Zero regression characterizations of natural exponential families generated by Levy stable laws - a complementary

Volume 13, No. 3 (2004), pp.

Allerton Press, Inc.

MATHEMATICAL

METHODS

OF

STATISTICS

ZERO REGRESSION CHARACTERIZATIONS OF NATURAL EXPONENTIAL FAMILIES GENERATED ´ BY LEVY STABLE LAWS — A COMPLEMENTARY Sh. K. Bar-Lev1 , D. Bshouty2 , and F. A. Van der Duyn Schouten3 1 Department

of Statistics, University of Haifa Haifa 31905, Israel E-mail: [email protected]

2 Department

of Mathematics, Technion-Israel Institute of Technology Haifa 32000, Israel

3 Center

for Economic Research, University of Tilburg 5000 LE Tilburg, The Netherlands

Bar-Lev and Stramer [2] presented a series of characterizations by zero regression properties of the class of steep natural exponential families having power variance functions, a subclass of which coincides with that of natural exponential families generated by L´ evy stable laws (NEFGSL’s) with stable index ρ ∈ (0, 1) ∪ {2}. The purpose of this note is to provide zero-regression characterizations for the complementary subclass of NEF-GSL’s with stable index ρ ∈ [1, 2). This will then complete the zero-regression characterizations of the whole class of NEF-GSL’s with stable index ρ ∈ (0, 2]. Key words: constant regression, characterization, exponential dispersion model, natural exponential family, stable distribution. 2000 Mathematics Subject Classification: 62E10.

1. Introduction Since the seminal study of Laha and Lukacs [11] characterizing all distributions for which the regression of a quadratic form on the sample mean is constant, numerous works on zero (or constant) regression characterizations have appeared in the literature (e.g., Kagan, Linnik and Rao [9], Gordon [6], Heller [7], Seshadri [20], BarLev and Stramer [2] (hereafter, BLS), Wesolowski [22, 23], Rao and Shanbhag [19], and Fosam and Shanbhag [5]). In general, such characterizations have the following form. Let X = (X1 , . . . , Xn ) be a random sample with a common distribution F c °2003 by Allerton Press, Inc. Authorization to photocopy individual items for internal or personal use, or the internal or personal use of specific clients, is granted by Allerton Press, Inc. for libraries and other users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service, provided that the base fee of $50.00 per copy is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923.

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Sh. K. Bar-Lev, D. Bshouty, and F. A. Van der Duyn Schouten

and let Si = Si (X), i = 1, 2, be two (possibly vector-valued) statistics such that the regression of S2 on S1 is zero (or constant). If F is the only distribution for which such a property holds, we say that F is characterized by the zero regression of S2 on S1 . In many of the characterizations of this type, S1 is either the sample mean or the sum of sample elements and S2 is a polynomial in the Xi ’s. Other authors (e.g., Heller [7]) have derived further characterizations of the type considered by Laha and Lukacs [11] by using higher sample moments. Zero regression characterizations when S1 is non-polynomial have been obtained in several works, e.g., Seshadri [20] derived a series of characterizations of the inverse Gaussian distribution; Wesolowski [22, 23] derived zero regression characterizations for the normal, Poisson-type, binomial, and negative binomial distributions. BLS presented a series of characterizations of steep natural exponential families (NEF’s) having power variance functions (NEF-PVF’s), a subclass of which coincides with that of NEF’s generated by L´evy stable laws (NEF-GSL’s) with stable index ρ ∈ (0, 1) ∪ {2}. The purpose of this note is to provide zero regression characterizations for the complementary subclass of NEF-GSL’s with stable index ρ ∈ [1, 2). This will then complete the zero regression characterizations of the whole class of NEF-GSL’s with stable index ρ ∈ (0, 2]. This note is organized as follows. In Section 2 we provide some preliminaries on NEF’s and their associated variance functions (VF’s) and describe the relations between the two classes of NEF-PVF’s and NEF-GSL’s. In Section 3 we consider the subclass of NEF-GSL’s with stable index ρ ∈ [1, 2) and derive some required cumulants’ relations and expressions for the corresponding characteristic functions (c.f.’s). The main characterization theorems are presented in Section 4. 2. NEF’s, NEF-PVF’s, and NEF-GSL’s 2.1. NEF’s and their associated VF’s. Let ν be a non-degenerate positive measure on R with Laplace transform Z Lν (θ) = eθx ν(dx). R

Denote by Dν the effective domain of ν, i.e., Dν = {θ ∈ R : Lν (θ) < ∞}. Assume that Θ = Θν = int Dν 6= ∅ and let kν (θ) = log Lν (θ) be the cumulant transform of Lν . The NEF F generated by ν is defined by probabilities n ¡ o ¢ F = F(ν) = P θ, ν(dx) = exp{θx − kν (θ)}ν(dx), θ ∈ Θν . The cumulant transform kν is strictly convex and real analytic on Θν and kν0 (θ), θ ∈ Θν , is the mean function of F. The open interval ΩF = kν0 (Θν ) is called the mean domain of F. Since the mapping θ 7−→ kν0 (θ) is one-to-one, its inverse function ψν : ΩF −→ Θν is well defined. Hence the mapping m 7−→ P (m, F) = P (ψν (m), ν) is one-to-one from ΩF onto F and is called the mean domain parametrization of F. The VF of the NEF F is defined as follows. The variance of the probability P (m, F) is VF (m) = 1/ψν0 (m) = kν00 (θ). The mapping m 7−→ VF (m) from ΩF into

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R+ is called the VF of F. In fact the VF of an NEF F is a pair (VF , ΩF ). It uniquely determines an NEF within the class of NEF’s. The mean domain ΩF is the largest open interval on which VF is positive real analytic. Given a VF, θ and kν are obtained as the primitives of 1/VF (m) and m/VF (m), respectively, i.e., Z Z ¡ ¢ ¡ ¢ (1) θ = dm/VF (m) and kν ψν (m) = m/VF (m) dm. Morris [18] characterized all NEF’s having quadratic VF’s and Letac and Mora [12] all NEF’s with strict cubic VF’s. The latter reference contains also a valuable and rigorous description of NEF’s and their associated VF’s. Let Cν denote the convex hull of the support of ν (hereafter, convex-support). Then Cν is also the common convex-support of all members of F. The NEF F is said to be steep if and only if Cν = ΩF . In the sequel, when no confusion is made, we shall suppress the dependence of Θν , kν , ψν , Cν , VF , and ΩF on ν and F and write Θ, k, ψ, C, V , and Ω instead. 2.2. The class of NEF-PVF’s. An NEF F is said to have a power VF V if (2)

V (m) = αmγ

for some

α > 0,

γ ∈ R,

where γ is termed the power parameter. The class of NEF-PVF’s was discovered in different contexts by Tweedie [21], Bar-Lev and Enis [1], and Jorgensen [8]. This large class contains the following families: normal (γ = 0), Poisson type (γ = 1), gamma (γ = 2), compound Poisson NEF’s generated by all gamma distributions (1 < γ < 2), and NEF’s generated by L´evy stable laws with stable index ρ ∈ (0, 1) ∪ (1, 2) (γ ∈ (2, ∞) ∪ (−∞, 0), respectively). The inverse Gaussian NEF corresponds to ρ = 1/2. No NEF’s exist if γ ∈ (0, 1). 2.3. NEF-GSL’s. L´evy stable laws are mainly distinguished by their stable index ρ ∈ (0, 2] (see Lukacs [15]). NEF-GSL’s with stable index ρ ∈ (0, 1) ∪ (1, 2] coincide with NEF-PVF’s with power parameter γ = (2 − ρ)/(1 − ρ). Accordingly, for completing the classification of NEF-GSL’s, it remains to consider the L´evy stable law with index ρ = 1. Such a L´evy stable law does not generate an NEFPVF, but rather an NEF with exponential VF (NEF-EVF) of the form (3)

(V, Ω) = (αem , R),

α > 0.

The NEF-EVF is discussed in the context of exponential dispersion models by Jorgensen [8] and Burridge [3]. It also appears as a limit of power VF’s, VFn (m) = (1 + m/n)n as n → ∞ (see Mora [17] and also Letac and Mora [12]). The following Table 1 displays some summarizing features regarding NEF-GSL’s. These results are mostly extracted from Bar-Lev and Enis [1] and Jorgensen [8] as related to NEF-PVF’s. Bar-Lev and Enis [1], in their characterizations of reproducible NEF’s, studied steep NEF’s only. Consequently, they were unable to reveal the non-steep NEFPVF’s with power parameter γ < 0 (or, equivalently, NEF-GSL’s with stable index ρ ∈ (1, 2)), neither the steep NEF-EVF (NEF-GSL with index ρ = 1), which

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Sh. K. Bar-Lev, D. Bshouty, and F. A. Van der Duyn Schouten

Table 1. Some classifications of NEF-GSL’s ρ ρ ∈ (0, 1) ρ=1 ρ ∈ (1, 2) ρ=2 NEF’s name NEF-PVF NEF-EVF NEF-PVF NEF-PVF (or Normal NEF) Ω R+ R R+ R C R+ R R R Θ R− R− R+ R Steepness steep steep non-steep steep does not posses a PVF. Based on Bar-Lev and Enis [1] characterizations, BLS presented a series of characterizations by zero regression properties of steep NEFPVF’s (NEF-GSL’s with ρ ∈ (0, 1) ∪ {2}). Consequently, in order to accomplish the goal of characterizing all NEF-GSL’s by zero regression properties, we have to provide such characterizations for NEF-GSL’s with stable index ρ ∈ [1, 2). In the next section we focus on such NEF-GSL’s and derive the corresponding c.f.’s as well as some general relations between their cumulants. Such relations are required for zero regression characterization aspects. 3. NEF-GSL’s with Index ρ ∈[1,2): cumulants’ relations and c.f.’s 3.1. General aspects. The particular choice of cumulants’s relations, which hold for certain NEF’s, is crucial for solving zero regression characterization problems for those NEF’s. In general, for a given NEF F with VF (V, Ω), the corresponding cumulants are derived successively by (4)

κj = κj (m) = κ2 (m)κ0j−1 (m),

κ1 (m) = m,

κ2 (m) = V (m),

j ≥ 2.

This may yield various relations between the cumulants of F. Such relations can always be given the general form g(κi1 , . . . , κir ) = 0, where (κi1 , . . . , κir ) is a given set of r cumulants. Among these forms one should choose those which are polynomial in the κij ’s. If g is such a polynomial then one can construct an unbiased polynomial statistic gb for g in a manner similar to that described in Lukacs Pn and Laha [16], pp. 99–100. By requiring gb to have zero regression on L1 = i=1 Xi , one obtains a differential equation, one solution of which is the c.f. associated with F. To show that the latter c.f. is the only feasible solution, one may have to impose some additional conditions. The number of conditions added depends on g. For further details the reader is referred to BLS. We have found, as in BLS, that cumulants’ relations of the type g(κj , κj+1 , κj+2 ) = 0, embracing three consecutive cumulants, lead to relatively easily solvable differential equations in terms of the derivatives of the associated c.f. Consequently, for the subclass of NEF-GSL’s with index ρ ∈ [1, 2), we shall aim at choosing cumulants’ relations of such a type. We first consider the case of NEF-GSL’s with index ρ ∈ (1, 2) and then with ρ = 1.

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3.2. NEF-GSL’s with index ρ ∈ (1, 2). The results for this case are mostly extracted from BLS. These authors derived cumulants’ relations (BLS, Eq. 2.3) and provided expressions (BLS, Eqs. 2.2 and 2.5) for the c.f.’s of NEF-GSL’s with index ρ ∈ (0, 1) in terms of (κr , κr+1 ), r ≥ 1, instead of the original parameters (α, θ). Fortunately, such relations and expressions are valid for the present case too, with one distinction indicated below. Accordingly, the appropriate cumulants’ relations are . Sr (γ) = κr κr+2 − βr (γ)κ2r+1 = 0,

(5) where

γ < 0,

r ≥ 1,

£ ¤±£ ¤ βr (γ) = rγ − (r − 1) (r − 1)γ − (r − 2) > 0.

In contrast to NEF-GSL’s with index ρ ∈ (0, 1), where all cumulants are positive, the cumulants of NEF-GSL’s with index ρ ∈ (1, 2) have successive alternate signs satisfying κ2r > 0, κ2r+1 < 0, for r ≥ 1. Let fρ (t) denote the c.f. of an NEF-GSL distribution with index ρ and set hρ (t) = log fρ (t). Then, expressions for hρ (t), ρ ∈ (1, 2), in terms of (α, θ) and (κr , κr+1 ), r ≥ 1, are given, respectively, by (6)

hρ (t) = k(θ + it) − k(θ) = [α(2 − γ)]−1 [α(1 − γ)θ]ρ [(1 + it/θ)ρ − 1]

with ρ ∈ (1, 2), γ ∈ R− , θ ∈ R+ , and (7)

hρ (t) =

µ r−1 Y j=0

¶ βjj+1

1 κr+1 r 2 − γ κrr+1

·µ

1−γ κr+1 1 + it (r − 1)γ − (r − 2) κr

¶ρ

¸ −1 ,

where β0 ≡ 1 and βr = βr (γ). 3.3. An NEF-GSL with index ρ = 1. Considering the VF in (3) and employing (1) gives k(θ) = θ(1 − log α) − θ log(−θ),

θ ∈ R− ,

and h1 (t) = log f1 (t) = it(1 − log α) + θ log(−θ) − (θ + it) log(−(θ + it)), Using (3) and (4) yields an expression for the rth cumulant as (8)

kr = (r − 2)!αr−1 exp{(r − 1)m},

implying that (9)

kr = (r − 2)kr−1 k2 ,

r ≥ 3.

r ≥ 2,

θ ∈ R− .

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Sh. K. Bar-Lev, D. Bshouty, and F. A. Van der Duyn Schouten

Note that except for κ1 all other higher degree cumulants are positive. By using (8), we obtain the following cumulants’ relations (10)

. Q1 = k3 − k22 = 0,

. Qr = (r − 1)kr2 − (r − 2)kr−1 kr+1 = 0,

r ≥ 3.

In a manner similar to (6) and (7), we use (9) to express h1 (t) in terms of (κ1 , κ2 ) and (κ1 , κr−1 , κr ). This gives h1 (t) = it(1 + κ1 ) + κ−1 2 (1 − itκ2 ) log(1 − itκ2 )

(11) and, for r ≥ 3,

µ ¶ µ ¶ (r − 2)kr−1 1 kr 1 kr (12) h1 (t) = it(1+κ1 )+ 1−it log 1−it . kr (r − 2) kr−1 (r − 2) kr−1 4. Characterization Theorems If F is a distribution function, we denote by f (t), h(t) = log f (t), and κi the c.f., the cumulant c.f., and the ith cumulant, respectively, associated with F . The jth derivative of a function u is denoted by u(j) . A random sample taken from F is denoted, as before, by X = (X1 , . . . , Xn ). As outlined in Section 3.1, the main tools behind characterizations by zero regression properties were established in Laha and Lukacs [11] (see also Lukacs and Laha [16]). Their basic idea is to construct an unbiased polynomial statistic for a cumulants’ relation satisfied by the distribution to be characterized. For this they derived a general expression of an unbiased polynomial statistic Tbq for an arbitrary cumulant kq , q ≥ 1. Analogously, BLS derived a general expression of an unbiased polynomial statistic Tbq,p for a product of two arbitrary cumulants kq kp , 1 ≤ q ≤ p. The sample size n required to have meaningful Tbq and Tbq,p should satisfy n ≥ q and n ≥ q + p, respectively. Since the expressions for the cumulant parameters Sr (γ) in (5) and the Qi ’s in (10) contain a product of at most two cumulants, it follows that Tbq and Tbq,p suffice for constructing unbiased polynomial estimates, Sbr (γ) and b i , for these parameters. Consequently, we have Q (13)

Sbr (γ) = Tbr,r+2 − βr (γ)Tbr+1,r+1 , b 1 = Tb3 − Tb2,2 , Q

γ < 0,

r ≥ 1,

b r = (r − 1)Tbr,r − (r − 2)Tbr−1,r+1 , Q

r ≥ 3,

where, for example, denoting n(m) = n(n − 1) . . . (n − m + 1),

(14)

1X 3 3 X 2 X Tb3 = Xi − Xi Xj2 + Xi Xj Xk , n n(2) n(3) 1 X 2 X 1 X Tb2,2 = Xi Xj2 − Xi Xj Xk2 + Xi Xj Xk Xl , n(2) n(3) n(4)

and where the summations in (14) are taken over all distinct indices ranging between 1 to n. Further examples of Sbr (γ) for some specific values of r are given in BLS.

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The following lemma is a direct consequence of Lemma 3.1 of BLS. Its first part coincides with Corollary 3.1 of BLS, which was stated there for Sr (γ) with γ ≥ 1. The statement holds, however, also for Sr (γ) with γ < 0. Lemma 1. Let X = (X1 , . . . , Xn ) be a random sample taken from a distribution F and let Nδ be some δ-neighborhood of the origin. Consider the polynomial statistics defined in (13). Then for t ∈ Nδ , the following statements hold: (i) If the (r + 2)th cumulant of F exists and n ≥ 2r + 2, r ≥ 1, then n £ ¤2 o E(Sbr (γ)eitL1 ) = i−(2r+2) f n (t) h(r) (t)h(r+2) (t) − βr (γ) h(r+1) (t) . (ii) If the third cumulant of F exists and n ≥ 4, then n £ ¤ o b 1 eitL1 ) = f n (t) i−3 h(3) (t) − i−4 h(2) (t) 2 . E(Q (iii) If the (r + 1)th cumulant of F exists and n ≥ 2r, r ≥ 3, then n o £ ¤ b r eitL1 ) = i−2r f n (t) (r − 1) h(r) (t) 2 − (r − 2)h(r−1) (t)h(r+1) (t) . E(Q b i are unbiased for Moreover, when the conditions in (i)–(iii) hold, then Sbr (γ) and Q Sr (γ) and Qi , respectively. Theorems 1 and 2 below provide, up to translations or convolutions with a normal law, zero regression characterizations of NEF-GSL’s distributions with stable index ρ ∈ [1, 2). The two theorems treat NEF-GSL’s with indices ρ ∈ (1, 2) and ρ = 1, respectively. The proofs of both theorems provide only their ‘necessity’ parts, as the ‘sufficiency’ parts are easily verified. Theorem 1 resembles Theorems 3.2 and 3.3 in BLS but is formulated with the required changes needed for NEF-GSL’s with index ρ ∈ (1, 2). Some parts of its proof are, however, slightly different, especially due to the fact that NEF-GSL’s with index ρ ∈ (1, 2) are supported on R, whereas those with index ρ ∈ (0, 1) are supported on R+ . This fact prevents the use of a fundamental result by Kawata [10], Theorem 11.5.6 (see also BLS, Eq. 4.16) that has been utilized in BLS for the case ρ ∈ (0, 1). We shall need, however, an important result of Christensen [4], which is presented in the following lemma. Lemma 2. Let f (z) be an infinitely divisible c.f. regular in a half-plane Im z > Pk −α, α > 0, and set g(z) = f (z) exp{Pk (z)}, where Pk (z) = n=0 cn z n is a polynomial of degree k. If k > 3 or k = 3 and Im c3 ≤ 0 then g(z) cannot be a c.f. Accordingly, characterizations of NEF-GSL’s distributions with stable index ρ ∈ [1, 2) mean that the distributions being characterized possess c.f.’s of the form (15)

f (t; ρ) = exp{hρ (t) + Pk (t)},

where hρ (t) is the cumulant c.f. of an NEF-GSL distribution with stable index ρ ∈ [1, 2) and Pk (t) is a polynomial with degree k ≤ 2. This resembles characterization theorems of the form (15) obtained by Lukacs [13, 14] in which h is the cumulant

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Sh. K. Bar-Lev, D. Bshouty, and F. A. Van der Duyn Schouten

c.f. of a Poisson distribution. Clearly if k = 2, then (15) is the c.f. of the convolution of a stable distribution with index ρ and a normal law. Theorem 1. Let F be a non-degenerate distribution supported on R and let X = (X1 , . . . , Xn ) be a random sample taken from F . Then, we have the following statements. (i) Assume that F has a finite third moment such that κ1 > 0. For n ≥ 4 and ρ ∈ (1, 2) (or, equivalently, γ < 0), Sb1 (γ) has a zero regression on L1 iff F is an NEF-GSL distribution with stable index ρ. (ii) Assume that F has a finite fourth moment such that κ1 > 0. For n ≥ 6 and ρ ∈ (1, 2), Sb2 (γ) has a zero regression on L1 iff, up to a translation, F is an NEF-GSL distribution with stable index ρ. If in addition the first three cumulants of F satisfy the relation κ1 κ3 = γκ22 , then Sb2 (γ) has a zero regression on L1 iff F is an NEF-GSL distribution with stable index ρ. (iii) Assume that for a fixed r ≥ 3, F has a finite moment of order r + 2 such that κr+1 /κr < 0. Let n ≥ 2r + 2 and ρ ∈ (1, 2) (γ < 0). Then, for r = 3, Sb3 (γ) has a zero regression on L1 iff F is a convolution of an NEF-GSL distribution with stable index ρ and a normal law. If r > 3 and F satisfies r−2 Y

£ ¤r−3 r−2 (jγ − (j − 1))κ3 κr−3 κr , r+1 = (r − 1)γ − (r − 2)

j=2

then Sbr (γ) has a zero regression on L1 iff F is a convolution of an NEF-GSL distribution with stable index ρ and a normal law. Proof. Fix ρ ∈ (1, 2) (or γ < 0). We prove both parts of the theorem simultaneously and assume that Sbr (γ), r ≥ 1, has a zero regression on L1 . The proof can now be conducted exactly as in Theorems 3.2 and 3.3 in BLS up to the Eqs. 4.26. The latter equation presents h(t) as (16)

lr κr+1 r h(t) = 2 − γ κrr+1

t ∈ Nδ , where (17)

P0 j=1

·µ

1−γ κr+1 1+ it (r − 1)γ − (r − 2) κr

¶ρ

¸ −1 +

r−1 X cj (it)j j=1

j!

,

= 0 and

lr =

r−1 Y

βjj+1 ,

ρ(j) = ρ(ρ − 1) . . . (ρ − (j − 1)),

j=0

cj = κj − ρ(j)

(1 − γ)j lr κr+1−j r , 2 − γ [(r − 1)γ − (r − 2)]j κr−j r+1

for j = 1, . . . , r − 1. Since h(t) in (16) is analytic in a neighborhood of the origin, it can be extended to the whole real line. Consequently, h(z), z = t + iy, is analytic in the half-plane in which (r − 1)γ − (r − 2) κr = −α, α > 0. Im z > − 1−γ κr+1

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We now distinguish between three cases: r = 1, r = 2, and r ≥ 3, in accordance with the three parts of the theorem. If r = 1, then h(t) in (16) coincides with hρ (t) in (7) with r = 1, the c.f. of an NEF-GSL distribution with index ρ ∈ (1, 2). If r = 2, then it follows from (16) that (18)

h(t) =

γ 2 κ32 (2 − γ)κ23

·µ 1+

1 − γ κ3 it γ κ2

¶ρ

¶ ¸ µ γκ2 − 1 + it κ1 − 2 , κ3

where the first term on the right-hand side of (18) coincides with hρ (t) in (7) with r = 1. Hence the corresponding F is a translation of an NEF-GSL distribution with index ρ ∈ (1, 2). Furthermore, if the first three cumulants of F also satisfy κ1 κ3 = γκ22 , then F is an NEF-GSL distribution. This proves the second part of the theorem. For the third part let r ≥ 3. Then the exponent of h(t) in (16) is the product of a c.f. of a stable distribution with index ρ ∈ (1, 2) and the exponent of a polynomial. If r = 3, then the result follows immediately from (17) with c1 = κ1 − γ −1 (2γ − 1)2 κ33 /κ24

and c2 = κ2 − γ −1 (2γ − 1)κ23 /κ4 .

If r > 3, then by Lemma 2 cj = 0 for j ≥ 4, but since by (17) c3 = κ3 −

[(r − 1)γ − (r − 2)]r−3 κr−2 r Qr−2 r−3 , κ r+1 j=2 (jγ − (j − 1))

the assumption of the theorem implies that c3 = 0 and hence the desired result with r−(i−1)

ci = κi −

γ i−1 [(r − 1)γ − (r − 2)]r−i κr , Qr−2 κr−i r+1 j=1 (jγ − (j − 1))

i = 1, 2.

¤

Theorem 2. Let F be a non-degenerate distribution and let X = (X1 , . . . , Xn ) be a random sample taken from F . Then, we have the following statements. b 1 has a (i) Assume that F has a finite third moment with κ3 > 0. For n ≥ 4, Q zero regression on L1 iff F is an NEF-GSL distribution with stable index ρ = 1. (ii) Assume that F has a finite moment of order r + 1 (r ≥ 3) such that κi > 0, i = 3, . . . , r + 1 and let n ≥ 2r. Then, b 3 has a zero regression on L1 iff F is an NEF-GSL distribution with stable (a) Q index ρ = 1; (b) if r = 4, or if r > 4 and µ (19)

κ3 =

j−2 κj

¶j−4

κj−3 j−1

for some

j ≥ 5,

b r has a zero regression on L1 iff F is a convolution of an NEF-GSL distrithen Q bution having stable index ρ = 1 with a normal law.

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Sh. K. Bar-Lev, D. Bshouty, and F. A. Van der Duyn Schouten

b 1 has Proof. (i) Let Nδ be some δ-neighborhood of the origin and assume that Q a zero regression on L1 . Then by Lemma 1 (ii) ih(3) (t) − [h(2) (t)]2 = 0

or

ih(3) (t)/h(2) (t) = h(2) (t),

t ∈ Nδ ,

which by integrating becomes i log h(2) (t) = h(1) (t) + c1 . Set u = h(1) then u0 = c2 e−iu . Integration using the separation of variable technique leads to h(1) (t) = u = −i log(1 + c3 t) + c4 , and hence

i h(t) = − (1 + ct) log(1 + ct) + at + b, c where ci , i = 1, . . . , 4, a, b, and c stand for arbitrary constants with c 6= 0. Since h(0) = 0 and h(j) (0) = ij κj , we find that b = 0, a = i(κ1 + 1), and c = −iκ2 , so that (20)

h(t) =

1 (1 − iκ2 t) log(1 − iκ2 t) + i(κ1 + 1)t, κ2

t ∈ Nδ .

Hence it follows that h(t) in (20) can be extended to the whole real axis. The desired result now follows by comparing (20) with (11). (ii) In what follows we use a, b, c, d, A, B, C, D, cj , j = 1, 2, . . . , to denote b r , r ≥ 3, has a zero regression on L1 . Then by arbitrary constants. Assume that Q Lemma 1 (iii) we have (r − 1)[h(r) (t)]2 − (r − 2)h(r−1) (t)h(r+1) (t) = 0, or

t ∈ Nδ ,

h(r) (t) r − 2 h(r+1) (t) = , (r−1) r − 1 h(r) (t) h (t)

which by integrating yields h(r) (t) = a[h(r−1) (t)](r−1)/(r−2) . Setting u = h(r−1) , so that u0 = au(r−1)/(r−2) , gives r−2

u(t) = h(r−1) (t) = b/ (1 − ct)

(21)

.

By integrating (21) (r − 1) times we get (22) h(t) = A(1−ct) log(1−ct)+Pr−2 (t) = A(1−ct) log(1−ct)+c0 +c1 t+

r−2 X

cj tj ,

j=2

where cj , j = 0, . . . , r−2, are arbitrary constants. Since h(0) = 0 and h(j) (0) = ij κj for j = 1, 2, . . . , we obtain (23)

c0 = 0,

c1 = cA + iκ1 ,

and

c2 = −

c2 A + iκ2 , 2

Zero Regression Characterizations

11

so that (24)

c=

i κr r − 2 κr−1

and

A=

(r − 2)r κrr−1 . (r − 2)! κr−1 r

Since c is purely imaginary, h(t) is analytic for all real t and therefore f (t) = exp(h(t)) is also analytic on the real line. Now, if r = 3 then by substituting the appropriate constants of (23) in (22) and comparing the resulting h(t) with h1 (t) in (12), we find that the corresponding F is an NEF-GSL distribution with stable index ρ = 1. This proves part (a). If r = 4, by repeating similar arguments, we obtain that F is a convolution of an NEF-GSL distribution having stable index ρ = 1 with a normal law. This proves the part of (b) with r = 4. Thus assume that r ≥ 5 and let z = t + iy. Hence the analytic continuation of f is given by © ª f (z) = exp A(1 − cz) log(1 − cz) + Pr−2 (z) , and in particular, it is analytic in a half-plane containing the real line. But since f (z) = g(z) exp{Pr−2 (z)}, where g(z) is a c.f. of an NEF-GSL distribution having stable index ρ = 1, it follows by Lemma 2 that Pr−2 is at most a third degree polynomial. Since h(j) (0) = ij κj , employing (22) yields Ac3 + 6c3 = i3 κ3 ,

(25)

Acj = ij κj ,

j ≥ 4.

From (25) we have c= so that Ac3 =

i κj j − 2 κj−1

for arbitrary

µ ¶j−4 ij−1 κj−1 3 j−2 = i κj−3 j−1 , cj−4 κj

j ≥ 5,

j ≥ 5.

Thus c3 = 0 if and only if condition (19) holds. Consequently, f is the c.f. of the convolution of an NEF-GSL distribution having stable index ρ = 1 with a normal law. This completes the proof. ¤ Acknowledgement. We thank two referees for their helpful comments. References [1] S. K. Bar-Lev and P. Enis, Reproducibility and natural exponential families with power variance functions, Ann. Statist., 14 (1986), 1507–1522. [2] S. K. Bar-Lev and O. Stramer, Characterizations of natural exponential families with power variance functions by zero regression properties, Probab. Theory Rel. Fields, 76 (1987), 509–522. [3] J. Burridge, Discussion on the paper by B. Jorgensen, “Exponential dispersion models”, J. Roy. Statist. Soc. Ser. B, 49 (1987), 150–152. [4] I. F. Christensen, Some further extension of a theorem of Marcinkiewicz, Pacific J. Math., 12 (1962), 59–67. [5] E. B. Fosam and D. N. Shanbhag, An extended Laha–Lukacs characterization results based on a regression property, J. Statist. Plann. Infer., 63 (1997), 173–186.

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[Received February 2004]