
Define E to be the set of all survivor functions G real analytic and strictly decreasing in [0, ∞), with kth derivative (k) G being strictly monotone in some interval [0, k ) for each k 1. Denote G (x) = 1/(1 + x), x > 0, where ∈ = (0, ∞). Further, define the operator A on E by
1 n AG(x) = (n − 1) G (ux)(1 − u)n−2 du, x > 0, G ∈ E. 0
It is seen that A is an intensively monotone operator on E and the family {G }∈ is strongly E-positive by Lemma 5. Moreover, AG = G for all ∈ by the necessity part of the theorem. Finally, by Lemma 4 and identity (19) (which means AG∗ = G∗ ), we conclude that G∗ = G for some ∈ . The condition F (0) = 21 then implies = 1 and completes the proof. Finally, we consider the case of random sample size. Let X, X1 , X2 , . . . , be a sequence of independent and identically distributed random variables. Moreover, let N, independent of {Xn }∞ n=1 , be a geometric random variable, namely, n−1 P (N = n) = p(1 − p) , n = 1, 2, . . . , where p ∈ (0, 1). As before, denote X1,N = min{X1 , X2 , . . . , XN }. Then Kakosyan et al. (1984, Theorem 3.1.6) obtained the following characterization result by the method of intensively monotone operator. Theorem 7. Let X be a random variable with continuous distribution function F satisfying F (0) = 21 . Assume that the d
limit limx→∞ F (x)ex exists and is finite. Under the above setting for N, if X = X1,N − log p, then F is the standard logistic distribution function. The next result, a variant of Theorem 7, is due to Galambos (1992). Theorem 8. Let X be a random variable with distribution function F satisfying F (0) = 21 . In addition to the setting d
for N, assume that for two values p1∗ and p2∗ of p such that (log p1∗ )/ log p2∗ is irrational, X = X1,N − log p. Then F is the standard logistic distribution function. Acknowledgements The authors thank the Associate Editor and the referee for helpful comments which improve significantly the presentation of the paper. Especially, Theorems 5 and 6 that avoid the need to impose conditions on the cf are suggested by the Associate Editor. This work is partially supported by the National Science Council of the Republic of China under Grant NSC 92-2118-M-001-021. References Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1992. A First Course in Order Statistics. Wiley, New York. Balakrishnan, N., 1992. Handbook of the Logistic Distribution. Marcel Dekker, New York. Balakrishnan, N., Nevzorov, V.B., 2003. A Primer on Statistical Distributions. Wiley, New Jersey. Berkson, J., 1944. Application of the logistic function to bioassay. J. Amer. Statist. Assoc. 39, 357–365. David, H.A., Nagaraja, H.N., 2003. Order Statistics. third ed. Wiley, New Jersey. Galambos, J., 1992. Characterizations. In: Balakrishnan, N. (Ed.), Handbook of the Logistic Distribution. Marcel Dekker, New York, pp. 169–188. Galambos, J., Kotz, S., 1978. Characterizations of Probability Distributions. Lecture Notes in Mathematics, vol. 675. Springer, New York. Gather, U., Kamps, U., Schweitzer, N., 1998. Characterizations of distributions via identically distributed functions of order statistics. In: Balakrishnan, N., Rao, C.R. (Eds.), Order Statistics: Theory and Methods. Handbook of Statistics, vol. 16. North-Holland, Amsterdam, pp. 257–290. George, E.O., Mudholkar, G.S., 1980. Some relationships between the logistic and the exponential distributions. In: Taillie, C., Patil, G., Baldessari, B.A. (Eds.), Statistical Distributions in Scientific Work, vol. 4. D. Reidel Publishing Company, Boston, pp. 401–409.
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George, E.O., Mudholkar, G.S., 1981. A characterization of the logistic distribution by a sample median. Ann. Inst. Statist. Math. 33, 125–129. George, E.O., Mudholkar, G.S., 1982. On the logistic and exponential laws. Sankhy¯a Ser. A 44, 291–293. Gumbel, E.J., 1944. Ranges and midranges. Ann. Math. Statist. 15, 414–422. Gupta, S.S., Balakrishnan, N., 1992. Logistic order statistics and their properties. In: Balakrishnan, N. (Ed.), Handbook of the Logistic Distribution. Marcel Dekker, New York, pp. 17–48. Hirschman, I.I., Widder, D.V., 1955. The Convolution Transform. Princeton University Press, New Jersey. Hwang, J.S., Lin, G.D., 1984. Characterizations of distributions by linear combinations of moments of order statistics. Bull. Inst. Math. Acad. Sinica 12, 179–202. Johnson, N.L., Kotz, S., Balakrishnan, N., 1995. Continuous Univariate Distributions. vol. 2, second ed. Wiley, New York. Kakosyan, A.V., Klebanov, L.B., Melamed, J.A., 1984. Characterization of Distributions by the Method of Intentively Monotone Operators. Springer, New York. Kamps, U., 1998. Characterizations of distributions by recurrence relations and identities for moments of order statistics. In: Balakrishnan, N., Rao, C.R. (Eds.), Order Statistics: Theory and Methods. Handbook of Statistics, vol. 16. North-Holland, Amsterdam, pp. 291–311. Kotz, S., 1974. Characterizations of statistical distributions: a supplement to recent surveys. Internet. Statist. Rev. 42, 39–65. Kotz, S., Kozubowski, T.J., Podgórski, K., 2001. The Laplace Distribution and Generalizations. A Revisit with Applications to Communications, Economics, Engineering, and Finance. Birkhäuser, Boston. Lin, G.D., 1988. Characterizations of distributions via relationships between two moments of order statistics. J. Statist. Plann. Inference 19, 73–80. Mudholkar, G.S., George, E.O., 1978. A remark on the shape of the logistic distribution. Biometrika 65, 667–668. Nevzorov, V.B., 2001. Records: Mathematical Theory. Translations of Mathematical Monographs, vol. 194. American Mathematical Society, Rhode Island. Nevzorov, V.B., Balakrishnan, N., Ahsanullah, M., 2003. Simple characterizations of Student’s t2 -distribution. Statistician 52, 395–400. Shah, B.K., 1966. On the bivariate moments of order statistics from a logistic distribution. Ann. Math. Statist. 37, 1002–1010. Steutel, F.W., van Harn, K., 2004. Infinite Divisibility of Probability Distributions on the Real Line. Monographs and Textbooks in Pure and Applied Mathematics, vol. 259. Marcel Dekker, New York. Verhulst, P.F., 1838. Notice sur la loi que la population pursuit dans son accroissement. Correspondance Mathématique et Physique 10, 113–121. Verhulst, P.F., 1845. Recherches mathématiques sur la loi d’accroissement de la population. Nouveaux Memoires de l Academie Royale des Sciences et Belles-Lettres de Bruxelles 18, 1–45. (Mathematical researches into the law of population growth increase.)
Further Reading Balakrishnan, N., Rao, C.R., 1998. Order Statistics: Theory and Methods. Handbook of Statistics, vol. 16. North-Holland, Amsterdam.
On characterizations of the logistic distribution Gwo Dong Lina,∗ , Chin-Yuan Hub a Institute of Statistical Science, Academia Sinica, Taipei 11529, Taiwan, ROC b Department of Business Education, National Changhua University of Education, Changhua 50058, Taiwan, ROC
Received 31 October 2004; received in revised form 2 September 2006; accepted 17 April 2007 Available online 3 July 2007
Abstract We modify and extend George and Mudholkar’s [1981. A characterization of the logistic distribution by a sample median. Ann. Inst. Statist. Math. 33, 125–129] characterization result about the logistic distribution, which is in terms of the sample median and Laplace distribution. Moreover, we give some new characterization results in terms of the smallest order statistics and the exponential distribution. © 2007 Elsevier B.V. All rights reserved. MSC: primary 62E10; 62G30; 60E10 Keywords: Characterization; Order statistics; Logistic distribution; Exponential distribution; Laplace distribution
1. Introduction Almost 170 years ago, Verhulst (1838, 1845) applied the logistic distribution in his pioneering work on demography. Gumbel (1944) might be the first one who gave a genesis of the distribution: the logistic distribution arises in a purely statistical manner as the limiting distribution of standardized midranges of random samples from a symmetric distribution of exponential-type, where the midrange of a random sample means the average of the largest and the smallest sample values. It has found several important applications in many fields including survival analysis, growth model and public health (see Berkson, 1944; Johnson et al., 1995). The logistic distribution is very similar in shape to the normal distribution because its density function is symmetric and has a bell shape. Besides, the maximum difference between two distribution functions can be less than 0.01 (Mudholkar and George, 1978). So the logistic distribution has a close approximation to the normal distribution. However, Kotz in 1974 made the following comment: while the normal distribution with a more complex distribution function is well characterized, the characteristic property of the logistic distribution has not been thoroughly developed. This comment still remains valid nowadays, although some characterization results are recently available in the literature. In the next section we first review basic properties of the logistic distribution and some existing characterization results. In Section 3, we modify and extend George and Mudholkar’s (1981) characterization result about the logistic distribution, which is in terms of the sample median and Laplace distribution. Finally, in Section 4 we give some new characterization results in terms of the smallest order statistics and the exponential distribution. ∗ Corresponding author.
E-mail address: [email protected] (G.D. Lin). 0378-3758/$ - see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2007.04.030
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2. Basic properties We need some notations in the sequel. Let X be a random variable with distribution function F and characteristic function (cf) . Let {Xk }nk=1 be a random sample from F, and denote the corresponding order statistics by X1,n X2,n · · · Xn,n . The distribution function and cf of Xk,n are denoted by Fk,n and k,n , respectively. Especially, for odd sample size n = 2m − 1, denote the sample median Xm,2m−1 by X(m) and its distribution function and cf by F(m) and (m) , respectively. As usual, define the quantile function of F by F −1 (t) = inf{x: F (x) t}, t ∈ (0, 1). d
Moreover, let R be the real line and denote equality in distribution by =. We now review some basic properties of the logistic distribution. Most of the following results are available in the literature (see, e.g., Balakrishnan, 1992; Balakrishnan and Nevzorov, 2003, Chapter 22) except the general form 1,n in (h) which can be proved directly by using beta function. Proposition 1. Let X have the standard logistic distribution function F (x) = (1 + e−x )−1 , x ∈ R, and let U obey the uniform distribution on [0, 1]. Then the following properties hold. (a) The quantile function of F is F −1 (t) = log[t/(1 − t)], t ∈ (0, 1), namely, the logit transformation of t, and hence d
X = log[U/(1 − U )]. (b) The moment generating function of X is M(s) = (1 + s)(1 − s), s ∈ (−1, 1). 2 −1 = it/ sin(it) = t/ sinh(t), t ∈ R. (c) The cf (t) = M(it) = (1 + it)(1 − it) = ∞ j =1 [1 + (t/j ) ] d
(d) X = V1 − V2 , where V1 and V2 are independent random variables having the standard extreme-value distribution function H (x) = exp(−e−x ), x ∈ R. 2k ) = 2(2k), k = 1, 2, . . . , where is the Riemann zeta function defined by (e) The moments ∞ E(X −x (x) = j =1 j for x > 1. In particular, E(X 2 ) = 2 /3. (f) The distribution function F is equal to its hazard rate function, namely, F (x) = f (x)/(1 − F (x)), x ∈ R, where f is the density function of F. 2 2 (g) For n = 2m − 13, the cf of the sample median X(m) is (m) (t) = m−1 j =1 (1 + t /j )(t), t ∈ R. In particular, (2) (t) = (1 + t 2 )(t), t ∈ R. (h) The cf of the smallest order statistic X1,n is 1,n (t) = (1 + it)(n − it)/(n) = n−1 j =1 (1 − it/j )(t), t ∈ R. In particular, 1,2 (t) = (1 − it)(t) and 1,3 (t) = (1 − it/2)(1 − it)(t), t ∈ R. d
(i) For n = 3, the sample median is distributed as the midrange, namely, X2,3 =(X1,3 + X3,3 )/2. (j) The product moment is of the form: E(Xi,n Xj,n ) = (j ) + [(i) − (n + 1)][(j ) − (n − j + 1)] +
∞ B(n − i + 1, i + k) k=1
kB(i, n − i + 1)
[(j + k) − (n − j + 1)],
where the function (x) = (x)/(x) for x > 0 and B(a, b) = (a)(b)/(a + b). (k) Another representation of the product moment is 2 E(Xi,n Xj,n ) = E(Xj,n )+
j −1 k−1
(−1)
k=i =1
× E(Xj +−k,n+−k ) +
k+i
k − 1 n j − k + − 1 B(, n − k + 1) i−1 k
n j −i−1 n−i i {− (n − j + 1) (−1)k k i+k i k=0
+ [(n − j + 1) − (n − i − k + 1)][(j − i − k) − (n − j + 1)]}. () The moments of order statistics satisfy the recurrence relation k E(Xj,n ) = E(Xjk−1,n ) +
kn E(Xjk−1 −1,n−1 ). (j − 1)(n − j + 1)
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Propositions 1(j) and 1(k) are essentially due to Shah (1966), but here we have modified the arguments of the digamma and trigamma functions ( and ) (see Gupta and Balakrishnan, 1992). Moreover, in Shah’s paper, the numerator i of the fraction i/(i + k) in (k) was missing. Applying the Müntz–Szász Theorem, we are able to characterize the logistic distribution by using an infinite set of recurrence relations () (see Lin, 1988). d The logistic distribution is infinitely divisible, because, by Proposition 1(c), we have the identity: X = ∞ j =1 Lj /j , where the Lj are independent and standard Laplace random variables. This fact also follows from Proposition 1(d) and the infinite divisibility of the extreme-value distribution (see, e.g., Steutel and van Harn, 2004, p. 209). The question whether the median-midrange identity in Proposition 1(i) characterizes the logistic distribution remains open (see Arnold et al., 1992, p. 150; Kotz et al., 2001, p. 127; Nevzorov et al., 2003). We next review some existing characterizations of the logistic distribution. Theorem A given by Galambos and Kotz (1978, p. 27) applies the conditional distribution. Theorem B applying a monotone transformation of the underlying distribution is due to Galambos (1992). Theorem C follows from a well-known characterization for general distributions (see also Galambos, 1992, p. 176). The results by properties of sample medians and of minimal order statistics are investigated in the next two sections. For other characterization results, see the survey works by Galambos (1992), Gather et al. (1998) and Kamps (1998).
Theorem A. Let X be a random variable with continuous distribution function F symmetric at zero. Then the following statements are equivalent: (a) F (x) = (1 + e−x )−1 , x ∈ R, where > 0 is a constant. (b) P (−x X x|X x) = 1 − e−x for all x > 0, where > 0 is a constant. (x+y) F (x+y) (c) [1−F1−F (x)][1−F (y)] = F (x)F (y) for all x, y 0. Theorem B. Let X be a random variable and define T = 1 + e−X . If P (T uv|T u) = P (T v) for all u 1 and v 1, then X has the standard logistic distribution. Theorem C. Let X be a random variable with E(|X(k0 ) |) < ∞ for some integer k0 1. If EX(m) =0 and EXm,2m =− m1 for all mk0 , then X has the standard logistic distribution. Remark. Theorem C is a consequence of the following facts: (a) The set of expected order statistics {EX (m) , EXm,2m }∞ m=k0 uniquely determines the larger set {EX k,n : k0 k n − k0 + 1, n2k0 − 1}, which in turn characterizes the distribution of X. [In fact, we can delete finitely many elements from {EX(m) , EXm,2m }∞ m=k0 .] (b) If X and Y obey the standard logistic and exponential distributions, respectively, then EX (m) = 0 and EXm,2m = 2m 1 1 1 EY m,2m − EY m+1,2m = 2m i=m+1 i − i=m i = − m . 3. Characterizations by properties of sample medians George and Mudholkar (1981) claimed that the identity for (2) in Proposition 1(g) is a characteristic property of the logistic distribution under the assumptions: F (0) = 21 ,
f (−∞) = f (−∞) = 0
and the function t(t) is integrable,
where f is the density function of F. Their proof however requires implicitly a stronger assumption that the function t 2 (t) is integrable over R. On the other hand, the boundary conditions on the density f and its derivative are actually consequences of the stronger condition. We now modify their result as follows. Theorem 1. Let X be a random variable with distribution function F satisfying F (0) = 21 and with cf . Assume further that the function t 2 (t) is absolutely integrable over R. Then a necessary and sufficient condition for F to be the standard logistic distribution function is that the cf (2) of the sample median X2,3 satisfies the equation (2) (t) = (1 + t 2 )(t)
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for all t ∈ R, or equivalently, d
X = X2,3 + L, where X2,3 and L are independent and L has the standard Laplace distribution with cf L
(1) (t) = 1/(1 + t 2 ),
t ∈ R.
We shall give an alternative proof of Theorem 1. For convenience, denote the left and the right extremities of F by F and rF , respectively. Namely, for any > 0, we have that F (F − ) = 0, F (F + ) > 0, F (rF − ) < 1 while F (rF ) = 1. It is well known that Proposition 1(f) is a characteristic property of the logistic distribution up to location parameter. This fact will be used in the sequel and we restate it as follows. Lemma 1. Let F be an absolutely continuous distribution function with density f satisfying f (x) = F (x)(1 − F (x)), x ∈ (F , rF ). Then for some ∈ R, F (x) = (1 + e−(x− ) )−1 , x ∈ R. ∞ Proof of Theorem 1. From the absolute integrability of the function t 2 (t) it follows that −∞ |(t)| dt < ∞ and ∞ 2 −∞ |t(t)| dt < ∞ because is continuous on R and |(t)| |t(t)| |t (t)| for |t| 1. By the inversion formula for cfs, F has a continuous density function
∞ 1 e−itx (t) dt, x ∈ R, (2) f (x) = 2 −∞ and ∞ f (∞) ≡ limx→∞ f (x) = 0, f (−∞) ≡ limx→−∞ f (x) = 0 due to the Riemann–Lebesgue lemma. Since −∞ |t(t)| dt < ∞, the function f in (2) is differentiable by Lebesgue’s Dominated Convergence Theorem, and we can take differentiation under integral to obtain
−i ∞ −itx e t(t) dt, x ∈ R, f (∞) = f (−∞) = 0. f (x) = 2 −∞ ∞ Similarly, by the assumption −∞ |t 2 (t)| dt < ∞, we have
−1 ∞ −itx 2 f (x) = e t (t) dt, x ∈ R, (3) 2 −∞ which is uniformly continuous on R. The necessity part of the theorem is exactly Proposition 1(g). To prove the sufficiency part, suppose that equality (1) holds. Then the cf (2) is absolutely integrable and we have by the inversion formula for cfs
∞ 1 (x) = e−itx (2) (t) dt, x ∈ R. (4) F(2) 2 −∞ = f − f , or equivalently, F = F − F , by the boundary condition f (∞) = 0. Combining (1) through (4) yields F(2) (2) On the other hand,
F (x) t (1 − t) dt = 3F 2 (x) − 2F 3 (x), x ∈ R, F(2) (x) = 6 0
and hence it remains to solve the equation 2F 3 − 3F 2 + F − F = 0.
(5)
Setting G = F (1 − F ), we have G = (1 − 2F )F and F = G(1 − 2F ) by (5). These together imply that F F = GG , or equivalently, (F )2 = G2 by the boundary condition f (∞) = 0. Therefore, F (x) = G(x) = F (x)(1 − F (x)) > 0 for x ∈ (F , rF ) because f (x)0. Lemma 1 together with the condition F (0) = 21 completes the proof. We now extend Theorem 1 to the following. Theorem 2. Let X be a random variable with distribution function F satisfying F (0) = 21 and for some fixed integer m 2, let (m−1) be the cf of the sample median X(m−1) . Assume further that the function t 2 (m−1) (t) is absolutely
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integrable over R. Then a necessary and sufficient condition for F to be the standard logistic distribution function is 2 that (m) (t) = [1 + t 2 ](m−1) (t) for all t ∈ R holds, or equivalently, (m−1)
d
X(m−1) = X(m) + L/(m − 1),
(6)
where X(m) and L are independent and L has the standard Laplace distribution with cf L (t) = 1/(1 + t 2 ), t ∈ R. To prove Theorem 2, we need the following two lemmas, in which the first one follows from the remarkable formula for distribution functions of order statistics (see, e.g., Hwang and Lin, 1984, pp. 179, 184; Nevzorov, 2001, p. 6). Lemma 2. Let n = 2m − 1. Then the distribution function of the sample median X(m) is 2F (x)−1 2m − 1 m F(m) (x) = 2m−1 (1 − v 2 )m−1 dv for all x ∈ R. m 2 −1 Lemma 3. For m 2, the distribution functions of sample medians satisfy the relation 1 2m − 2 F(m) (x) = F(m−1) (x) + [F (x)(1 − F (x))]m−1 (2F (x) − 1) for all x ∈ R. 2 m−1 Proof. Using integration by parts, we have
u 1 2(m − 1) u (1 − v 2 )m−1 dv = (1 − v 2 )m−2 dv, u(1 − u2 )m−1 + 2m − 1 2m − 1 −1 −1 The required recurrence relation follows from Lemma 2 and the identity above.
u ∈ [−1, 1].
Proof of Theorem 2. The necessity part follows immediately from Proposition 1(g). To prove the sufficiency part, suppose that equality (6) holds. Then by the inversion formula for cfs, we have F(m) (x) = F(m−1) (x) −
1 (m − 1)2
F(m−1) (x),
x ∈ R,
(7)
and f(m−1) (∞) = F(m−1) (∞) = 0, f(m−1) (∞) = 0. It follows from (7) and Lemma 3 that 1 2m − 2 1 F(m−1) (x) = [F (x)(1 − F (x))]m−1 (2F (x) − 1) for all x ∈ R. − 2 m−1 (m − 1)2
On the other hand, Lemma 2 implies d m − 1 2m − 3 [1 − (2F (x) − 1)2 ]m−2 (2F (x) − 1), F(m−1) (x) = 2m−3 m−1 2 dx
x ∈ R.
(8)
(9)
Multiplying (8) by (9) and simplifying the result leads to 1 2m − 3 m − 1 2m − 2 − [1 − (2F − 1)2 ]2m−3 (2F − 1)(2F − 1) . F F = (m−1) (m−1) m−1 24m−4 m − 1 (m − 1)2 Therefore, we have, by the boundary condition f(m−1) (∞) = 0, 2 1 2m − 3 2 1 = 2m−2 [1 − (2F − 1)2 ]2m−2 . F(m−1) m−1 m−1 2 This implies 1 1 = 2m−2 F(m−1) m−1 2
2m − 3 [1 − (2F − 1)2 ]m−1 . m−1
(10)
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Combining (9) and (10) yields that d 1 (2F (x) − 1) = [1 − (2F (x) − 1)2 ], x ∈ (F , rF ), dx 2 or equivalently, f (x) = F (x)(1 − F (x)), x ∈ (F , rF ). Lemma 1 together with the condition F (0) = 21 completes the proof. In view of Proposition 1(g) and the inversion formula for cfs, the logistic density function f satisfies the equality: for m2, m−1
1−
j =1
D2 j2
f (x) =
(2m − 1)! {F (x)(1 − F (x))}m−1 f (x) ≡ (x), [(m − 1)!]2
x ∈ R,
(11)
where the operator symbol D stands for differentiation. In order to extend Theorem 1, George and Mudholkar (1980, Theorem 8) tried to solve Eq. (11). However, their proof was incomplete because it was ignored that the function in the RHS of (11) should be independent of the unknown f when applying Hirschman and Widder’s (1955) Theorem 9.3 on the uniqueness of the solution of a differential equation. (For example, each differentiable function on R satisfies the equation Df (x) = f (x), x ∈ R; namely, the solution of this equation is not unique.) We have instead the following corollary to Theorem 2, in which the logistic distribution is characterized by two identities given in Proposition 1(g). It is not clear whether one such identity with m 3 is enough to characterize the distribution. Corollary 1. Let X be a random variable with distribution function F satisfying F (0) = 21 and with cf . For some integer k 4, let the function t 2k−2 (t) integrable over R. Assume further that for m = k − 1, k, the be absolutely 2 /j 2 )(t), t ∈ R, or equivalently, following equality holds: (m) (t) = m−1 (1 + t j =1 d
X = X(m) +
m−1
Lj /j ,
j =1
where X(m) and Lj , j = 1, . . . , m − 1, are independent random variables and each Lj obeys the standard Laplace distribution. Then F is the standard logistic distribution function. 4. Characterizations by properties of minimal order statistics The next theorem, also due to George and Mudholkar (1982), gives another characterization of the logistic distribution in terms of the minimal order statistic of a random sample from the underlying distribution. Theorem 3. Let X be a random variable with distribution function F satisfying F (0) = 21 and with cf . Assume further that the function t(t) is absolutely integrable over R. Then a necessary and sufficient condition for F to be the standard logistic distribution function is that the cf 1,2 of the smallest order statistic X1,2 satisfies the equation d
1,2 (t) = (1 − it)(t), t ∈ R, or equivalently, X = X1,2 + Y , where X1,2 and Y are independent and Y has the standard exponential distribution with cf Y (t) = 1/(1 − it), t ∈ R. We give another interesting result below. Theorem 4. Let X have a symmetric distribution function F satisfying the conditions of Theorem 1. Assume further that Y1 and Y2 , independent of the random sample {Xk }3k=1 from F, are two independent random variables having the standard exponential distribution. Then a necessary and sufficient condition for F to be the standard logistic distribution function is that the smallest order statistic X1,3 satisfies the distribution equation d
X1,3 + 21 Y2 + Y1 = X.
(12)
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Proof. The necessity part follows from Proposition 1(h). To prove the sufficiency part, suppose that the relation (12 ) holds, namely, 1,3 (t) = (1 − 21 it)(1 − it)(t),
t ∈ R.
(13)
On the other hand, by the symmetry property of F, the cf of the largest order statistic X3,3 satisfies the equation 3,3 (t) = 1,3 (t) = (1 + 21 it)(1 + it)(t),
t ∈ R,
(14)
where w denotes the conjugate of a complex number w. Recall further the identity 1,3 + 2,3 + 3,3 = 3
(15)
which follows from the fact F1,3 + F2,3 + F3,3 = 3F. Equalities (13)–(15) together imply that 2,3 for t ∈ R. Theorem 1 finally completes the proof.
(t) = (1 + t 2 )(t)
In the next result, a variant of Theorem 3 above, we assume the differentiation of the distribution function instead of the integrability of the cf. Theorem 5. Let X be a random variable with distribution function F satisfying F (0) = 21 . Assume further that F is differentiable on R. Then a necessary and sufficient condition for F to be the standard logistic distribution function is d that X = X1,2 + Y , where X1,2 and Y are independent and Y has the standard exponential distribution. d
Proof. It is enough to prove the sufficiency part. Suppose X = X1,2 + Y , where X1,2 and Y are independent and Y has the standard exponential distribution. Then we want to prove that X obeys the standard logistic distribution. Let V = eX d
and let U obey the uniform distribution on [0, 1]. Then by the assumption and the fact that Y = − log U , we have d
V = V1,2 /U ,
(16)
where V1,2 and U are independent and V1,2 is the smallest order statistic of a random sample (of size 2) from the distribution of V. Let G be the distribution of V and G = 1 − G the survival function of V. Then it follows from (16) that
1 2 G(x) = G (ux) du, x > 0, 0
namely,
x
xG(x) =
2
G (y) dy,
x > 0.
0
2
Differentiating yields xG (x) + G(x) = G (x), x > 0, or equivalently, xH (x) = H (x), x ∈ (G , rG ), where H (x) = 1/G(x) − 1. Therefore, H (x) = x, x ∈ (G , rG ), for some constant > 0. This in turn implies that G(x) = 1/(1 + x), x > 0, or equivalently, F (x) = 1 − (1 + ex )−1 , x ∈ R. The assumption F (0) = 21 finally completes the proof. To give a general result, we impose some smoothness conditions on the exponential transformation of the underlying distribution. Theorem 6. Let n > 2 be an integer and let X be a random variable with continuous distribution function F satisfying F (0) = 21 . Assume further that the distribution function G∗ of eX is real analytic and strictly increasing in [0, ∞) and (k) that for each k 1, its kth derivative G∗ is strictly monotone in some interval [0, k ). Then a necessary and sufficient condition for F to be the standard logistic distribution function is that d
X = X1,n +
n−1
Yj /j ,
j =1
where X1,n and Yj , j = 1, . . . , n − 1, are independent and each Yj has the standard exponential distribution.
(17)
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To prove Theorem 6, we shall apply the method of intensively monotone operator introduced by Kakosyan et al. (1984). Definition 1. Let C=C[0, ∞) be a space of all real-valued functions defined and continuous in the interval [0, ∞). The notation f g for f, g ∈ C means that f (t)g(t) for all t ∈ [0, ∞). Let A be an operator mapping some set E ⊂ C into C. We say that the operator A is intensively monotone, if for any f1 and f2 , belonging to E, the condition f1 ( ) f2 ( ) for all ∈ (0, t) implies that (Af 1 )( ) (Af 2 )( ) for ∈ (0, t) and, in addition, the condition f1 ( ) > f2 ( ) for all ∈ (0, t) implies that (Af 1 )(t) > (Af 2 )(t). Definition 2. Let E ⊂ C and {f }∈ be a family of elements of E. We say that the family {f }∈ is strongly E-positive if the following conditions hold: (1) for any f ∈ E there are t0 > 0 and 0 ∈ such that f (t0 ) = f0 (t0 ); (2) for any f ∈ E and any ∈ either f (t) = f (t) for all t ∈ [0, ∞), or there is > 0 such that the difference f (t) − f (t) does not vanish (preserves its sign) in the interval (0, ]. Lemma 4. Let A be an intensively monotone operator on E ⊂ C and let {f }∈ be a strongly E-positive family. Assume that Af = f for all ∈ . Then the condition Af = f , where f ∈ E, implies that there is ∈ such that f = f . In other words, all solutions of the equation Af = f , belonging to E, coincide with elements of the family {f }∈ . Proof. See Kakosyan et al. (1984, pp. 2–3).
Lemma 5. Let f and g be two functions real analytic and strictly monotone in [0, ∞). Assume that for each n 1, the nth derivatives f (n) and g (n) are strictly monotone in some interval [0, n ). Let {xn }∞ n=1 be a sequence of positive real numbers converging to zero. If f (xn ) = g(xn ), n = 1, 2, . . . , then f = g. Proof. Note first that f (0) = g(0) by the assumption f (xn ) = g(xn ) and the continuity of f and g. Next, since f and g are real analytic in [0, ∞), we have the Taylor’s expansions: f (x) =
∞ f (n) (0) n=0
n!
xn
and
g(x) =
∞ g (n) (0) n=0
n!
xn,
x 0.
We shall prove that f (n) (0) = g (n) (0), n = 1, 2, . . . , under the assumptions. By the Generalized Mean-Value Theorem, there exists yn ∈ (0, xn ) such that f (yn )(g(xn ) − g(0)) = g (yn )(f (xn ) − f (0)), n = 1, 2, . . . . Therefore, f (yn ) = g (yn ), n = 1, 2, . . . , and yn converges to zero as n goes to infinity. This in turn implies that f (0) = g (0). Again, there exists zn ∈ (0, yn ) such that f (zn )(g (yn ) − g (0)) = g (zn )(f (yn ) − f (0)), n = 1, 2, . . . . This implies that f (zn ) = g (zn ) for sufficiently large n, due to the monotone property of f and g in the interval [0, 1 ). Moreover, zn converges to zero as n goes to infinity and f (0) = g (0). By induction, we conclude that f (n) (0) = g (n) (0), n1, which completes the proof. Lemma 6. Let Y1 , Y2 , . . . , Yn be independent random variables having the standard exponential distribution. Then d Yn,n = nj=1 Yj /j . Proof. See, e.g., David and Nagaraja (2003, pp. 17–18). Proof of Theorem 6. The necessity part is exactly Proposition 1(h). To prove the sufficiency part, suppose that the condition (17) holds. Then we want to prove that X obeys the standard logistic distribution. Let V = eX and Uj = e−Yj , j = 1, . . . , n − 1, each Uj obeying the uniform distribution on [0, 1]. It follows from condition (17) and Lemma 6 that d
V = V1,n /U1,n−1 ,
(18)
G.D. Lin, C.-Y. Hu / Journal of Statistical Planning and Inference 138 (2008) 1147 – 1156
1155
where V1,n is the smallest order statistic of a random sample (of size n) from the distribution of V, and is independent of U1,n−1 = min{U1 , . . . , Un−1 }. By identity (18), the survival function G∗ of V satisfies
1 n G∗ (x) = (n − 1) G∗ (ux)(1 − u)n−2 du, x > 0. (19) 0
Define E to be the set of all survivor functions G real analytic and strictly decreasing in [0, ∞), with kth derivative (k) G being strictly monotone in some interval [0, k ) for each k 1. Denote G (x) = 1/(1 + x), x > 0, where ∈ = (0, ∞). Further, define the operator A on E by
1 n AG(x) = (n − 1) G (ux)(1 − u)n−2 du, x > 0, G ∈ E. 0
It is seen that A is an intensively monotone operator on E and the family {G }∈ is strongly E-positive by Lemma 5. Moreover, AG = G for all ∈ by the necessity part of the theorem. Finally, by Lemma 4 and identity (19) (which means AG∗ = G∗ ), we conclude that G∗ = G for some ∈ . The condition F (0) = 21 then implies = 1 and completes the proof. Finally, we consider the case of random sample size. Let X, X1 , X2 , . . . , be a sequence of independent and identically distributed random variables. Moreover, let N, independent of {Xn }∞ n=1 , be a geometric random variable, namely, n−1 P (N = n) = p(1 − p) , n = 1, 2, . . . , where p ∈ (0, 1). As before, denote X1,N = min{X1 , X2 , . . . , XN }. Then Kakosyan et al. (1984, Theorem 3.1.6) obtained the following characterization result by the method of intensively monotone operator. Theorem 7. Let X be a random variable with continuous distribution function F satisfying F (0) = 21 . Assume that the d
limit limx→∞ F (x)ex exists and is finite. Under the above setting for N, if X = X1,N − log p, then F is the standard logistic distribution function. The next result, a variant of Theorem 7, is due to Galambos (1992). Theorem 8. Let X be a random variable with distribution function F satisfying F (0) = 21 . In addition to the setting d
for N, assume that for two values p1∗ and p2∗ of p such that (log p1∗ )/ log p2∗ is irrational, X = X1,N − log p. Then F is the standard logistic distribution function. Acknowledgements The authors thank the Associate Editor and the referee for helpful comments which improve significantly the presentation of the paper. Especially, Theorems 5 and 6 that avoid the need to impose conditions on the cf are suggested by the Associate Editor. This work is partially supported by the National Science Council of the Republic of China under Grant NSC 92-2118-M-001-021. References Arnold, B.C., Balakrishnan, N., Nagaraja, H.N., 1992. A First Course in Order Statistics. Wiley, New York. Balakrishnan, N., 1992. Handbook of the Logistic Distribution. Marcel Dekker, New York. Balakrishnan, N., Nevzorov, V.B., 2003. A Primer on Statistical Distributions. Wiley, New Jersey. Berkson, J., 1944. Application of the logistic function to bioassay. J. Amer. Statist. Assoc. 39, 357–365. David, H.A., Nagaraja, H.N., 2003. Order Statistics. third ed. Wiley, New Jersey. Galambos, J., 1992. Characterizations. In: Balakrishnan, N. (Ed.), Handbook of the Logistic Distribution. Marcel Dekker, New York, pp. 169–188. Galambos, J., Kotz, S., 1978. Characterizations of Probability Distributions. Lecture Notes in Mathematics, vol. 675. Springer, New York. Gather, U., Kamps, U., Schweitzer, N., 1998. Characterizations of distributions via identically distributed functions of order statistics. In: Balakrishnan, N., Rao, C.R. (Eds.), Order Statistics: Theory and Methods. Handbook of Statistics, vol. 16. North-Holland, Amsterdam, pp. 257–290. George, E.O., Mudholkar, G.S., 1980. Some relationships between the logistic and the exponential distributions. In: Taillie, C., Patil, G., Baldessari, B.A. (Eds.), Statistical Distributions in Scientific Work, vol. 4. D. Reidel Publishing Company, Boston, pp. 401–409.
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George, E.O., Mudholkar, G.S., 1981. A characterization of the logistic distribution by a sample median. Ann. Inst. Statist. Math. 33, 125–129. George, E.O., Mudholkar, G.S., 1982. On the logistic and exponential laws. Sankhy¯a Ser. A 44, 291–293. Gumbel, E.J., 1944. Ranges and midranges. Ann. Math. Statist. 15, 414–422. Gupta, S.S., Balakrishnan, N., 1992. Logistic order statistics and their properties. In: Balakrishnan, N. (Ed.), Handbook of the Logistic Distribution. Marcel Dekker, New York, pp. 17–48. Hirschman, I.I., Widder, D.V., 1955. The Convolution Transform. Princeton University Press, New Jersey. Hwang, J.S., Lin, G.D., 1984. Characterizations of distributions by linear combinations of moments of order statistics. Bull. Inst. Math. Acad. Sinica 12, 179–202. Johnson, N.L., Kotz, S., Balakrishnan, N., 1995. Continuous Univariate Distributions. vol. 2, second ed. Wiley, New York. Kakosyan, A.V., Klebanov, L.B., Melamed, J.A., 1984. Characterization of Distributions by the Method of Intentively Monotone Operators. Springer, New York. Kamps, U., 1998. Characterizations of distributions by recurrence relations and identities for moments of order statistics. In: Balakrishnan, N., Rao, C.R. (Eds.), Order Statistics: Theory and Methods. Handbook of Statistics, vol. 16. North-Holland, Amsterdam, pp. 291–311. Kotz, S., 1974. Characterizations of statistical distributions: a supplement to recent surveys. Internet. Statist. Rev. 42, 39–65. Kotz, S., Kozubowski, T.J., Podgórski, K., 2001. The Laplace Distribution and Generalizations. A Revisit with Applications to Communications, Economics, Engineering, and Finance. Birkhäuser, Boston. Lin, G.D., 1988. Characterizations of distributions via relationships between two moments of order statistics. J. Statist. Plann. Inference 19, 73–80. Mudholkar, G.S., George, E.O., 1978. A remark on the shape of the logistic distribution. Biometrika 65, 667–668. Nevzorov, V.B., 2001. Records: Mathematical Theory. Translations of Mathematical Monographs, vol. 194. American Mathematical Society, Rhode Island. Nevzorov, V.B., Balakrishnan, N., Ahsanullah, M., 2003. Simple characterizations of Student’s t2 -distribution. Statistician 52, 395–400. Shah, B.K., 1966. On the bivariate moments of order statistics from a logistic distribution. Ann. Math. Statist. 37, 1002–1010. Steutel, F.W., van Harn, K., 2004. Infinite Divisibility of Probability Distributions on the Real Line. Monographs and Textbooks in Pure and Applied Mathematics, vol. 259. Marcel Dekker, New York. Verhulst, P.F., 1838. Notice sur la loi que la population pursuit dans son accroissement. Correspondance Mathématique et Physique 10, 113–121. Verhulst, P.F., 1845. Recherches mathématiques sur la loi d’accroissement de la population. Nouveaux Memoires de l Academie Royale des Sciences et Belles-Lettres de Bruxelles 18, 1–45. (Mathematical researches into the law of population growth increase.)
Further Reading Balakrishnan, N., Rao, C.R., 1998. Order Statistics: Theory and Methods. Handbook of Statistics, vol. 16. North-Holland, Amsterdam.