COMMUNICATIONS IN STATISTICS
Theory and Methods
Vol. 33, No. 5, pp. 1007–1029, 2004
On the Power-Variance Family of
Probability Distributions
Vladimir Vinogradov*
Department of Mathematics, Ohio University,
Athens, Ohio, USA
ABSTRACT
We derive new properties for the power-variance family of
probability distributions, which are also referred to as Tweedie laws.
They emerge in the theory of generalized linear models and have a
wide range of applications. We present decomposition criteria for
Tweedie distributions which generalize Cochran’s and Raikov’s
theorems. These criteria are interpreted in terms of the additivity
of a shape parameter. The intersection of the power-variance family
with the class of distributions which have no indecomposable
components is determined. We reveal continuity of this family and
obtain convenient common formulas for the cumulants.
Key Words: Cumulant; Decomposition criterion; Indecomposable
component; Power-variance family; Tweedie law; Weak convergence.
*Correspondence: Vladimir Vinogradov, Department of Mathematics, Ohio
University, 321 Morton Hall, Athens, OH 45701, USA; Fax: (740) 593-9805;
E-mail: [email protected].
1007
DOI: 10.1081/STA-120029821
Copyright # 2004 by Marcel Dekker, Inc.
0361-0926 (Print); 1532-415X (Online)
www.dekker.com
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1.
DESCRIPTION AND BASIC PROPERTIES OF
TWEEDIE DISTRIBUTIONS
This paper pertains to the three-parametric power-variance family of
distributions, which are also referred to as Tweedie probability laws. This
family is very heterogeneous (see below). We establish new algebraic and
topological properties for members of this class, which incorporate
famous results derived previously for certain individual representatives
of the Tweedie family. In addition, we obtain common formulas for
the cumulants and standardized cumulants of Tweedie distributions.
The power-variance family is characterized in this section. We
formulate and clarify our main results in Sec. 2. The same section
contains the simple proofs. For convenience of the reader, the major
proofs are deferred to the concluding Sec. 3.
Note that although the properties of the family considered are
interesting and elegant in their own right, but this family is currently
attracting a growing attention in view of various applications
(see Hougaard, 2000; Vinogradov, 2002, Remarks 1.1.i and 2.5.ii). Also,
each member of this family is obtained as a weak limit for a wide class of
r.v.’s. This stipulates an applicability of the family of Tweedie distributions for approximation. See Jørgensen (1997, Ch. 4) or Jørgensen and
Vinogradov (2002) for more detail.
Next, let us introduce the power-variance family of probability
distributions. Hereinafter, we denote both such distributions and the
r.v.’s possessing such distributions by Twp ðm; lÞ. Each particular Tweedie
law is uniquely characterized by specific values of power parameter
p 2 D :¼ R1 nð0; 1Þ, scaling parameter l 2 L :¼ ð0; þ1Þ, and location
(or mean) parameter m. The latter parameter is such that m 2 ½0; þ1Þ if
p 2 ð 1; 0Þ; m 2 R1 if p ¼ 0; m 2 ð0; þ1Þ if p 2 ½1; 2Š, and m 2 ð0; þ1Š
if p 2 ð2; þ1Þ. Hereinafter, we refer to the values of parameters specified
above as their admissible values. For simplicity of notation, we denote
the above domains of the location parameter by Op . The probabilistic
meaning of parameters p, m and l is clarified by (1.6). Also, the boundary
@D of D is f0; 1g.
It is known that Tweedie laws with the same value of p satisfy the
following scaling relationship:
d
c Twp ðm; lÞ ¼ Twp ðc m; cp
2
lÞ;
ð1:1Þ
where c > 0 is an arbitrary fixed real (see Jørgensen, 1997, formula (4.7)).
d
Hereinafter, sign ‘¼’ is understood in the sense that the distributions
(of r.v.’s) coincide. Also, each Tweedie distribution Twp ðm; lÞ can be
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characterized by its cumulant-generating function zp;m;l ðsÞ :¼
log E expfs Twp ðm; lÞg, where s belongs to a certain subset of R1 .
Now, set
1
l m1 p ;
yð p; m; lÞ :¼
ð1:2Þ
j1 pj
where p 2 Dnf1g. For each fixed p 2 ð2; þ1Þ and l 2 L, set
yð p; 1; lÞ :¼ lim yð p; m; lÞ ¼ 0:
ð1:3Þ
m!þ1
Also, for p 2 Dnf1; 2g, define
Bp;l :¼
j1
Proposition 1.1.
pjð2 pÞ=ð1
j2 pj
pÞ
l1=ð p
1Þ
Assume that p 2 D, m 2 Op and l 2 L. Then
(i) For p 2 ð 1; 0Š,
n
zp;m;l ðsÞ ¼ Bp;l ðyð p; m; lÞ þ sÞð2
pÞ=ð1 pÞ
yð p; m; lÞð2
pÞ=ð1 pÞ
o
;
pÞ=ð1 pÞ
o
;
yð p; m; lÞ if p 2 ð 1; 0Þ and s 2 R1 if p ¼ 0.
where s (ii)
ð1:4Þ
:
z1;m;l ðsÞ ¼ m l ðes=l
For p 2 ð1; 2Þ,
n
zp;m;l ðsÞ ¼ Bp;l ðyð p; m; lÞ
1Þ, where s 2 R1 .
(iii)
sÞð2
pÞ=ð1 pÞ
yð p; m; lÞð2
where s < yð p; m; lÞ.
(iv)
z2;m;l ðsÞ ¼
l logð1
For p 2 ð2; þ1Þ,
n
zp;m;l ðsÞ ¼ Bp;l yð p; m; lÞð2
s=yð2; m; lÞÞ, where s < yð2; m; lÞ.
(v)
pÞ=ð1 pÞ
ðyð p; m; lÞ
sÞð2
where s yð p; m; lÞ.
z3
pÞ=ð1 pÞ
o
;
(vi)
zp;m;l ðsÞ ¼ þ1 for all the remaining values of s.
(vii)
For arbitrary fixed admissible values of p, m and l, zp;m;l ðsÞ and
sÞ are the inverse functions.
p;1=m;l ð
Proof of Proposition 1:1. (i)–(vi) All these representations for zp;m;l ðÞ
can be derived from Jørgensen (1997, formula (4.16)).
(vii)
The proof is straightforward.
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Remark 1.1. (i) Part (vii) of Proposition 1.1 is new, although it is
well known for a few special cases (see Tweedie, 1984, p. 580). It yields
new results on the first passage times for a class of Lévy processes
constructed starting from the Tweedie laws with p 0 (see Vinogradov,
2002, Sec. 4).
(ii) For an arbitrary fixed p 2 D, Tweedie laws Twp ðm; lÞ; m 2 Op ;
l 2 Lg described in Proposition 1.1 comprise a two-parametric family of
distributions (with respect to parameters m 2 Op and l 2 L). Each such
family constitutes a reproductive exponential dispersion model in the
same sense as in Jørgensen (1997, Ch. 3). Moreover, the family of Tweedie
laws characterized in Proposition 1.1 is a unique class of the probability
distributions that for each fixed p 2 D, simultaneously satisfy (1.1) and
constitute a reproductive exponential dispersion model (see Jørgensen,
1997, Th. 4.1 and Prop. 4.2). Also, the just quoted theorem yields that
each Tweedie law Twp ðm; lÞ 2 ID, where ID denotes the set of all
univariate infinitely divisible distributions.
d
Next, an application of (1.1) with c ¼ 1=m yields that Twp ðm; lÞ ¼
2 p
m Twp ð1; l m Þ. This motivates the consideration of the following
shape parameter, which is hereinafter denoted by fp :
fp :¼ l m2 p :
ð1:5Þ
Clearly, fp 0. Also, it is obvious that fp ¼ 0 if and only if either
f p 2 ð 1; 0Š; m ¼ 0g or f p 2 ð2; þ1Þ; m ¼ þ1g. These two cases correspond to stable distributions with index a 2 ð0; 1Þ [ ð1; 2Š (see below).
However, it is not customary to refer to fp as a ‘‘shape parameter’’ in
these two cases. Parameter fp is of importance for the other members
of the family of Tweedie laws (see (2.10)). It follows from (1.5)–(1.6)
that fp equals the reciprocal of the squared coefficient of variation of
r.v. Twp ðm; lÞ. In the case when p ¼ 3 (the inverse Gaussian family), f3
was employed by Chhikara and Folks (1989).
Now, Tweedie laws satisfy the variance-to-mean relationship of a
d
power type. Namely, the variance of r.v. Y ¼ Twp ðm; lÞ is as follows:
VarðY Þ ¼ mp =l
ð1:6Þ
(see Jørgensen, 1997, formula (4.2)). Property (1.6) justifies referring to
the class of Tweedie distributions as the power-variance family (compare
to Hougaard, 2000, Subsecs. 7.5.1 and 8.6.2). Also, set
a ¼ að pÞ :¼ ð2
pÞ=ð1
pÞ:
ð1:7Þ
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It is clear that parameters p 2 D and a 2 ð 1; 1Þ [ ð1; 2Š are in one-toone correspondence.
The three-parametric family described above was introduced
independently by Tweedie (1984), Hougaard (1986), and Bar-Lev and Enis
(1986). Recall that the family of Tweedie laws is quite heterogeneous.
It includes continuous normal distributions ( p ¼ 0), gamma distributions
( p ¼ 2) and inverse Gaussian distributions ( p ¼ 3), discrete scaled
Poisson distributions ( p ¼ 1) and mixed non-central gamma distributions
with zero degrees of freedom ( p ¼ 3=2). In addition, Tweedie
distributions with p 2 ð2; þ1Þ are derived by an exponential tilting of
positive stable laws with index a ¼ að pÞ 2 ð0; 1Þ. The corresponding
value of the exponential tilting parameter equals yð p; m; lÞ (see (1.2)).
Namely, it is true that for each fixed p 2 ð2; þ1Þ, the probability
densities fp;m;l ðÞ and fp;1;l ðÞ of r.v. Twp ðm; lÞ and positive stable r.v.
Twp ð1; lÞ, respectively, are related as follows:
fp;m;l ðxÞ e
ðyð p;m;lÞxþzp;1;l ð yð p;m;lÞÞÞ
fp;1;l ðxÞ:
ð1:8Þ
Here, x 2 R1þ , and function zp;1;l ðÞ is given in Proposition 1.1.v. Note
that both these densities are equal to zero for negative values of x.
Evidently, positive stable laws with index a 2 ð0; 1Þ are included in
the power-variance family corresponding to f p 2 ð2; þ1Þ; m ¼ þ1g.
Various properties of their densities fp;1;l ðÞ can be found in Zolotarev
(1986, Sec. 2.5), where a different parametrization was used.
Next, Tweedie distributions with p 2 ð 1; 0Þ are derived by an
exponential tilting of extreme stable laws with index a ¼ að pÞ ¼
ð2 pÞ=ð1 pÞ 2 ð1; 2Þ and skewness parameter b ¼ 1. The corresponding value of the exponential tilting parameter equals yð p; m; lÞ
(see (1.2)). By analogy to (1.8), one derives that for each p 2 ð 1; 0Þ,
the probability densities fp;m;l ðÞ and fp;0;l ðÞ of r.v. Twp ðm; lÞ and extreme
stable r.v. Twp ð0; lÞ, respectively, are related as follows:
fp;m;l ðxÞ eyð p;m;lÞx
zp;0;l ðyð p;m;lÞÞ
fp;0;l ðxÞ:
ð1:9Þ
Here, x 2 R1 , and function zp;0;l ðÞ is given in Proposition 1.1.i. It is clear
that extreme stable laws with index a 2 ð1; 2Þ and skewness parameter
b ¼ 1 are included in the power-variance family corresponding to
f p 2 ð 1; 0Þ; m ¼ 0g. Various properties of their densities fp;0;l ðÞ can
be found in Zolotarev (1986, Sec. 2.5). In particular, the lower tail of each
such density is of a power type, whereas its upper tail is lighter than that
of a normal density.
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We refer to Tweedie distributions Twp ðm; Þ for which either
f p 2 ð 1; 0Þ; m ¼ 0g or f p 2 ð2; þ1Þ; m ¼ þ1g as non-normal stable
Tweedie laws. The union of this subfamily with the family of normal
distributions fTw0 ðm; lÞ; m 2 O0 ; l 2 Lg is termed the class of stable
Tweedie laws. All the other representatives of the power-variance
family are referred to as non-stable Tweedie laws. They pertain to
f p 2 Dnf0g; m 2 ð0; þ1Þg.
Now, each Tweedie r.v. Twp ðm; lÞ with p 2 ð1; 2Þ has a mixed,
compound Poisson-gamma distribution (see Hougaard, 2000). In particular, this distribution has an absolutely continuous component, while
PfTwp ðm; lÞ ¼ 0g ¼ expf fp =ð2 pÞg > 0.
Recall that each member of the power-variance family is infinitely
divisible. Proposition 1.2, Theorem 1.1 and Remark 1.2 provide Lévy
representations for the logarithm of the characteristic function of
Tweedie distributions. However, we first should introduce some auxiliary
notation. Let us denote the analytic continuation of the gamma function
onto Cnf0; 1; 2; . . .g by GðÞ. This continuation relies on the next
formula: Gðz 1Þ :¼ GðzÞ=ðz 1Þ. In addition, for arbitrary fixed real g
and x > 0, we consider the complement of the incomplete gamma function,
which is hereinafter denoted by Gðg; xÞ. Set
Z 1
Gðg; xÞ :¼
yg 1 e y dy
ð1:10Þ
x
(compare to Johnson et al., 1992, formula (1.85)).
The following result provides the density pp;m;l ðÞ of Lévy measure of
r.v. Twp ðm; lÞ with respect to Lebesgue measure in the case when
p 2 Dn@D. It will be employed in the proof of Theorems 1.1 and 2.2.i.
Proposition 1.2 (Compare to Küchler and Sørensen, 1997, Formula
(2.1.17)). Assume that r.v. Twp ðm; lÞ is such that p 2 Dn@D, m 2 Op and
l 2 L. Then
ðiÞ In the case when p 2 ð1; þ1Þ, Twp ðm; lÞ is spectrally positive,
there exists density pp;m;l ðÞ of its Lévy measure with respect to Lebesgue
measure, and for an arbitrary x > 0,
pp;m;l ðxÞ ¼
j p 1j1=ð1 pÞ 1=ð p
l
jGð1=ð p 1Þj
1Þ
j xj
ð1þð2 pÞ=ð1 pÞÞ
e
yð p;m;lÞjxj
:
ð1:11Þ
Here, yð p; m; lÞ is given by (1.2)–(1.3).
(ii) In the case when p 2 ð 1; 0Þ, Twp ðm; lÞ is spectrally negative,
and there exists density pp;m;l ðÞ of its Lévy measure with respect to
Lebesgue measure that admits representation (1.11) for an arbitrary x < 0.
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Proof of Proposition 1:2. Following Küchler and Sørensen (1997,
Sec. 2.1), we will use cumulant-generating functions of probability
distributions instead of the logarithm of their characteristic functions.
This enables one to employ Lévy representation of the cumulantgenerating function of an infinitely divisible distribution (see Küchler
and Sørensen, 1997, formula (2.1.8)). It is convenient to consider the next
four cases separately.
(i) If p 2 ð1; 2Þ then Proposition 1.2 follows from the formulas
given in Küchler and Sørensen (1997, p. 12).
(ii) Assume that p 2 ð2; þ1Þ. Then an application of a formula
given in Bertoin (1996, p. 73) and some scaling arguments yield that
Z 1
zp;1;l ðsÞ ¼
ðesx 1Þ pp;1;l ðxÞ dx;
0þ
where s 0. The rest follows from Küchler and Sørensen (1997, formula
(2.1.13)).
(iii) Assume that p 2 ð 1; 0Þ. By (1.7), a ¼ að pÞ 2 ð1; 2Þ. Then it is
straightforward to demonstrate that for such a,
Z 1
Gð1 aÞ ð sÞa ¼
ðesx 1 s xÞ dð x a Þ:
0
Here, s 0 and the analytic continuation of the gamma-function is used.
A combination of this representation with simple scaling arguments
implies that
Z 0
zp;0;l ðsÞ ¼
ðesx 1 s xÞ pp;0;l ðxÞ dx;
1
where s 0. The rest follows from Küchler and Sørensen (1997, formula
(2.1.13)).
(iv) If p ¼ 2 then Proposition 1.2 follows from a representation of
z2;m;l ðÞ in terms of the Frullani integral (see Bertoin, 1996, p. 73).
&
Evidently, Proposition 1.2 implies that for each p 2 Dn@D, Lévy
measure n p;m;l ðÞ of r.v. Twp ðm; lÞ is as follows:
Z
pp;m;l ðxÞ dx:
ð1:12Þ
n p;m;l ðAÞ :¼
A
Here, A R1 nf0g is an arbitrary Lebesgue-measurable set.
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Vinogradov
The next result provides the classical form of Lévy representations
for the logarithm of characteristic function cp;m;l ðuÞ of r.v. Twp ðm; lÞ.
In addition, it contains the explicit formulas for the corresponding Lévy
measure. It is obvious that
log cp;m;l ðuÞ ¼ zp;m;l ði uÞ;
ð1:13Þ
where zp;m;l ði uÞ denotes the analytic continuation of the corresponding
cumulant-generating function of r.v. Twp ðm; lÞ, which is defined in
Proposition 1.1.
Theorem 1.1.
(i)
Assume that p 2 Dn@D, m 2 Op and l 2 L. Then
For each p 2 ð 1; 0Þ,
log cp;m;l ðuÞ ¼
Z
0
ðeiux
1
i u xÞ n p;m;l ðdxÞ:
ð1:14Þ
1
Here, Le´vy measure n p;m;l ðÞ is such that for each y < 0,
j p 1j1=ð1 pÞ 1=ð p
n p;m;l fð 1; yŠg ¼
l
jGð1=ð p 1Þj
(ii)
1Þ
2
G
1
p
; yð p; m; lÞ j yj :
p
ð1:15Þ
For each p 2 ð1; þ1Þ,
log cp;m;l ðuÞ ¼
Z
1
ðeiux
1Þ n p;m;l ðdxÞ:
ð1:16Þ
0þ
Here, Le´vy measure n p;m;l ðÞ is such that for each y > 0,
j p 1j1=ð1 pÞ 1=ð p
n p;m;l f½ y; 1Þg ¼
l
jGð1=ð p 1Þj
1Þ
G
2
1
p
; yð p; m; lÞ y :
p
ð1:17Þ
Proof of Theorem 1:1. (i) In order to derive (1.14), we apply the
arguments used in the proof of Proposition 1.2.ii as well as representations (1.12)–(1.13). In turn, a combination of (1.10)–(1.12) implies (1.15).
(ii) The validity of (1.16) is easily obtained by combining the
arguments used in points (i), (ii) and (iv) of the proof of Proposition 1.2
with (1.12)–(1.13). In addition, representation (1.17) easily follows from
a combination of (1.10)–(1.12).
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Remark 1.2. (i) Lévy representation for the logarithm of the characteristic function of normal r.v. Tw0 ðm; lÞ is well known. Namely,
log c0;m;l ðuÞ ¼ m i u u2 =ð2 lÞ.
(ii) It follows from Proposition 1.1.ii that in the case when p ¼ 1,
Lévy representation for the logarithm of the characteristic function of
scaled Poisson r.v. Tw1 ðm; lÞ is as follows:
Z 1
log c1;m;l ðuÞ ¼
ðeiux 1Þ n 1;m;l ðdxÞ:
1
Here, n 1;m;l ðÞ ¼ m l d1=l . Hence, in this case Lévy measure is degenerate
being a multiple of the Dirac point mass d1=l concentrated at 1=l.
(iii) It follows from (1.16)–(1.17) that all Tweedie distributions with
p 2 ð1; þ1Þ belong to the class of infinitely divisible distributions on R1þ .
In contrast, (1.14)–(1.15) imply that all Tweedie laws with p 2 ð 1; 0Þ
are spectrally negative.
2.
RESULTS
We start with a technical result that is essential for the formulation
and proof of our main Theorem 2.1.
Lemma 2.1. Fix p 2 Dnf0; 1g. Let all li ’s, 1 i n, belong to L.
Suppose that all mi ’s, 1 i n, belong to Op nf0g. Fix arbitrary real
ci > 0, where 1 i n. Assume that
c1 1 l1 m11
p
¼ ¼ cn 1 ln mn1 p ;
ð2:1Þ
and set
m :¼
n
X
ci mi :
ð2:2Þ
i¼1
Let l 2 L. Then
l m1
p
¼ c1 1 l1 m11
p
ð2:3Þ
if and only if
n
X
ð p 2Þ=ð p 1Þ
ci
!p
1
:
ð2:4Þ
Proof of Lemma 2:1 is straightforward.
&
l¼
i¼1
1=ð p 1Þ
li
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Our main theorem can be regarded as a decomposition criterion for
Tweedie laws with p 2 Dn@D.
Theorem 2.1.
1 i n.
Let p 2 Dn@D, and fix arbitrary real ci > 0, where
d
(i) Assume that r.v. W ¼ Twp ðm; lÞ, where m 2 Op and l 2 L. Consider
d
independent r.v.’s fWi ; 1 i ng such that Wi ¼ Twp ðmi ; li Þ. Here, mi ’s
and li ’s, 1 i n, are certain constants which belong to Op and L,
respectively. Then the validity of decomposition
d
W¼
n
X
ci Wi
ð2:5Þ
i¼1
implies the fulfillment of conditions (2.1)–(2.3).
(ii) Let independent r.v.’s Wi ’s have the same distributions as in part
(i). Suppose that condition (2.1) is fulfilled, and define r.v. W by formula
d
(2.5). Then W ¼ Twp ðm; lÞ, where the values of parameters m and l are
given by formulas (2.2) and (2.4), respectively.
Proof of Theorem 2.1 is deferred to Sec. 3.
d
Remark 2.1. (i) In the case when r.v. W ¼ Twp ðm; lÞ is not non-normal
stable, (1.6) yields that a combination of relationships (2.1) and (2.3) can
be interpreted as the property that r.v.’s W and ci Wi possess common
ratios of mean to variance. Here, 1 i n.
(ii) The assumption of independence of r.v.’s fWi ; 1 i ng
imposed in Theorem 2.1 is not necessary. This follows from two
counter-examples to Cochran’s theorem, which is a special case of
Theorem 2.1 and is given below as Theorem 2.5. The reader is referred
to James (1952) and Dykstra and Hewett (1972) for these counterexamples.
(iii) In the case when p ¼ 2, Theorem 2.1.ii is stated in Johnson
et al. (1994, p. 340). In addition, in the case when p ¼ 3, the validity of
Theorem 2.1 is mentioned in Chhikara and Folks (1989, p. 13, Property 2).
The problem of existence and construction of decompositions of r.v.
d
W ¼ Twp ð; Þ into a finite sum of independent random components such
that at least one of these components is not distributed according to
Tweedie law Twp ð; Þ with the same value of power parameter p is of
interest for the arithmetic of probability distributions. In this respect,
let us give a few relevant definitions.
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Definition 2.1 (cf., e.g., Linnik, 1964, p. 79). R.v. Y and its distribution
function F ðÞ are said to be indecomposable if the relationship
F ¼ F1 F2
ð2:6Þ
implies that F1 or F2 is a degenerate distribution function. Here, F1 and
F2 are certain distribution functions which are commonly referred to as
the factors of F , and sign ‘ ’ denotes the operation of convolution.
It follows from (2.6) that if F1 is a factor of F then for each a 2 R1 ,
F1 Ea is also a factor of F . Hereinafter, Ea denotes the degenerate
distribution function which is concentrated at point a. Factor F1 Ea is
commonly referred to as that which is equivalent to F1 , whereas each
component of the type Ea will be hereinafter termed a trivial factor of
decomposition (2.6). Any other component that can emerge on the
right-hand side of (2.6) will be called a proper factor. In the sequel, we will
occasionally limit our consideration to decompositions into proper
factors only (see Theorems 2.2.ii and 2.3.i) or into the components which
have no trivial factors (see Theorem 2.4.i).
Definition 2.2 (see Linnik and Ostrovskii, 1977, p. 55). The set of all
distribution functions on R1 which have no indecomposable components
is hereinafter denoted by I0 .
It is known that I0 is a proper subclass of ID (cf., e.g., Linnik and
Ostrovskii, 1977, Th. 3.5.1).
Definition 2.3 (see Lukacs, 1970, p. 245). A family of probability
distributions is called factor-closed if the factors of every element of the
family belong necessarily to the family. Hereinafter, each such family is
referred to as an FC-family.
Remark 2.2. (i) An FC-family does not necessarily belong to class ID.
Thus, it is known that the class of binomial distributions is factor-closed
(cf., e.g., Teicher, 1954). However, each binomial distribution is not
infinitely divisible.
(ii) Assume that each member of a certain family of univariate
probability distributions belongs to IDnI0 . That is, each member of the
family is infinitely divisible, but it has an indecomposable component.
Then such family cannot be factor-closed, since this indecomposable
component does not belong to ID. Hence, it is not included in the family.
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Vinogradov
Let us consider another family of infinitely divisible distributions,
which is slightly more general than the class of scaled Poisson distributions.
Definition 2.4. Consider a family of univariate infinitely divisible distributions, which are described by the following characteristic function:
Rm;g;h ðtÞ ¼ exp i t m þ g ðeith 1Þ :
ð2:7Þ
Here, m 2 R1 , whereas g and h are arbitrary non-negative real numbers.
Hereinafter, we will term the three-parametric family of distributions
defined by (2.7) the family of Poisson-type distributions.
The intersection of the power-variance family with I0 is described by
the following result.
Theorem 2.2. (i) Consider a particular Tweedie distribution Twp ðm; lÞ,
where p 2 D, l 2 L, and m 2 Op . Then Twp ðm; lÞ 2 I0 if and only if p 2 @D.
(ii) The family of normal distributions Tw0 ðm; lÞ; m 2 R1 ; l 2 L is
factor-closed provided that the trivial factors are excluded.
(iii)
The family (2.7) of Poisson-type distributions is factor-closed.
Proof of Theorem 2:2.
(i)
The proof is deferred to Sec. 3.
(ii)
The proof can be found in Lukacs (1970, Sec. 8.2).
(iii)
See Lukacs (1970, Corollary to Theorem 8.2.2).
&
The next two statements can be regarded as decomposition criteria
for Tweedie laws with p 2 @D. Theorem 2.3 is a version of the famous
Cramér’s theorem (see Cramér, 1970, Th. 19 and also Johnson et al.,
1994, pp. 102–103). In turn, Theorem 2.4 slightly generalizes Raikov’s
theorem (see Raikov, 1938 and also Johnson et al., 1992, p. 173).
Theorem 2.3 (Cramér’s Theorem).
1 i n.
Fix arbitrary real ci > 0, where
d
(i) Assume that r.v. W ¼ Tw0 ðm; lÞ, where m 2 O0 and l 2 L.
Consider independent r.v.’s fWi ; 1 i ng such that decomposition
(2.5) holds, and neither of r.v.’s Wi is a trivial factor. Then each
d
Wi ¼ Tw0 ðmi ; li Þ, where mi ’s and li ’s, 1 i n, are certain constants
which belong to O0 and L, respectively, and conditions (2.2) and (2.4)
(with p ¼ 0) are fulfilled.
(ii) Let independent r.v.’s Wi ’s have the same distributions as in part
d
(i). Define r.v. W by formula (2.5). Then W ¼ Tw0 ðm; lÞ, where the values
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Power-Variance Family
1019
of parameters m and l are given by formulas (2.2) and (2.4) (with p ¼ 0),
respectively.
d
Proof of Theorem 2:3. (i) Fix r.v. W ¼ Tw0 ðm; lÞ, where m 2 O0 and
l 2 L. By Theorem 2.2.ii, all its proper components Wi ’s are normally
distributed. Next, Remark 1.2.i yields that the values mi ’s and li ’s which
characterize the distributions of normal r.v.’s fWi ; 1 P
i ng may be
chosen arbitrarily, provided that (2.2) is fulfilled, and ni¼1 c2i =li ¼ 1=l.
It remains to note that in the case when p ¼ 0, this relationship coincides
with (2.4).
(ii) The proof is obtained by a combination of (1.1) and
Proposition 1.1.i.
&
Theorem 2.4 (Raikov’s Theorem).
1 i n.
Fix arbitrary real ci > 0, where
d
(i) Assume that r.v. W ¼ Tw1 ðm; lÞ, where m 2 O1 and l 2 L.
Consider independent r.v.’s fWi ; 1 i ng such that decomposition (2.5)
holds, and the distributions of all Wi ’s do not contain trivial factors. Then
d
each Wi ¼ Tw1 ðmi ; li Þ, where mi ’s and li ’s, 1 i n, are certain constants
which belong to O1 and L, respectively, and condition (2.2) along with
conditions (2.1) and (2.3) (with p ¼ 1) are fulfilled.
(ii) Let independent r.v.’s Wi ’s have the same distributions as in part
(i). Suppose that condition (2.1) is fulfilled (with p ¼ 1), and define r.v. W
d
by formula (2.5). Then W ¼ Tw1 ðm; lÞ, where the value of parameter m is
given by formula (2.2), and that of l is given by condition (2.3) (with p ¼ 1).
Proof of Theorem 2:4. (i) By (1.1), it suffices to prove this statement in
the case when c1 ¼ ¼ cn ¼ 1.
Note that the class fTw1 ðm; lÞ; m 2 O1 ; l 2 Lg of scaled Poisson
laws is a proper subclass of the family (2.7) of Poisson-type distributions.
This is easily derived by a combination of Proposition 1.1.ii with (2.7).
In particular, given m 2 O1 and l 2 L, one should set m ¼ 0, g ¼ m l
and h ¼ 1=l in formula (2.7) for ch.f. Rm;g;h ðtÞ.
d
Next, fix r.v. W ¼ Tw1 ðm; lÞ that has ch.f. R0;ml;1=l ðtÞ. Consider
its decomposition into independent components Wi ’s, 1 i n, such
that neither of these components has a trivial factor. It then follows
from Theorem 2.2.iii that for each 1 i n, ch.f. RðiÞ ðtÞ of r.v. Wi
satisfies (2.7). Subsequently, one concludes that parameter m in representation (2.7) for RðiÞ ðtÞ equals zero, since the distribution of each Wi
is assumed to not contain a trivial factor. Therefore, we obtain that for
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1020
Vinogradov
each 1 i n,
RðiÞ ðtÞ ¼ exp gi ðeithi
1Þ ;
ð2:8Þ
where all gi ’s and hi ’s are certain positive real numbers.
Now, a combination of Proposition 1.1.ii with (1.1) yields that
d
Tw1 ðm; lÞ ¼ l
1
d
Tw1 ðm l; 1Þ ¼ l
1
Poissðm lÞ:
ð2:9Þ
Here, PoissðnÞ denotes a Poission r.v. with mean n. A subsequent
combination of (2.8)–(2.9) with Proposition 1.1.ii implies that each
d
Wi ¼ Tw1 ðgi =hi ; hi 1 Þ. Hence, it has a scaled Poisson distribution. Finally,
an application of Linnik (1964, Th. 5.1.1) along with the induction
arguments yields that all hi ’s are the same and equal to l. The rest
is trivial.
(ii) The proof is obtained by a combination of (1.1) with Proposition 1.1.ii.
&
Remark 2.3. (i) Apparently, the results of Theorem 2.2 and Remark
2.2.ii reveal a reason behind the difference in the sets of conditions
imposed in the decomposition criteria given by Theorems
2.1 and
2.3–2.4. Namely, each subfamily of Tweedie distributions Twp ðm; lÞ;
m 2 Op ; l 2 Lg with a fixed value of p from the interior of D is not
factor-closed.
In contrast,
each subfamily of Tweedie distributions
Twp ðm; lÞ; m 2 Op ; l 2 L with a fixed value of p belonging to the
boundary of D is factor-closed.
(ii) In the case when p ¼ 0, a combination of (2.1)–(2.3) is sufficient
but not necessary for decomposition (2.5) to be held. Sufficiency follows
by combining (1.1) with Theorem 2.3. The refutation of necessity can be
done by constructing trivial counter-examples. The details are left to the
reader.
Now, we proceed with a series of statements pertaining to Theorems
2.1–2.3. The following result can be interpreted as the additivity property
of shape parameter fp , which is defined by (1.5). Also, in the sequel we
will employ the nth partial sum of the sequence of independent
r.v.’s
P
fWi ; 1 i ng. On this reason, it is convenient to set Sn :¼ ni¼1 Wi :
Proposition 2.1. Let p 2 D, and fix arbitrary real ci > 0, where 1 i n.
d
Assume that r.v. W ¼ Twp ðm; lÞ, where m 2 Op , and l 2 L. Consider
d
independent r.v.’s fWi ; 1 i ng such that Wi ¼ Twp ðmi ; li Þ. Here, mi ’s and
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Power-Variance Family
1021
li ’s, 1 i n, are certain constants which belong to Op and L, respectively.
Suppose that (2.2) is fulfilled, and set fpðiÞ :¼ li mi2 p . Then
(i) Under these assumptions, the validity of conditions (2.1) and (2.3)
implies the fulfillment of the next property:
fp ¼
n
X
fðiÞ
p :
ð2:10Þ
i¼1
(ii)
Suppose that (2.10) is satisfied, and either
min1in fðiÞ
p > 0 and
VarðW Þ ¼ VarðSn Þ:
Pn
i¼1
fðiÞ
p ¼ 0, or
ð2:11Þ
Then conditions (2.1) and (2.3) are fulfilled.
Proof of Proposition 2.1 is deferred to Sec. 3.
Remark 2.4. (i) Let us clarify the reason of imposing an assumption
on the properties of
shape parameter in Proposition 2.1.ii. Namely,
Pthe
n
ðiÞ
f
in the case when
i¼1 p > 0, one of the terms that emerge on the
right-hand side of (2.10), say fð1Þ
p , can be zero. This would imply
the violation of (2.1). Hence, we should exclude the case when
min1in fðiÞ
p ¼ 0.
(ii) It is clear that in the case when min1in fðiÞ
p > 0, each Wi has a
finite variance, and both terms of Eq. (2.11) are well defined. Moreover,
condition (2.11) that emerges in Proposition 2.1.ii is essential. Thus, one
d
can easily construct three gamma variables Wi ¼ Tw2 ðmi ; li Þ, 1 i n,
such that m1 ¼ m2 þ m3 , l1 ¼ l2 þ l3 , but l2 =m2 6¼ l3 =m3 . Suppose that
r.v.’s W2 and W3 are independent. Then the probability density of the
sum W2 þ W3 , which can be derived from Johnson et al. (1994, formula
(17.110)) is not that of a gamma variable. Hence, it differs from that of
r.v. W .
The original proof of the next celebrated result was given by
Cochran (1934, Th. II). Below we will present a simpler derivation of
his statement, which relies on a combination of Theorem 2.1 with
Proposition 2.1. Recall that r.v. Sn is defined above Proposition 2.1.
d
Theorem 2.5 (Cochran’s Theorem). Let r.v. W ¼ Tw2 ðm; m=2Þ, where
m > 0 is an integer. Assume that independent r.v.’s fWi ; 1 i ng are such
ORDER
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1022
Vinogradov
d
that Wi ¼ Tw2 ðmi ; li Þ, where all li ’s belong to L, and all mi ’s are positive
integers. Then the distributions of r.v.’s W and Sn coincide if and only if
d P
m ¼ ni¼1 mi and each li ¼ mi =2.
Proof of Theorem 2:5. First, let us not impose a constraint that mi ’s
and m are integers. Then the statement follows from Theorem 2.1 and
Proposition 2.1. Thus, a combination of (2.1) with (2.3) implies that
l1 =m1 ¼ ¼ ln =mn ¼ l=m ¼ 1=2. Subsequently,
a combination
of this
P
P
equation with (2.10) yields that m=2 ¼ l ¼ ni¼1 li ¼ ni¼1 mi =2.
Now, assume that all mi ’s and m can only take on positive integer
values. Then a combination of Proposition 1.1.iv with the above
d
d
arguments yields that r.v.’s W ¼ Tw2 ðm; m=2Þ and Wi ¼ Tw2 ðmi ; mi =2Þ are
chi-square P
distributed with m and mi degrees of freedom, respectively,
n
such that
i¼1 mi ¼ m. The latter relationship coincides with Cochran
(1934, cond. (2)).
&
Next, we present a continuity property of the Tweedie family.
d
Hereinafter, sign ‘!’ is understood as weak convergence.
Theorem 2.6. Fix p0 2 D, m0 2 Op0 , and l0 2 L. Consider a certain
subfamily of Tweedie laws fTwp ðm; lÞg, where p 2 D, m 2 O, and l 2 L
d
are such that p ! p0 , m ! m0 and l ! l0 . Then fTwp ðm;lÞg! Twp0 ðm0 ;l0 Þ.
Proof of Theorem 2:6.
First, let us demonstrate that
d
Twp ðm0 ; l0 Þ ! Twp0 ðm0 ; l0 Þ
ð2:12Þ
as p ! p0 . It is easily seen that unless p0 2 f1; 2g, (2.12) follows from
parts (i), (iii) and (v) of Proposition 1.1 and an obvious continuity of
function zp;m;l ðÞ with respect to p. Hence, it remains to consider the
following cases:
(i) p ! 2. Then a combination of parts (iii) and (v) of
Proposition 1.1 implies that for arbitrary fixed m 2 O2 and l 2 L, and
either for an arbitrary fixed s yð p; m0 ; l0 Þ if p > 2 or for an arbitrary
fixed s < yð p; m0 ; l0 Þ if p < 2,
zp;m0 ;l0 ðsÞ ¼
fp
2
2
exp
1
p
p
logð1
p
s=yð p; m0 ; l0 ÞÞ
1 :
ð2:13Þ
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Power-Variance Family
1023
Subsequently, the expression
2 p
log ð1 s=yð p; m0 ; l0 ÞÞ ¼ expfð2
exp
1 p
pÞ Constg ð1 þ oð1ÞÞ
as p ! 2. Hence, it is easy to show that the expression on the right-hand
side of (2.13) is equivalent to ð1 pÞ 1 fp log ð1 s=yð p; m0 ; l0 ÞÞ !
l logð1 s=yð2;m0 ;l0 ÞÞ as p ! 2. By Proposition 1.1.iv, this limit coincides
with z2;m0 ;l0 ðsÞ.
(ii) p # 1. The proof can be derived by employing certain analytical techniques similar to those used in part (i) (compare to Hougaard,
2000, p. 506).
Next, consider the general case
when p ! p0 , m! m0 and l ! l0 .
Evidently, zp;m;l ðsÞ zp;m0 ;l0 ðsÞ þ zp;m;l ðsÞ zp;m0 ;l0 ðsÞ . A combination
of Proposition 1.1.i–vi with (1.2)–(1.4) yields that for an arbitrary fixed
p 2 D, zp;m;l ðsÞ ! zp;m0 ;l0 ðsÞ as m ! m0 and l ! l0 . The rest follows from
(2.12).
&
Remark 2.5. (i) Theorem 2.6 is of the same spirit as the continuity
property of the family of stable laws (see Zolotarev, 1986, Property 2.4).
For p0 2 @D, the validity of Theorem 2.6 is mentioned in Küchler and
Sørensen (1997, p. 13). See also Hougaard (1986, Lm. 2.a) for the case
when p0 2 ½2; 1Þ.
(ii) Theorem 2.6 provides a basis for the derivation of new results
on a near-equilibrium behavior of both random movements of equities
and rational prices of stock options in the presence of downward jumps
in the logarithmic returns on equities (see Vinogradov, 2002, Sec. 3)).
In order to formulate the next statement, we need to introduce some
additional notation. Let kp;m;l ðlÞ denote the lth cumulant of Twp ðm; lÞ,
where l 1 is an integer. For l 2, denote rp;m;l ðlÞ :¼ kp;m;l ðlÞ=kp;m;l ð2Þl=2 .
These coefficients are frequently referred to as the standardized cumulants (see McCullagh and Nelder, 1989, p. 351). In particular, rp;m;l ð3Þ
and rp;m;l ð4Þ are skewness g1 and kurtosis g2 of r.v. Twp ðm; lÞ, respectively. Also, consider a parametric family of special functions of an
integer argument (compare to Tweedie, 1984, formula (23)):
Pp ðlÞ :¼
l 2
Y
ðm ð p
1Þ þ 1Þ:
m¼0
Here, l 2 is an integer, and p 2 D. In particular, P2 ðlÞ ¼ GðlÞ.
ð2:14Þ
ORDER
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1024
Vinogradov
The following theorem stipulates that the cumulants of each Tweedie
law can be expressed by a common, easy-to-use computational formula.
Also, it emphasizes the importance of shape parameter fp defined by
(1.5).
d
Theorem 2.7. Fix an arbitrary Tweedie r.v. W ¼ Twp ðm; lÞ, where p 2 D,
l 2 L, and m 2 Op . Assume that l 2 is an arbitrary fixed integer. Then (i)
kp;m;l ðlÞ ¼ Pp ðlÞ ml fp ðl
1Þ
ð2:15Þ
;
where Pp ðlÞ is defined by (2.14).
(ii)
d
In the case when r.v. W ¼ Twp ðm; lÞ is not non-normal stable,
rp;m;l ðlÞ ¼ Pp ðlÞ fp ðl
2Þ=2
:
ð2:16Þ
Proof of Theorem 2:7. (i) (2.15) is obtained by induction. Its derivation is based on a successive differentiation of function zp;m;l ðsÞ, which
is given in Proposition 1.1. The induction base corresponds to l ¼ 2
and coincides with (1.6).
d
(ii) Assume that r.v. W ¼ Twp ðm; lÞ is not non-normal stable. Then
a combination of (1.1) and (1.5) implies that for an arbitrary real c > 0,
fp ðc W Þ ¼ fp ðW Þ ¼ fp :
ð2:17Þ
Subsequently, a combination of (1.5)–(1.6) implies that the standard
deviation sW ¼ m fp 1=2 < 1. Set Y :¼ W =sW . Then (1.1) yields that
d
p=2
Y ¼ Twp ðf1=2
p ; fp Þ. By (2.17), fp ðY Þ ¼ fp ðW Þ. In order to obtain
(2.16), it remains to note that lth cumulant of r.v. Y equals lth
standardized cumulant of r.v. W .
&
Remark 2.6. (i) By (2.16), each Tweedie law Twp ðm; lÞ which is
not non-normal stable, has skewness g1 ¼ p=f1=2
and kurtosis
p
g2 ¼ p ð2p 1Þ=fp . Hence, Tweedie distributions with positive (respectively, negative) p are skewed to the right (respectively, to the left). They
are leptokurtic, with the exception of the normal Tweedie laws Tw0 ðm; lÞ.
(ii) In the case when p ¼ 0, (2.15) is consistent with the formulas
for cumulants of normal distribution with mean m and variance 1=l.
Also, in the case when p ¼ 1, (2.16) can be derived by combining (2.9)
with Johnson et al. (1992, formula (4.9)).
ORDER
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Power-Variance Family
1025
In the case when p ¼ 2, (2.15) can be obtained by applying Johnson
et al. (1994, formula (17.9)). Also, in the case when p ¼ 3, (2.15) is
consistent with Chhikara and Folks (1989, formula (2.8)).
(iii) In the case when p 2 Dnð 1; 0Þ, (2.16) can be derived by a
combination of Tweedie (1984, formulas (22)–(23)).
(iv) In the cases when either f p 2 ð2; þ1Þ; m ¼ þ1g or
f p 2 ð 1; 0Þ; m ¼ 0g, which correspond to non-normal stable Tweedie
distributions, (2.15) should be understood in a generalized sense,
whereas (2.16) does not hold. In these two cases, (2.15) implies that
for an arbitrary integer l 2, kp;m;l ðlÞ ¼ 1. In the former case, this is
also true for l ¼ 1.
3.
MAJOR PROOFS
Proof of Theorem 2.2.i. First, it is known that both normal and
Poisson families Tw0 ðm; lÞ and Tw1 ðm; 1Þ belong to I0 (see Cramér,
1970, Theorem 19; Raikov, 1938, Sec. 3; or Johnson et al., 1992, p. 172).
Next, we derive that for an arbitrary fixed l 2 L, subfamily
fTw1 ðm; lÞ; m 2 O1 g 2 I0 . This follows from the validity of this property
in the special case when l ¼ 1 by combining (2.9) with Linnik (1964,
Th. 5.1.1) and applying the induction arguments.
The rest of proof relies on the use of Proposition 1.2. In the case
when p 2 ð 1; 0Þ [ ½3=2; þ1Þ, Linnik (1964, Th. 12.1.1) yields that the
distribution, whose density of Lévy measure with respect to Lebesgue
measure is given by Proposition 1.2, does not belong to I0 . Finally,
in the case when p 2 ð1; 3=2Þ, Linnik and Ostrovskii (1977, Th. 6.7.2)
implies that the distribution whose density of Lévy measure with respect
to Lebesgue measure is given by Proposition 1.2 does not belong to I0 .
&
Proof of Theorem 2:1. By (1.1), it suffices to prove the theorem in the
case when c1 ¼ ¼ cn ¼ 1.
Proof of (ii). First, assume that n ¼ 2. Subsequently, if at least one of
r.v.’s fW1 ; W2 g is extreme stable then the other one should also be
extreme stable. This is easily derived from a combination of (2.1) with
(1.2)–(1.3). The rest follows from Samorodnitsky and Taqqu (1994,
Property 1.2.1). Observe that in this case, their conditions coincide with
(2.2) and (2.4).
Now, assume that both independent Tweedie r.v.’s fW1 ; W2 g are not
stable. Then a combination of (2.1) with (1.2) yields that yð p; m1 ; l1 Þ ¼
yð p; m2 ; l2 Þ. Subsequently, let us employ Proposition 1.1 and keep in
ORDER
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1026
Vinogradov
mind (1.4). One obtains that cumulant-generating function of r.v.
W1 þ W2 admits the representation given in Proposition 1.1 with the
d
same p, m ¼ m1 þ m2 , and l defined by (2.4). Hence, W ¼ Twp ðm; lÞ.
In the case when n > 2, the proof is easily obtained by induction in n.
Proof of (i). The proof relies on equating cumulant-generating
functions of r.v.’s W and Sn , which is defined above Proposition 2.1. It
is based on specific representations for zp;m;l ðÞ given in Proposition 1.1
and hence, should be carried out separately in four distinct cases, which
correspond to p 2 ð 1; 0Þ, p 2 ð1; 2Þ, p ¼ 2 and p 2 ð2; þ1Þ. Here, we
only consider the case when p 2 ð2; þ1Þ, since the method of proof
employed for each of these cases is essentially the same.
The proof is carried out by induction in n. The induction base
corresponds to n ¼ 1 and is trivial. Next, suppose that n 2 and prove
the induction step. To this end, we equate cumulant-generating functions
of r.v.’s W and Sn , where the latter r.v. is defined above Proposition 2.1.
Thus,
n
zp;m;l ðsÞ ¼ Bp;l yðp;m;lÞð2
¼
n
X
i¼1
pÞ=ð1 pÞ
n
Bp;li yð p; mi ;li Þð2
ðyð p;m;lÞ
pÞ=ð1 pÞ
sÞð2
ðyðp; mi ; li Þ
pÞ=ð1 pÞ
sÞð2
o
pÞ=ð1 pÞ
o
;
ð3:1Þ
where s yðp; m;lÞ. In addition, the cumulant-generating functions of
both r.v.’s should be equal to þ1 for s > yðp; m; lÞ.
It is easily seen that log E exp fs Sn g ¼ þ1 for s > min1in yð p; mi ; li Þ
and is given by the expression that emerges on the right-hand side of
(3.1) for s min1in yð p; mi ; li Þ. Hence,
min yð p; mi ; li Þ ¼ yð p; m; lÞ:
ð3:2Þ
1in
It is clear that we can assume without loss of generality that
min1in yð p; mi ; li Þ ¼ yð p; mn ; ln Þ.
Now, if m ¼ þ1 then mn ¼ þ1. This follows from the fact that
0 ¼ yð p; m; lÞ ¼ yð p; mn ; ln Þ. Note that zp;1;l ðsÞ and all zp;mi ;li ðsÞ are finite
and negative for s 2 ð 1; 0Þ. Hence, Bp;l > Bp;ln , where Bp;l is defined
by (1.4). Therefore,
ð^
lÞ1=ð p
1Þ
:¼ l1=ð p
1Þ
ln1=ð p
1Þ
> 0:
ð3:3Þ
ORDER
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Power-Variance Family
1027
Subsequently, an application of (3.1) yields that
zp;1;^l ðsÞ ¼ zp;1;l ðsÞ zp;1;ln ðsÞ
n 1
X
¼
i¼1
n
Bp;li yðp;mi ;li Þð2
pÞ=ð1 pÞ
ðyðp;mi ;li Þ sÞð2
pÞ=ð1 pÞ
o
;
where ^
l is defined by (3.3) and s yðp;1; ^
lÞ ¼ 0. The rest follows from
the induction hypothesis.
Next, suppose that m 2 ð0; 1Þ, which implies that 0 < mn < m.
Otherwise, one gets a contradiction, since n 2 and min1in 1 mi > 0.
Then a combination of (1.2) and (3.2) yields the validity of (3.3). Set
^ :¼ m mn 2 ð0; þ1Þ. It is easily checked that yð p; m
^; ^
m
lÞ ¼ yð p; m; lÞ ¼
yð p; mn ; ln Þ: Hence, one may ascertain that
zp;^m;^l ðsÞ ¼ zp;m;l ðsÞ
¼
n 1
X
i¼1
zp;mn ;ln ðsÞ
n
Bp;li yðp;mi ;li Þð2
pÞ=ð1 pÞ
ðyðp;mi ;li Þ
sÞð2
pÞ=ð1 pÞ
o
;
^; ^
where s yðp; m
lÞ. Also, the cumulant-generating functions of r.v.’s
^
^; ^
m; lÞ and Sn 1 are equal to þ1 for s > yðp; m
lÞ. The rest follows
Twp ð^
from the induction hypothesis.
&
Proof of Proposition 2:1. By analogy to (2.17), it suffices to prove
Proposition 2.1 in the case when c1 ¼ ¼ cn ¼ 1.
(i) The proof of the fact that a combination ofP(2.1) and (2.3) yields
the validity of (2.10) is straightforward, since m ¼ ni¼1 mi .
(ii) In the case when fp ¼ 0, the derivation of (2.1) and (2.3) is
trivial. Evidently, each fðiÞ
p 0 and hence, it is equal to zero.
Now, assume that fp ¼ l m2 p > 0. Subsequently, a combination of
(1.6) and (2.11) implies that
mp =l ¼
n
X
mpi =li :
ð3:4Þ
i¼1
Next, let us multiply the left- and right-hand sides of Eqs. (2.10) and
(3.5), and keep in mind the validity of (2.2). One gets that
n
X
i¼1
m2i
!2
¼ m2 ¼
n
X
i¼1
mip =li n
X
j¼1
lj m2j p :
ORDER
REPRINTS
1028
Vinogradov
After simple algebra, one derives that the latter formula yields the
following equation:
X X lj p 2 p li p 2 p X X
mi mj ¼
m mj þ mj mi
2
:
li i
lj
1i<jn
1i<jn
Note that under the conditions imposed, all mi ’s are positive and finite.
Hence, the latter formula can be rewritten as follows:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2
X X
lj p 2 p
li p 2 p
mi mj
m mi
¼ 0:
li
lj j
1i<jn
Evidently, this implies that l1 m11
p
¼ ¼ ln m1n p . The rest is trivial.
&
ACKNOWLEDGMENTS
I thank G.M. Feldman, A. Feuerverger and Y.-H. Wang for their
advice and an anonymous referee for very useful suggestions.
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