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Communications in Statistics—Theory and Methods, 36: 253–263, 2007
Copyright © Taylor & Francis Group, LLC
ISSN: 0361-0926 print/1532-415X online
DOI: 10.1080/03610920600974534
Distributions, Models, and Applications
On Infinitely Divisible Exponential Dispersion Model
Related to Poisson-Exponential Distribution
VLADIMIR VINOGRADOV
Department of Mathematics, Ohio University, Athens, Ohio, USA
We construct a univariate exponential dispersion model comprised of discrete
infinitely divisible distributions. This model emerges in the theory of branching
processes. We obtain a representation for the Lévy measure of relevant distributions
and characterize their laws as Poisson mixtures and/or compound Poisson
distributions. The regularity of the unit variance function of this model is
employed for the derivation of approximations by the Poisson-exponential model. We
emphasize the role of the latter class. We construct local approximations relating
them to properties of special functions and branching diffusions.
Keywords Bessel function; Branching process; Compound Poisson distribution;
Confluent hypergeometric function; Lévy measure; Local approximation;
Poisson mixture; Pólya-Aeppli distribution; Power-variance family; Rao damage
process; Tweedie exponential dispersion models; Unit variance function; Weak
convergence.
Mathematics Subject Classification Primary 60E07, 60F05; Secondary 60J80,
62E20.
1. Introduction
This work is concerned with specific univariate natural exponential families and
exponential dispersion models. Hereinafter, these classes of probability laws are
denoted by NEFs and EDMs, respectively. The NEFs are used in many areas
including the maximum likelihood inference (cf., e.g., Küchler and Sørensen, 1997).
On the other hand, a (univariate) EDM can be viewed as a certain two-parametric
family of probability distributions which is constructed starting from a particular
NEF.
Our main results pertain to the introduction of a certain EDM, the
establishment of infinite divisibility of its members, as well as the derivation of
numerous approximations. Although some of them are of the integral type, these
Received January 6, 2006; Accepted May 1, 2006
Address correspondence to Vladimir Vinogradov, Department of Mathematics, Ohio
University, Athens, OH, 45701, USA; Fax: (740) 593-9805; E-mail: [email protected]
253
254
Vinogradov
are our local approximations which we perceive to be of a particular interest. Thus,
we associate them with properties of confluent hypergeometric and Bessel functions
(see Secs. 2, 3).
This article emphasizes the role of EDM comprised of Poisson-exponential
distributions. This EDM belongs to the class of Tweedie EDMs and emerges in
various approximations (see below). This family was considered, among others,
by Vinogradov (2004). Since the author is unaware of a systematic treatment of
properties of this important class being available, he presented a short summary of
the results pertaining to the Poisson-exponential EDM in Sec. 2.
In Sec. 3, we demonstrate that this EDM serves as an appropriate
approximation for purely discrete Pólya-Aeppli EDM, which is comprised of
Poisson-geometric distributions. The range of the latter distributions is the set
Z+ of non negative integers. Such distributions are frequently termed “contagious”
(cf., e.g., Johnson et al., 2005, p. 382). Some approximations for the “contagious”
distributions, which are of the same spirit as ours, have already been used for
fitting population biology data (cf., e.g., Kendal, 2004). This suggests that our
approximations for EDMs have a potential to provide a better understanding of
interrelations between certain experimental data and the theory of birth-and-death
processes.
Now, let us make a few general comments on EDMs per se. Each EDM
possesses simple structural properties. Their consideration was partly motivated by
the necessity to develop the distribution theory for (not necessarily normal) random
errors arising in the theory of generalized linear models. In this article, we will
concentrate on NEFs and EDMs comprised of infinitely divisible distributions. A
systematic coverage of properties of the univariate NEFs and EDMs can be found
in Jørgensen (1997, Chs. 2, 3). To keep the length of the article within a reasonable
limit, we opted to refer the reader to the latter work for the required background
on univariate NEFs and EDMs.
It is of interest that certain non trivial-classes of distributions (which turned
out to comprise EDMs) were introduced well outside the mainstream of their
theory. Thus, we demonstrate that some of our EDMs emerge in the studies on
the evolution of branching-fluctuating particle systems (see Sec. 3, Dawson and
Vinogradov, 1994, Proof of Prop. 1.10 and formula (116 ); Dawson et al., 1995,
Prop. 1.2.1). The author made an attempt to bridge the distribution theory and
the theory of stochastic processes together, herewith enriching both of them. For
instance, he points out to a connection between certain results on branchingdiffusing populations and Rao damage processes. (The description of the latter
processes can be found in Johnson et al., 2005, Sec. 9.2.) Some of our methods
are related to those employed by Panjer and Willmot (1992) and by Smyth and
Jørgensen (2002) for investigating stochastic models of Property and Casualty
Insurance.
2. Poisson-Exponential EDM: Properties and Convergence
In this section, we summarize the
of the Poisson information on properties
exponential reproductive EDM Tw3/2 ∈ R+1 ∈ R+1 . We follow the
notation developed in Vinogradov (2004) whenever possible. Consider
√
3/2 = 3/2 = 2 · / (2.1)
EDM Related to Poisson-Exponential Law
255
and
3/2 = ·
√
(2.2)
which constitute the exponential tilting and shape parameters, respectively (compare
to Vinogradov, 2004, formulas (1.2) and (1.5)).
d
Next, let the sign ‘=’ mean that the distributions of r.v.’s coincide. Then the
d
probability law of r.v. U3/2 = Tw3/2 can be characterized by its cumulantgenerating function (or c.g.f.) 3/2 s = log E exps · U3/2 . It takes on the
following form:
3/2 s = 2 · 3/2 · s · 3/2 − s−1 (2.3)
where the argument s < 3/2 . In turn, (2.3) implies that VarU3/2 = −1 · 3/2 . It
is known that r.v. U3/2 is compound Poisson. This is because it can be represented
as the following Poisson random sum of i.i.d.r.v.’s:
d
U3/2 =
L
(2.4)
Gk k=1
d
Here, r.v. L = Poisson2 · 3/2 , whereas independent r.v.’s Gk k ≥ 1 do not
−1
. Hence, the
depend on L and have common exponential distribution with mean 3/2
distribution of non-negative r.v. U3/2 has a positive atom at zero:
PU3/2 = 0 = PL = 0 = exp−2 · 3/2 > 0
(2.5)
Next, a combination of standard arguments which involve conditioning with
Abramowitz and Stegun (1975, formulas (9.6.6) and (9.6.10)) and Hougaard (2000,
formula (A.26)) ascertain that the distribution of r.v. U3/2 has an absolutely
continuous component in R+1 whose density f3/2 x is as follows:
f3/2 x = x−1 · exp−3/2 · x + ·
4 · 2 · xk
k! · k
k=1
√
2·
= √ · exp−3/2 · x + · I1 4 · · x
x
(2.6)
Here, I1 · denotes the modified Bessel function of the first kind. Various
computational aspects which are related to representation (2.6) were studied in
Dunn and Smyth (2005).
A subsequent application of Abramowitz and Stegun (1975, formula (9.7.1))
yields that
√
√ · x−3/4 · exp −3/2 · x − 2
(2.7)
f3/2 x ∼
2·
as x → . Here, 3/2 is given by formula (2.1).
It is noteworthy that (2.7) implies the validity of the next saddlepoint
approximation.
256
Vinogradov
Lemma 2.1. The exact asymptotics for f3/2 x given by (2.7) holds true if parameter
∈ R+1 and real-valued argument x > 0 remain fixed, whereas → .
Proof of Lemma 2.1. It is relatively easily obtained by combining (2.6)–(2.7) with
d
the next scaling property: c · U3/2 = U3/2c·/√c (compare to Jørgensen, 1997,
formula (4.7) or Vinogradov, 2004, formula (1.1)). It also follows by observing that
the expression that emerges on the right-hand side of (2.7) coincides with Hougaard
(2000, formula (A.21)).
It has already been said that the members of the Poisson-exponential class
emerge as the limits for families of EDMs. We illustrate this statement by virtue of
d
the next result. Hereinafter, the sign ‘→’ denotes weak convergence. Also, although it
is convenient to formulate the following assertion in terms of a family Y ∈
∈ of r.v.’s that constitutes a reproductive EDM, but the result pertains only
to the two-parametric family of their distribution functions. Hereinafter, and stand for the domains of the location and scaling parameters and , respectively
(compare to Jørgensen, 1997, Ch. 4).
Theorem 2.1 (see Jørgensen et al., 1994, Th. 4.2). Suppose that a two-parametric
family of r.v.’s Y ∈ ∈ constitutes a reproductive EDM. Assume that the
unit variance function of this EDM has a power asymptotics at zero or infinity with the
value of the power parameter p = 3/2:
V ∼ 0 · 3/2
(2.8)
as ↓ 0 or as → . Here, real 0 > 0 is a constant. Fix arbitrary values of the
location and scaling parameters 0 ∈ and 0 ∈ . Then (i) the one-parameter family
of r.v.’s
√
d
c−1 · Yc · 0 0 / c → Tw3/2 0 0 /0 (2.9)
as c ↓ 0 or as c → , respectively.
Here, the convergence is through the values of c
√
for which c · 0 ∈ and 0 / c ∈ .
(ii) The fulfillment of (2.9) as c → implies that the EDM considered is comprised
of infinitely divisible distributions.
In the next section, we will employ (2.9) in the case when (2.8) is fulfilled as
→ .
3. Pólya-Aeppli EDM: Application and Convergence
Here we demonstrate that the class of Pólya-Aeppli laws constitutes an additive
infinitely divisible EDM and present properties of this family. In particular, we
emphasize a relation with the Poisson-exponential class and applications in the
theory of branching processes.
Each member of this two-parametric family can be represented as a Poisson
sum of i.i.d. geometric r.v.’s (see formula (3.1)) as well as a Poisson mixture.
In the latter case, the mixing distribution should belong to the Poisson-exponential
family described in the previous section. The former, compound Poisson approach is
EDM Related to Poisson-Exponential Law
257
reviewed in Johnson et al. (2005, Sec. 9.7). On the other hand, the latter derivation,
which is based on the Poisson mixture representation, was used by Hougaard et al.
(1997, Sec. 2.2), Kokonendji et al. (2004, Sec. 3), and Johnson et al. (2005, Subsec.
11.1.2).
By analogy to Johnson et al. (2005, Sec. 9.7), we say that a (generic) r.v. WQ has
the Pólya-Aeppli distribution with the values of parameters ∈ R+1 and Q ∈ 0 1 if
d
WQ =
L
Hk (3.1)
k=1
d
Here, r.v. L = Poiss, whereas i.i.d. (unshifted) geometric r.v.’s Hk k ≥ 1 do not
depend on L and are such that for each n ∈ Z+ , PHk = n = Q · 1 − Qn .
Now, consider c.g.f. Q · of r.v. WQ . By Johnson et al. (2005, p. 411),
Q
es − 1
−
1
=
·
Q s = ·
1 − 1 − Q · es
1/1 − Q − es
(3.2)
where the argument s < − log1 − Q. Evidently, this representation implies that
the subclass of all Pólya-Aeppli distributions with fixed Q is closed with respect
to convolution. Representation (3.2) yields that each Pólya-Aeppli law is infinitely
divisible, since nth root of r.v. WQ also has a Pólya-Aeppli distribution with n =
/n and the same value of Q. In order to find the Lévy measure of r.v. WQ , observe
that Johnson et al. (2005, formulas (9.134)–(9.135)), imply that the Poisson sum of
i.i.d. unshifted geometric r.v.’s can be represented as another Poisson sum of i.i.d.
shifted geometric r.v.’s:
d
WQ =
L
k H
(3.3)
k=1
k k ≥ 1 do
Here, r.v. L = Poiss · 1 − Q, whereas i.i.d. shifted geometric r.v.’s H
k = n = Q · 1 − Qn−1 . An
not depend on L and are such that for each n ∈ N, PH
application of Sato (1999, Prop. 4.5) stipulates that the Lévy representation for c.g.f.
Q · of r.v. WQ does not contain the drift and diffusion components. Its Lévy
measure WQ is concentrated on set N, and for each n ∈ N,
d
k = n = · Q · 1 − Qn WQ n = · 1 − Q · PH
(3.4)
Next, straightforward arguments which involve a combination of (3.4) with Küchler
and Sørensen (1997, Prop. 2.1.3), imply that any admissible exponential tilting
transformation of a Pólya-Aeppli distribution also has a Pólya-Aeppli law. The
details are left to the reader.
Evidently, this argument implies that NEF generated by an arbitrary fixed
Pólya-Aeppli law is comprised of the distributions which belong to the same class. A
subsequent combination of this statement with the above observation that nth root
of r.v. WQ also has a specific Pólya-Aeppli distribution, the convolution property
of this class and standard properties of EDMs ascertain that the (infinitely divisible)
additive EDM generated by r.v. WQ is comprised of Pólya-Aeppli laws only. The
proof of the fact that this EDM includes all Pólya-Aeppli laws is straightforward.
258
Vinogradov
Also, this result can be recovered from Kokonendji et al. (2004, Prop. 3.1), in the
case when p = 3/2.
The common range of this family is Z+ , and it belongs to the class of
“contagious” distributions (compare to Douglas, 1980). Set pQ n = PWQ = n,
where n ∈ Z+ . Then
pQ 0 = e−·1−Q (3.5)
In the case when n ≥ 1, one obtains that
pQ n = pQ 0 · 1 − Qn ·
n
1
n−1
·
· · Qk k
−
1
k!
k=1
(3.6)
A subsequent combination of (3.5)–(3.6) with the definition of the confluent
hypergeometric function
1 F1 a b z
= 1 +
a · a + 1 z2
a z
· +
·
+ ···
b 1! b · b + 1 2!
given in Slater (1960, formula (1.1.8)), along with some algebra imply that for an
arbitrary n ∈ Z+ ,
pQ n = e− · · Q · 1 − Qn · 1 F1 n + 1 2 · Q
(3.7)
Proposition 3.1. The upper tail of the probability function pQ n of Pólya-Aeppli r.v.
WQ possesses the following asymptotics as n → :
pQ n =
√ · Q1/4 −·2−Q/2 −3/4
·n
· exp 2 · · Q · n · 1 − Qn · 1 + O1/ n √ ·e
2· Proof of Proposition 3.1. It follows by combining (3.7) with Douglas (1980, p. 320).
Remark 3.1. It is easy to see that this asymptotic formula is of the same character
as representation (2.7) for the density of the absolutely continuous component of
a Poisson-exponential distribution. This resemblance in the asymptotic behavior
between Pólya-Aeppli and Poisson-exponential laws extends much further (see
Theorems 3.1–3.2 and Corollary 3.1).
Now, consider the reproductive version of the Pólya-Aeppli additive EDM,
which is hereinafter denoted by YP−A ∈ P−A ∈ P−A . This reproductive
EDM is obtained from the additive version by the division by . This transition
involves replacing the exponential tilting parameter with the location parameter .
Let us specify the unit variance function VP−A · of the Pólya-Aeppli reproductive
EDM.
Lemma 3.1. (i) The unit variance function VP−A · that characterizes the PólyaAeppli reproductive EDM is as follows:
1/2
VP−A = + 3/2 · C0 − 2 · −1 ·
1 + C0 · − 1
EDM Related to Poisson-Exponential Law
259
where location parameter ∈ R+1 and C0 > 0 is a certain real constant that depends on
the choice of the initial Pólya-Aeppli law to be used for generating the model.
(ii) Function VP−A · admits representation (2.8) as → with 0 = C0 .
Proof of Lemma 31. (i) The first step involves the application of non-negativity
of r.v. WQ and the already established infinite divisibility of the class of PólyaAeppli distributions to show that P−A = P−A = R+1 . The rest follows from
Kokonendji et al. (2004, Prop. 2). Observe that formula (11) therein involves the
solution to the corresponding mean-value equation. It is easy to show that in the
case when the NEF considered belongs to the Pólya-Aeppli (or Poisson-Tweedie
with p = 3/2) EDM, one is able to solve this equation explicitly.
(ii) The proof is straightforward.
The next assertion constitutes an integral limit theorem on weak convergence
to the Poisson-exponential distribution.
Theorem 3.1. Fix arbitrary √
0 and 0 belonging to R+1 . Then the one-parameter family
−1
of r.v.’s c · YP−A c · 0 0 / c comprised of the members of the reproductive version
of Pólya-Aeppli EDM YP−A ∈ R+1 ∈ R+1 converges weakly to Poissonexponential r.v. Tw3/2 0 0 / C0 as c → .
Proof of Theorem 31. It follows by combining Lemma 3.1 with Theorem 2.1.i. It is instructive to describe a problem pertaining to the evolution of a branchingfluctuating particle system where the above class of integer-valued probability laws
emerges. This illustrates the applicability of Theorem 3.1 on weak convergence
of EDMs to the theory of branching processes. To this end, we develop some
ideas presented in Dawson and Vinogradov (1994, pp. 226–227, 231–233) and
Dawson et al. (1995, Prop. 1.2.1).
We start with a Poisson number Poiss with mean of independent particles
located at the origin at the initial time instant t = 0. Each particle is assigned the
same mass 1/ and is assumed to perform an independent random spatial motion.
We also assume that at an exponentially distributed time instant with mean 1/,
the particle either dies out with probability 1/2 or splits into two offspring with the
same probability 1/2. Each newly-born particle is an identical copy of its parent and
immediately starts to perform the same spatial motion. The motions, lifetimes, and
branchings of all particles are assumed to be independent of each other. Hereinafter,
we denote this branching-fluctuating particle system (or BPS) by Yt . The total mass
process of BPS Yt is hereinafter denoted by Mt . This process disregards specific
locations of particles and represents the total number of particles alive at time t
multiplied by their common mass 1/. The consideration of Mt is important for
the study of evolution of branching diffusions.
Let Lt denote the initial number of particles of this BPS having alive
d
descendents at time t. Recall that L0 = Poiss. The latter condition implies that
Lt t ≥ 0 constitutes a Rao damage process (compare to Johnson et al., 2005,
Sec. 9.2). In turn, the theory of Rao damage processes ascertains that for each real
t > 0,
d
Lt = Poisst/2 + −1 −1 (3.8)
260
Vinogradov
Set
2/ · t
1 + 2/ · t
Qt =
(3.9)
which represents the probability of survival of descendents of a particular initial
particle by time t. It is customary to say that each initial particle that has surviving
descendents at time t gives rise to a cluster of its offspring, which are alive at time t.
We denote the size of ith cluster by Kt i. Obviously, r.v.’s Kt i 1 ≤ i ≤
Lt
are independent and identically distributed. It can also be shown that the
distribution of the size of cluster Kt i is shifted geometric with Q = Qt .
Now, one should combine the equivalence of representations (3.1) and (3.3),
formulas (3.8)–(3.9) and the above arguments to derive that the total number of
particles · Mt alive at time t admits representation (3.1) with EL = 2/t and Q =
Qt , where the latter parameter takes on the value given by formula (3.9). Thus,
d
· Mt
=
L
Hk (3.10)
k=1
Here, r.v. Mt stands for the total mass process introduced above. Evidently, (3.10)
yields that r.v. · Mt is Pólya-Aeppli distributed.
Next, a combination of Johnson et al. (2005, formulas (9.134)–(9.135)), formula
(3.2), and the above observation implies that c.g.f. · of r.v. Mt can be derived
from formula (3.2) with = 2/t and Q = Qt . In particular,
s = 2/tQ s/ =
t
es/ − 1
2
·
t 1 + 2/ · t − es/
(3.11)
Here, s < log1 + 2/ · t. Fix an arbitrary real t > 0 and take the limit of the
expression that emerges on the right-hand side of (3.11) as → . One gets that
s →
2
s
·
= 3/211/t s
t 2/t − s
where 3/2 · denotes c.g.f. of Poisson-exponential r.v. U3/2 given by (2.3). This
proves that the univariate distributions of the total mass process Mt of BPS Yt
converge to those of the total mass process Mt of a certain superprocess as → (compare to Dawson et al., 1995, Prop. 1.2.1). Obviously, the pointwise convergence
of c.g.f.’s established above implies that
Mt
d
d
→ Mt = Tw3/2 1 1/t
(3.12)
as → . Recall that here, t > 0 is an arbitrary fixed real.
Remark 3.2. (i) It is easy to derive (3.12) directly from Theorem 3.1. To this end,
it suffices to set 0 = 1, 0 = 1/t, and c = →. Various convergence results
derived within the framework of the theory of measure-valued processes, which are
similar to (3.12), are usually termed the high-density limits. The author extends this
EDM Related to Poisson-Exponential Law
261
terminology for the characterization of the behavior of the whole class of EDMs
which satisfy the conditions of Theorem 2.1. See Corollary 3.1.
(ii) The combination of the fact that the distribution of the size Kt i of the
surviving cluster is shifted geometric with Jørgensen (1997, Exercise 3.22) on weak
convergence of the negative binomial reproductive EDM (with c = ) implies that
the mass of a cluster alive at time t converges to an exponential r.v. with mean t/2.
This is an additional application of the convergence theory for EDMs. Also, the
above argument gives an alternative derivation for the well-known cluster structure
of a superprocess. It is such that the number of clusters alive at time t is Poisson
with mean 2/t with each surviving cluster having an exponential mass with mean
t/2.
Now, let us present a local counterpart of Theorem 3.1. To some extent, the
next Theorem 3.2 can be regarded as an approximation for the probability function
pQ n that corresponds to the triangular array of integer-valued compound Poisson
r.v.’s WQ . This approximation is given in two parts, which involve discrete
and absolutely continuous components of the limiting (mixed) Poisson-exponential
distribution (see formulas (3.15) and (3.16), respectively). The proof of this result
involves a relationship between the confluent hypergeometric function 1 F1 · · · that
emerges in formula (3.7) and the modified Bessel function of the first kind I1 x
(see (2.6) and (3.17)). Also, recall that pQ n = PWQ = n, whereas r.v. U3/2
is defined by (2.4). Note that the density f3/2 x of its absolutely continuous
component is given by formula (2.6).
Theorem 3.2. Consider the class of Pólya-Aeppli distributed r.v.’s W Q that depend
on parameter such that → 0 , and Q ∼ Q0 / as → . Here, 0 > 0 and
Q0 > 0 are certain real constants. Set
0 = 0 /Q0
(3.13)
and
0 =
0 · Q0 /2
(3.14)
Then
i p Q 0 ∼ PU3/20 0 = 0 = exp −2 · 0 · 0
(3.15)
as → .
(ii) Let real x > 0 be fixed and the product x · take on integer values only. Then
p Q x · ∼ −1 · f3/20 0 x
(3.16)
as → .
Proof of Theorem 3.2. Part (i) is easily obtained by combining
formulas (2.2), (2.5),
(3.5), (3.13), and (3.14), since · 1 − Q → 0 = 2 · 0 · 0 as → .
(ii) The proof relies on the use of representations (3.7) and (2.6), which involve
the confluent hypergeometric function and modified Bessel function of the first kind,
262
Vinogradov
respectively. Also, it employs the next result, which can be derived from Slater (1960,
formula (4.4.1)):
1 F1 a 2 y/a
∼ I1 2 ·
√
√
y/ y
(3.17)
as a → . Here, y > 0 is an arbitrary fixed real. By (3.7),
p Q x · = exp− · · Q · 1 − Q x· · 1 F1 x · + 1 2 · Q A subsequent combination of the above assumptions, (3.17) and analyticity of the
confluent hypergeometric function along with some algebra yield that the latter
expression is equivalent to
1
·2·
0 · Q0 /2
0 · Q0 √
· x
· exp−x · Q0 + 0 · I1 4 ·
√
2
x
(3.18)
as → . The combination of (2.1)–(2.2), (2.6), (3.13)–(3.14), and (3.18) implies
(3.16).
We now present a corollary to Theorem 3.2.
Corollary 3.1. Suppose that x > 0 is a fixed real such that the product x · takes on
integer values only, and → . Then the total mass Mt of BPS Yt , that has a nonnegative discrete distribution and is given by formula (3.10), is such that x belongs to
its range, whereas its probability function possesses the following asymptotics.
PMt
= 0 ∼ PMt = 0 = e−2/t
and
PMt
= x ∼ −1 · f3/211/t x
(3.19)
as → , where the latter factor represents the density of the absolutely continuous
component of the distribution of the total mass Mt of the limiting superprocess.
Proof of Corollary 3.1. It easily follows from Theorem 3.2.ii by multiplying both
Mt and x, which emerge in (3.19), by . Set = ELt and Q = Qt , where the
latter quantities are given by formulas (3.8) and (3.9), respectively. Evidently, all the
conditions of Theorem 3.2 are fulfilled with 0 = Q0 = 2/t. A subsequent application
of formulas (3.13)–(3.14) ascertains that the location and scaling parameters which
characterize the limiting Poisson-exponential density are equal to 1 and 1/t,
respectively.
To conclude, note that Jørgensen (1997, pp. 149–150) refers to convergence
results having the same character as those given in Theorems 2.1, 3.1–3.2, and
Corollary 3.1 as the results on the infinitely divisible type of convergence. Although
the results on the high density limits do not represent a new form of convergence
as (3.12) demonstrates, but this seems to be an area where theory and applications
interact well. Moreover, the development of limit theorems for branching and
related processes suggests that one can sometimes discover new limits under
EDM Related to Poisson-Exponential Law
263
numerous constraints and weaken assumptions consequently. Such narrow classes
of distributions could have simple structural properties or even constitute EDMs.
Also, the approximations of Theorem 3.2 and Corollary 3.1 are analogous to the
local DeMoivre-Laplace theorem. It is plausible they could be extended to wider
classes of distributions.
Acknowledgments
I thank D. Dawson, A. Feuerverger, L. Mytnik, and T. Salisbury for help and the
anonymous referee for many useful suggestions. I appreciate the hospitality of the
Fields Institute, the University of Toronto, and York University.
References
Abramowitz, M., Stegun, I. A. (1975). Handbook of Mathematical Functions with Formulas,
Graphs, and Mathematical Tables. New York: Dover.
Dawson, D. A., Hochberg, K. J., Vinogradov, V. (1995). On path properties of super-2
processes II. In: Cranston, M. G., Pinsky, M. A., eds. Proceedings of Symposia in Pure
Mathematics Series. Providence: AMS, pp. 385–403.
Dawson, D. A., Vinogradov, V. (1994). Almost-sure path properties of 2 d superprocesses. Stoch. Proc. Appl. 51:221–258.
Douglas, J. B. (1980). Analysis with Standard Contagious Distributions. Burtonsville, MD:
International Co-operative Publishing House.
Dunn, P. K., Smyth, G. K. (2005). Series evaluation of Tweedie exponential dispersion model
densities. Stat. Comput. 15:267–280.
Hougaard, P. (2000). Analysis of Multivariate Survival Data. New York: Springer.
Hougaard, P., Lee, M.-L. T., Whitmore, G. A. (1997). Analysis of overdispersed count data
by mixtures of Poisson variables and Poisson processes. Biometrics 53:1225–1238.
Johnson, N. L., Kotz, S., Kemp, A. W. (2005). Univariate Discrete Distributions. 3rd ed.
Hoboken, NJ: Wiley.
Jørgensen, B. (1997). The Theory of Dispersion Models. London: Chapman & Hall.
Jørgensen, B., Martínez, J. R., Tsao, M. (1994). Asymptotic behaviour of the variance
function. Scand. J. Statist. 21:223–243.
Kendal, W. S. (2004). A scale invariant clustering of genes on human chromosome 7. BMC
Evolutionary Biol. 4(3).
Kokonendji, C. C., Dossou-Gbété, S., Demétrio, C. G. B. (2004). Some discrete exponential
dispersion models: Poisson-Tweedie and Hinde-Demétrio classes. SORT: Statist.
Operations Res. Trans. 28:201–214.
Küchler, U., Sørensen, M. (1997). Exponential Families of Stochastic Processes. New York:
Springer.
Panjer, H. H., Willmot, G. E. (1992). Insurance Risk Models. Schaumburg, IL: Society of
Actuaries.
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge: Cambridge
University Press.
Slater, L. J. (1960). Confluent Hypergeometric Functions. Cambridge: Cambridge University
Press.
Smyth, G. K., Jørgensen, B. (2002). Fitting Tweedie’s compound Poisson model to insurance
claims data: dispersion modeling. Astin Bull. 32:143–157.
Vinogradov, V. (2004). On the power-variance family of probability distributions. Commun.
Statist. Theor. Meth. 33:1007–1029 (Errata p. 2573).