A Characterization of the Compound Multiparameter Hermite Gamma Distribution via Gauss’s Principle

Abstract

We consider the class of those distributions that satisfy Gauss's principle (the maximum likelihood estimator of the mean is the sample mean) and have a parameter orthogonal to the mean. It is shown that this so-called “mean orthogonal class” is closed under convolution. A previous characterization of the compound gamma characterization of random sums is revisited and clarified. A new characterization of the compound distribution with multiparameter Hermite count distribution and gamma severity distribution is obtained.

1. Introduction

The topic of maximum likelihood characterizations of distributions has a long history and is an active field of contemporary mathematical sciences. It concerns the characterization of a (class of) probability distribution(s) through the structure of the maximum likelihood estimator (MLE) of one or more parameters of interest (e.g., location, scale, etc.). Starting point is a famous result by Gauss [1] on the foundation of least squares theory (see, e.g., [2], [3, afterword, pages 208 and 215]). Given is a location family with continuous derivative. If the maximum likelihood estimator of the location parameter is the sample mean, then the distribution is normal. This important result has been discussed by Poincaré [4, Chapter 10], Teicher [5], Ferguson [6, 7], Marshall and Olkin [8], Bondesson [9], and Azzalini and Genton [10], by many other authors. The property that the MLE of the mean is the sample mean has been called Gauss’s principle by Campbell [11] (see also [12, 13] and references therein). A brief account of the present contribution follows.

Within the framework of multiparameter distributions, consider the “mean orthogonal class” of those distributions that besides Gauss’s principle have a parameterization such that the mean is orthogonal to some parameter vector, a property which can always be satisfied by Amari [14], Section 8. This class has been considered by Sprott [15]. A characterization of the mean orthogonal class through the cumulant generating function (cgf) has been formulated by Hürlimann [16]. Extending a result by Hudson [17], it is shown in Theorem 3 that this class is closed under convolution. Section 3 is devoted to a characterization of random sums through the mean orthogonal class. Hürlimann [18] has established that the mean scaled severity in a compound model is necessarily gamma distributed provided that the count distribution and the distribution of the random sum belong to the mean orthogonal class, and some additional partial differential parameter equation can be solved. A followup to this construction for the individual model of risk theory is Hürlimann [19]. We clarify and simplify the original proof to obtain a characterization that is further used in Section 4. Based on a result by Puig and Valero [20], we derive in Theorem 17 a most stringent characterization, which allows compounding of the gamma distribution under a single count data family, namely, the multiparameter Hermite distribution. This one requires that the count distribution is closed under convolution and binomial subsampling.

2. Distributions with the Mean Orthogonal Property

Let 𝑋 be a random variable whose distribution depends upon a vector (𝜇,𝜃)=(𝜇,𝜃₁,…,𝜃_𝑚) of 𝑚+1 parameters, where the mean 𝜇 is functionally independent of 𝜃, that is, 𝜕𝜇/𝜕𝜃_𝑘=0, 𝑘=1,…,𝑚. The log likelihood of 𝑋 is denoted by ℓ(𝑥;𝜇,𝜃). We assume throughout that the cumulant generating function (cgf) 𝐶(𝑡;𝜇,𝜃)=ln{𝐸[exp(𝑡𝑋)]} exists and denotes the variance by 𝜎²=𝜎²(𝜇,𝜃). The standard regularity conditions for maximum likelihood estimation are supposed to hold. The vector 𝐗=(𝑋₁,…,𝑋_𝑛) denotes a random sample of size 𝑛, which realizes the random variable 𝑋, and 𝑋 denotes the sample mean. We are interested in the class 𝐶 of distributions that satisfy Gauss’s principle (the maximum likelihood estimator of the mean is the sample mean), that is, such that 𝜇=𝑋. A distribution belongs to this class if and only if there are functions 𝑔_𝑘=𝑔_𝑘(𝜇,𝜃), 𝑘=1,…,𝑚, ℎ=ℎ(𝜇,𝜃), such that the following equivalent partial differential equations hold (e.g., [15–17]): 𝜕ℓ𝑥;𝜇,𝜃+𝜕𝜇^𝑚_𝑘=1𝑔_𝑘⋅𝜕ℓ𝑥;𝜇,𝜃𝜕𝜃_𝑘+ℎ⋅𝑥−𝜇=0,𝜕𝐶+𝜕𝜇^𝑚_𝑘=1𝑔_𝑘⋅𝜕𝐶𝜕𝜃_𝑘+ℎ⋅𝜕𝐶𝜕𝑡−𝜇=0. (1)

Definition 1. The mean 𝜇 is called orthogonal to the parameter vector 𝜃, denoted by 𝜇⟂𝜃, if one has 𝐸[(𝜕²ℓ/𝜕𝜇𝜕𝜃_𝑘)]=𝐸[(𝜕ℓ/𝜕𝜇)(𝜕ℓ/𝜕𝜃_𝑘)]=0,𝑘=1,…,𝑚.

The original motivation for parameter orthogonality is improvement of maximum likelihood estimation by reparameterization. In the class 𝐶 the number of maximum likelihood equations is reduced by one and parameter orthogonality decreases the often high correlation between the MLEs of the parameters since the MLEs of orthogonal parameters are asymptotically uncorrelated. Indeed, the expectations in (2) are elements of the (expected) Fisher information matrix, which determines the asymptotic covariance matrix of (𝜇,𝜃). In this respect, one is interested in the subclass 𝐶^⟂ of 𝐶 of all distributions satisfying besides 𝜇=𝑋 the mean orthogonal property 𝜇⟂𝜃. This so-called mean orthogonal class is characterized as follows.

Theorem 2 (Characterization of the mean orthogonal class). Let 𝑋 be a random variable with cgf 𝐶(𝑡;𝜇,𝜃) satisfying the above assumptions. Then, one has 𝑋∈𝐶^⟂ if and only if the following quasi-linear partial differential equation is satisfied: 𝜎²⋅𝜕𝐶−𝜕𝜇𝜕𝐶𝜕𝑡−𝜇=0. (2)

Proof. This is shown in Hürlimann [16].

Hudson [17, Theorem 1] has shown that the class 𝐶 is closed under convolution. In fact, convolution invariance holds under the more stringent mean orthogonal property.

Theorem 3 (Convolution invariance of the mean orthogonal class). If 𝑋₁,𝑋₂∈𝐶^⟂ are independent, then 𝑋=𝑋₁+𝑋₂∈𝐶^⟂. More precisely, if 𝑋_𝑖∈𝐶^⟂ has cgf 𝐶_𝑖(𝑡;𝜇_𝑖,𝜃^(𝑖)), with 𝜇_𝑖⟂𝜃^(𝑖),𝑖=1,2, then the cgf 𝐶(𝑡;𝜇,𝛿,𝜃) of 𝑋=𝑋₁+𝑋₂, with 𝜇=𝜇₁+𝜇₂, 𝛿=|𝜇₁𝜎²₂−𝜇₂𝜎²₁|, and 𝜃=(𝜃⁽¹⁾,𝜃⁽²⁾), satisfies (2) and one has 𝜇=𝑋, 𝜇⟂(𝛿,𝜃).

Proof. Without loss of generality we assume that 𝛿=𝜇₁𝜎²₂−𝜇₂𝜎²₁>0. Since 𝜇=𝜇₁+𝜇₂ and 𝜎²=𝜎²₁+𝜎²₂, one can express (𝜇₁,𝜇₂) as a function of (𝜇,𝛿) through the parameter transformation 𝜇₁=(𝜇𝜎²₁+𝛿)/𝜎², 𝜇₂=(𝜇𝜎²₂−𝛿)/𝜎². Since 𝑋₁,𝑋₂∈𝐶^⟂ and 𝐶(𝑡;𝜇,𝛿,𝜃)=𝐶₁(𝑡;𝜇₁,𝜃⁽¹⁾)+𝐶₂(𝑡;𝜇₂,𝜃⁽²⁾), one obtains 𝜎²⋅𝜕𝐶𝑡;𝜇,𝛿,𝜃𝜕𝜇=𝜎²⋅𝜕𝜇₁⋅𝜕𝜇𝜕𝐶₁𝑡;𝜇₁,𝜃⁽¹⁾𝜕𝜇₁+𝜎²⋅𝜕𝜇₂⋅𝜕𝜇𝜕𝐶₂𝑡;𝜇₂,𝜃⁽²⁾𝜕𝜇₂=𝜎²₁⋅𝜕𝐶₁𝑡;𝜇₁,𝜃⁽¹⁾𝜕𝜇₁+𝜎²₂⋅𝜕𝐶₂𝑡;𝜇₂,𝜃⁽²⁾𝜕𝜇₂=𝜕𝐶₁𝑡;𝜇₁,𝜃⁽¹⁾𝜕𝑡−𝜇₁+𝜕𝐶₂𝑡;𝜇₂,𝜃⁽²⁾𝜕𝑡−𝜇₂=𝜕𝐶𝑡;𝜇,𝛿,𝜃𝜕𝑡−𝜇, (3) which implies the result by (2) of Theorem 2.

Example 4. Binomial random variables and their convolutions belong to the class 𝐶^⟂. For two binomials this is shown in Hürlimann [16, Example 2] (see also [21]). For an arbitrary number of binomials this is derived in the Appendix of Hudson [17].

3. Mean Orthogonal Characterization of the Compound Gamma Distribution

Consider random sums of the type 𝑋=^𝑁_𝑖=1𝑌_𝑖, (4) where the 𝑌_𝑖’s are independent and identically distributed nonnegative random variables, and 𝑁 is a counting random variable defined on the nonnegative integers, which is independent of the 𝑌_𝑖’s. The mean and variance of 𝑋, 𝑁, and 𝑌∼𝑌_𝑖 are denoted, respectively, by 𝜇, 𝜎², 𝜇_𝑁, 𝜎²_𝑁, and 𝜇_𝑌, 𝜎²_𝑌. The coefficient of variation of 𝑋 is denoted by 𝜅=𝜎/𝜇. In some applications, it is convenient to scale the severity 𝑌 by the mean 𝜇 such that the mean scaled severity 𝑍=𝑌/𝜇∼𝑍_𝑖=𝑌_𝑖/𝜇 has mean 1/𝜇_𝑁. The resulting sum 𝑋=𝜇⋅^𝑁_𝑖=1𝑍_𝑖 (5) is called mean scaled compound random sum. The mean scaled compound model has important insurance risk applications. It has been studied in Hürlimann [18], which establishes that the mean scaled severity is necessarily gamma distributed provided that the random variables 𝑁 and 𝑋 belong to the mean orthogonal class and some additional partial differential parameter equation can be solved. A followup to this construction for the individual model of risk theory is Hürlimann [19]. We clarify and simplify the original proof to obtain a characterization of (4), which will be used in Section 4. In particular, (3.20) in Hürlimann [18] is not a consequence but an assumption. Since this equation is satisfied in the provided examples, this error does not harm the obtained result but must be rectified from a mathematical logical point of view. Also, the proof of Lemma 7 there will be simplified (proof of Lemma 8 below).

Theorem 5 (Compound gamma characterization). Let 𝑁∈𝐶^⟂ be a counting random variable with cgf 𝐶_𝑁(𝑡;𝑝,𝜃_𝑁). Suppose there exists a one-to-one coordinate transformation mapping (𝑝,𝜃_𝑁) to (𝜇_𝑁,𝜃_𝑁) such that 𝜇_𝑁=𝜇_𝑁(𝑝,𝜃_𝑁)⟂𝜃_𝑁, and set 𝜆_𝑁=−lnPr(𝑁=0). Suppose the cgf 𝐶_𝑌(𝑡) of the severity 𝑌 exists, and let 𝐶(𝑡)=𝐶_𝑁(𝐶_𝑌(𝑡)) be the cgf of the random sum 𝑋=∑^𝑁_𝑖=1𝑌_𝑖. Assume the cgf of the mean scaled severity 𝐶_𝑍(𝑡)=𝐶_𝑌(𝑡/𝜇) is functionally independent of 𝜇, and set 𝛼=(𝜇_𝑁𝜅²)⁻¹, and 𝜅=𝜎/𝜇. If 𝑋∈𝐶^⟂ and 𝜎²⋅(𝜕𝜆_𝑁/𝜕𝜇)=𝜇, then 𝑌 is gamma distributed with cgf 𝐶_𝑌(𝑡)=𝛼⋅ln{𝛼/(𝛼−𝜇_𝑌𝑡)}.

To show this, some preliminaries are required. First, we review conditions under which 𝑁∈𝐶^⟂. Given the probability generating function (pgf) 𝑃(𝑠)=𝐸[𝑠^𝑁]=∑^∞_𝑛=0𝑝(𝑛)𝑠^𝑛 of 𝑁, it is very useful to consider the associated so-called cumulant pgf defined by 𝐺(𝑠)=ln{𝑃(𝑠)}+𝜆_𝑁=^∞_𝑘=1𝑔(𝑘)⋅𝑠^𝑘,𝜆_𝑁=−ln𝑃(0)=𝐺(1). (6) The given name stems from the following series representation of the cgf: 𝐶_𝑁(𝑡)=ln𝑃𝑒^𝑡=𝐺𝑒^𝑡−𝜆_𝑁=^∞_𝑘=1𝑒^𝑘𝑡−1⋅𝑔(𝑘). (7)

Remark 6. The sequence 𝑔(𝑘), 𝑘=1,2,…, is the unique solution of the system of equations (e.g., [22, Corollary 2], [23], and [24, Theorem 1]): 𝑛⋅𝑝(𝑛)=^𝑛_𝑘=1(𝑘)(),(0)𝑘⋅𝑔⋅𝑝𝑛−𝑘𝑛=1,2,…,𝑝=𝑒^−𝜆𝑁. (8) If 𝑔(𝑘)≥0, 𝑘=1,2,…, the distribution of 𝑁 is compound Poisson with parameter 𝜆_𝑁 and severity distribution ℎ(𝑘)=𝑔(𝑘)/𝜆_𝑁, 𝑘=1,2,…. Otherwise, one speaks of the so-called pseudo-compound Poisson representation of the distribution.

Lemma 7. Let 𝑁 be a counting random variable with cgf 𝐶_𝑁(𝑡;𝑝,𝜃_𝑁) of the form (7). Suppose there exists a one-to-one coordinate transformation mapping (𝑝,𝜃_𝑁) to (𝜇_𝑁,𝜃_𝑁), and set 𝜆_𝑁=−lnPr(𝑁=0). Then 𝑁∈𝐶^⟂ with 𝜇_𝑁=𝜇_𝑁(𝑝,𝜃_𝑁)⟂𝜃_𝑁 is equivalent to the following conditions: 𝜎²_𝑁⋅𝜕𝐶_𝑁(𝑡)𝜕𝜇_𝑁=𝜕𝐶_𝑁(𝑡)𝜕𝑡−𝜇_𝑁,𝜇_𝑁⋅𝜕𝐺(𝑠)=𝜕𝑝𝜕𝜆_𝑁𝜕𝑝⋅𝑠⋅𝐺^′(𝑠),𝜇_𝑁⋅𝜕𝑔(𝑘)=𝜕𝑝𝜕𝜆_𝑁(𝑘)𝜕𝑝⋅𝑘𝑔,𝑘=1,2,…. (9)(10)(11)

Proof. The condition (9) is a restatement of Theorem 2. Applying the chain rule of differential calculus, this condition transforms to 𝜎²_𝑁⋅𝜕𝐶_𝑁(𝑡)=𝜕𝑝𝜕𝜇_𝑁⋅𝜕𝑝𝜕𝐶_𝑁(𝑡)𝜕𝑡−𝜇_𝑁. (12) Now, by Lemma 8 below and the chain rule, one has 𝜎²_𝑁⋅𝜕𝜆_𝑁=𝜕𝑝𝜕𝜇_𝑁𝜕𝑝⋅𝜇_𝑁. (13) Inserting into (12) shows that 𝜇_𝑁⋅𝜕𝐶_𝑁(𝑡)=𝜕𝑝𝜕𝜆_𝑁⋅𝜕𝑝𝜕𝐶_𝑁(𝑡)𝜕𝑡−𝜇_𝑁. (14) The statements (10) and (11) follow by using the representation (7).

Lemma 8. If 𝑁∈𝐶^⟂, then the partial differential parameter equation 𝜎²_𝑁⋅(𝜕𝜆_𝑁/𝜕𝜇_𝑁)=𝜇_𝑁 holds.

Proof. The representation (7) implies that 𝜆_𝑁=𝐺(1)=∑^∞_𝑘=1𝑔(𝑘), 𝐶^′_𝑁(0)=∑^∞_𝑘=1𝑘𝑔(𝑘)=𝜇_𝑁. Now, using (7) one sees that (9) is equivalent to 𝜎²_𝑁⋅(𝜕𝑔(𝑘)/𝜕𝜇_𝑁)=𝑘𝑔(𝑘), 𝑘=1,2,…. It follows that 𝜎²_𝑁⋅(𝜕𝜆_𝑁/𝜕𝜇_𝑁)=∑^∞_𝑘=1𝜎²_𝑁⋅(𝜕𝑔(𝑘)/𝜕𝜇_𝑁)=∑^∞_𝑘=1𝑘𝑔(𝑘)=𝜇_𝑁.

Proof of Theorem 5. Let 𝑀_𝑌(𝑡)=exp{𝐶_𝑌(𝑡)} be the moment generating function of 𝑌. Expressed in terms of the mean scaled severity 𝑍=𝑌/𝜇 one has 𝑀_𝑌(𝑡)=𝑀_𝑍(𝜇𝑡). The relationship (7) for the cgf 𝐶_𝑁(𝑡) yields the series expansion: 𝐶(𝑡)=𝐶_𝑁ln𝑀_𝑍𝜇𝑡=^∞_𝑘=1𝑔(𝑘)⋅𝑀^𝑘_𝑍𝜇𝑡−𝜆_𝑁. (15) By Theorem 2 one has 𝑋∈𝐶^⟂ if and only if the equation 𝜎²⋅(𝜕𝐶/𝜕𝜇)−((𝜕𝐶/𝜕𝑡)−𝜇)=0 is satisfied. With the series representation for 𝐶(𝑡) and the assumption 𝜎²⋅(𝜕𝜆_𝑁/𝜕𝜇)=𝜇, this equation is equivalent to the following condition (use that 𝑀_𝑍(𝑡) does not depend on 𝜇): 𝜇−𝜎²𝑡⋅𝑀^′_𝑍𝜇𝑡⋅^∞_𝑘=1𝑘𝑔(𝑘)⋅𝑀_𝑍^𝑘−1𝜇𝑡=𝑀_𝑍𝜇𝑡⋅𝜎²⋅^∞_𝑘=1𝜕𝑔(𝑘)𝜕𝜇⋅𝑀_𝑍^𝑘−1𝜇𝑡. (16) Now, by Lemma 7 and (10), one has the identity (use the differential chain rule) 𝜇_𝑁⋅𝜕𝑔(𝑘)=𝜕𝜇𝜕𝜆_𝑁(𝑘),𝜕𝜇⋅𝑘𝑔 (17) which, together with 𝜎²⋅(𝜕𝜆_𝑁/𝜕𝜇)=𝜇, implies that 𝜎²⋅𝜕𝑔(𝑘)=𝜎𝜕𝜇𝜇²_𝑁⋅𝜕𝜆_𝑁(𝑘)=𝜇𝜕𝜇⋅𝑘𝑔𝜇_𝑁⋅𝑘𝑔(𝑘). (18) Inserted into the above expression one obtains the ordinary differential equation: 𝜇_𝑁⋅1−𝜅²𝑡⋅𝑀^′_𝑍(𝑡)=𝑀_𝑍(𝑡), (19) whose unique solution is 𝑀_𝑍(𝑡)=(1−𝜅²𝑡)^−𝛼. Since 𝜅²𝜇=𝜇_𝑌/𝛼, one sees that 𝐶_𝑌(𝑡)=ln𝑀_𝑍𝛼𝜇𝑡=𝛼⋅ln𝛼−𝜇_𝑌𝑡 (20) is the cgf of a gamma-distributed random variable. The proof is complete.

The proof uses the so-called natural parameterization (𝑝,𝜃_𝑁,𝜇_𝑌,𝛼) of the compound gamma distribution. It is interesting to obtain explicit parameters orthogonal to the means of 𝑁,𝑌, and 𝑋. By the assumption 𝑁∈𝐶^⟂ one has 𝜇_𝑁=𝜇_𝑁(𝑝,𝜃_𝑁)⟂𝜃_𝑁, and since 𝑌 is gamma distributed, one has 𝑌∈𝐶^⟂ with 𝜇_𝑌⟂𝛼. It remains to construct a parameter vector orthogonal to the mean of 𝑋 such that 𝜇=𝜇_𝑁𝑝,𝜃_𝑁⋅𝜇_𝑌⟂𝜃_𝑁,𝛼,𝜃, (21) where 𝜃=𝜃(𝑝,𝜃_𝑁,𝜇_𝑌,𝛼) must be determined. This task can be solved in a unified way for a lot of counting distributions (see [18, Section 4]). To illustrate the method, it suffices to consider here a single example.

Example 9 (compound negative binomial gamma distribution). Let 𝑁∼𝑁𝐵(𝜃_𝑁,𝑝), 𝜃_𝑁>0, 𝑝∈(0,1), be a negative binomial random variable. Its cumulant pgf (6) reads 𝐺(𝑠)=−𝜃_𝑁⋅ln(1−𝑝⋅𝑠), 𝜆_𝑁=−𝜃_𝑁⋅ln(1−𝑝). One has the following identity (see [18, equation (4.7)]): 𝑝𝜕𝐺(𝑠)𝜕𝑝=𝑠⋅𝐺^′(𝑠), (22) which implies for 𝑠=1 that 𝑝(𝜕𝜆_𝑁/𝜕𝑝)=𝜇_𝑁. Together, this shows that (10) is satisfied. Therefore, one has 𝑁∈𝐶^⟂ and 𝜇_𝑁=𝜃_𝑁⋅𝑝/(1−𝑝)⟂𝜃_𝑁. Now, by Theorem 5 the compound negative binomial will be a compound negative binomial gamma if 𝑋∈𝐶^⟂ and 𝜎²⋅(𝜕𝜆_𝑁/𝜕𝜇)=𝜇 is satisfied. Written in terms of the parameter 𝛼=(𝜇_𝑁𝜅²)⁻¹, 𝜅=𝜎/𝜇, the latter equation is equivalent to the condition (1/𝜇_𝑁)⋅(𝜕𝜆_𝑁/𝜕𝜇)=𝛼/𝜇. With 𝜕𝜆_𝑁/𝜕𝜇=(𝜕𝜆_𝑁/𝜕𝑝)⋅(𝜕𝑝/𝜕𝜇)=(𝜇_𝑁/𝑝)⋅(𝜕𝑝/𝜕𝜇) one obtains the differential equation (1/𝑝)⋅(𝜕𝑝/𝜕𝜇)=𝛼/𝜇, which has the solution 𝑝=𝜃⋅𝜇^𝛼 for some 𝜃. In the coordinates (𝑝,𝜃_𝑁,𝜇_𝑌,𝛼), this constant is equal to 𝜃=𝜃𝑝,𝜃_𝑁,𝜇_𝑌,𝛼=𝑝⋅𝜇^−𝛼=𝑝⋅𝜇_𝑌𝜃_𝑁𝑝1−𝑝^−𝛼. (23) Since 𝑋∈𝐶^⟂ one must have 𝜇=𝜇_𝑁(𝑝,𝜃_𝑁)⋅𝜇_𝑌⟂(𝜃_𝑁,𝛼,𝜃).

4. Mean Orthogonal Characterization of the Compound Multiparameter Hermite Gamma

The mean orthogonal characterization of the compound gamma distribution allows for a wide variety of count data distributions in the mean orthogonal class. In order to reduce further the possible set of count distributions that can be used, one can ask for characterizations in terms of additional assumptions. For example, Puig [25] and Puig and Valero [26] characterize count data distributions satisfying Gauss’s principle and several notions of additivity, which via Theorem 5 can be translated to characterizations of compound gamma distributions. Based on a result by Puig and Valero [20], we derive a most stringent characterization, which allows compounding of the gamma distribution under a single count data family, namely, the multiparameter Hermite distribution. To show this, some additional preliminaries are required.

Definition 10. Let 𝑁 be a counting random variable, let 𝐵₁(𝑝),𝐵₂(𝑝),…, be independent and identically distributed Bernoulli random variables with probability of success 𝑝∈(0,1], and let 𝑁 be independent of 𝐵_𝑖(𝑝),𝑖=1,2,…. Then 𝑁(𝑝)=∑^𝑁_𝑖=1𝐵_𝑖(𝑝)(𝑋(𝑝)=0if𝑁=0) is called an independent p-thinning of 𝑁.

Definition 11. Let F be a family of count distributions. It is called closed under binomial subsampling if, for any random variable 𝑁 with distribution in F, all its independent p-thinnings, for all 𝑝∈(0,1], have distributions in F.

Definition 12. Let F be a family of distributions. It is called closed under convolution if, for any two independent random variables 𝑋,𝑌 with distributions in F, the distribution of the sum 𝑋+𝑌 also belongs to F.

Definition 13. Let 𝑁 be an integer random variable with pgf 𝑃(𝑠) and factorial cumulant generating function (fcgf) 𝐶^𝑓_𝑁(𝑠)=ln𝑃(𝑠+1). For any integer 𝑛≥1 the 𝑛th factorial cumulant of 𝑁 is defined and denoted by 𝜅_(𝑛)=𝑑^𝑛𝐶^𝑓_𝑁(𝑠)/𝑑𝑠^𝑛|_𝑠=0.

There is only one count distribution family closed under convolution and binomial subsampling.

Theorem 14 (Characterization of the multiparameter Hermite distribution). Let 𝐅 be a family of count distributions parameterized by its 𝑟 first factorial cumulants 𝜃=(𝜅₍₁₎,…,𝜅_(𝑟)) and assume that its pgf 𝑃(𝑠) is continuous in 𝜃 over its parameter space. Then 𝐅 is closed under convolution and binomial subsampling if and only if the pgf is of the form 𝑃(𝑠)=exp^𝑟_𝑘=1𝜅_(𝑘)()𝑘!𝑠−1^𝑘. (24)

Proof. See Puig and Valero [20], proof of Theorem 1.

Some comments are in order. The case 𝑟=1 corresponds to the Poisson distribution, 𝑟=2 is the Hermite distribution (e.g., [27]). For arbitrary 𝑟, this distribution is called the multiparameter Hermite distribution of order 𝑟 by Milne and Westcott [28]. In terms of the cumulant pgf (6), the representation (24) can be rewritten as 𝑃(𝑠)=exp^𝑟_𝑘=1𝑔(𝑘)⋅𝑠^𝑘−1, (25) where 𝑔(𝑘),𝑘=1,2,…,𝑟, solves the system in (8), that is, 𝑛⋅𝑝(𝑛)=^{min(𝑟,𝑘)}_𝑘=1𝑘⋅𝑔(𝑘)⋅𝑝(𝑛−𝑘),𝑛=1,2,…,_𝑟𝑘=1𝑔(𝑘)=−ln𝑝(0). (26) The case 𝑟=2 of (26) is already in A.W. Kemp and C.D. Kemp [29], and for arbitrary 𝑟 this assertion is equivalent to Lemma 2 in Puig and Valero [20]. The special case 𝑔(1)>0, 𝑔(𝑟)≥0, 𝑔(𝑘)=0, 𝑘=2,…,𝑟−1 is the generalized Hermite by Gupta and Jain [30]. The multiparameter Hermite belongs also to the Kumar [31] family of distributions. In general, the conditions on the sequence 𝑔(𝑘), 𝑘=1,2,…,𝑟, under which (25) defines a true probability distribution have been identified in Lévy [32]. According to Lukacs [33, page 252] and Johnson et al. [34, page 356], this is the case provided that a negative value 𝑔(𝑘)<0 is preceded by a positive value and followed by at least two positive values. In particular, if at least 𝑔(1),𝑔(𝑟−1),𝑔(𝑟) are nonzero, then 𝑔(1)>0,𝑔(𝑟−1)>0,𝑔(𝑟)>0, are necessary conditions for (25) to be a pgf [28, Remark 1]. If 𝑔(𝑘)≥0 for 𝑘=1,2,…,𝑟, then the multiparameter Hermite is compound Poisson with parameter 𝜆_𝑁=∑^𝑟_𝑘=1𝑔(𝑘) and severity ℎ(𝑘)=𝑔(𝑘)/𝜆_𝑁, thus infinitely divisible by Feller [35, Section XII.2]. Due to the next result, the multiparameter Hermite is of interest in the context of Gauss’s principle, orthogonal parameters to the mean, and the related compound gamma characterization of random sums.

Lemma 15. Let 𝑐_𝑘(𝜃), 𝑘=1,2,…,𝑟, be continuous real functions in the parameter vector 𝜃 over some parameter space, and set 𝑔(𝑘)=𝑐_𝑘(𝜃)⋅𝑝^𝑘, 𝑘=1,2,…,𝑟, for a parameter 𝑝>0. Assume that the cumulant pgf 𝐺(𝑠)=∑^𝑟_𝑘=1𝑐_𝑘(𝜃)⋅(𝑝𝑠)^𝑘 defines a feasible multiparameter Hermite random variable 𝑁 of order 𝑟 over the parameter space. Then 𝑁∈𝐶^⟂ and 𝜇_𝑁=𝜇_𝑁(𝑝,𝜃)=∑^𝑟_𝑘=1𝑘⋅𝑐_𝑘(𝜃)⋅𝑝^𝑘⟂𝜃.

Proof. Set 𝜆_𝑁=𝐺(1)=∑^𝑟_𝑘=1𝑐_𝑘(𝜃)⋅𝑝^𝑘. Then one has (𝑠)𝑝⋅𝜕𝐺𝜕𝑝=𝑠⋅𝐺^′(𝑠),𝑝⋅𝜕𝜆_𝑁𝜕𝑝=𝜇_𝑁. (27) Together, this shows that (10) is satisfied. The result follows by Lemma 7.

Example 16 (Hermite distribution (𝑟=2)). Suppose the Hermite distribution is parameterized by its first two factorial cumulants 𝜅₍₁₎,𝜅₍₂₎. Since 𝜅₍₁₎=𝜇_𝑁, 𝜅₍₂₎=𝜎²_𝑁−𝜇_𝑁, it can equivalently be parameterized by its mean 𝜇_𝑁 and variance 𝜎²_𝑁. Consider a parameterization 𝑝>0, 𝜃_𝑁>0 such that 𝑔(𝑘)=𝜃_𝑁⋅𝑝^𝑘, 𝑘=1,2. There exists a one-to-one mapping between (𝑝,𝜃_𝑁) and (𝜇_𝑁,𝜃_𝑁). Since 𝜇_𝑁=𝑔(1)+2𝑔(2),𝜎²_𝑁=𝑔(1)+4𝑔(2), it is determined by the coordinate transformation: 1𝑝=2⋅𝜎²_𝑁−𝜇_𝑁𝜎²_𝑁+2𝜇_𝑁,𝜃_𝑁𝜎=2⋅^2𝑁+2𝜇_𝑁²𝜎²_𝑁−𝜇_𝑁. (28) Therefore, the cumulant pgf 𝐺(𝑠)=𝜃⋅(𝑝𝑠+𝑝²𝑠²) defines a feasible two-parameter Hermite distribution such that the corresponding random variable belongs to 𝐶^⟂ and 𝜇_𝑁=𝜇_𝑁(𝑝,𝜃_𝑁)=𝜃_𝑁⋅𝑝⋅(1+2𝑝)⟂𝜃_𝑁. Since 𝜎²_𝑁>𝜇_𝑁 one notes that the Hermite distribution is necessarily overdispersed. As noted by Puig and Valero [20] overdispersion holds for all infinitely divisible multiparameter Hermite distributions of arbitrary order 𝑟≥2. Therefore, it should be useful to analyze data with this property (e.g., claim number data in automobile insurance, up-to-date Hürlimann [36, page 802], and multiparameter Hermite for 𝑟≥3).

We are ready for the following new characterization result.

Theorem 17 (Compound multiparameter Hermite gamma characterization). Let 𝑁 be a counting random variable parameterized by its 𝑟 first factorial cumulants 𝜉=(𝜅₍₁₎,…,𝜅_(𝑟)) and assume that its cgf 𝐶_𝑁(𝑡;𝜉) is continuous in 𝜉 over its parameter space and set 𝜇_𝑁=𝜅₍₁₎, 𝜆_𝑁=−lnPr(𝑁=0). Suppose the cgf 𝐶_𝑌(𝑡) of the severity 𝑌 exists, and let 𝐶(𝑡)=𝐶_𝑁(𝐶_𝑌(𝑡)) be the cgf of the random sum 𝑋=∑^𝑁_𝑖=1𝑌_𝑖. Assume the cgf of the mean scaled severity 𝐶_𝑍(𝑡)=𝐶_𝑌(𝑡/𝜇) is functionally independent of 𝜇, and set 𝛼=(𝜇_𝑁𝜅²)⁻¹, 𝜅=𝜎/𝜇. Assume 𝑁 is closed under convolution and binomial subsampling, 𝑋∈𝐶^⟂ and 𝜎²⋅(𝜕𝜆_𝑁/𝜕𝜇)=𝜇. Then 𝑁 is a multiparameter Hermite distribution of order 𝑟, and 𝑌 is gamma distributed with cgf 𝐶_𝑌(𝑡)=𝛼⋅ln{𝛼/(𝛼−𝜇_𝑌𝑡)}. Furthermore, there exists a parameterization (𝑝,𝜃_𝑁) of N such that its cumulant pgf reads 𝐺(𝑠)=∑^𝑟_𝑘=1𝑐_𝑘(𝜃_𝑁)⋅(𝑝𝑠)^𝑘. One has 𝑁,𝑌,𝑋∈𝐶^⟂ with 𝜇_𝑁=𝜇_𝑁(𝑝,𝜃_𝑁)=∑^𝑟_𝑘=1𝑘⋅𝑐_𝑘(𝜃_𝑁)⋅𝑝^𝑘⟂𝜃_𝑁, 𝜇_𝑌⟂𝛼, 𝜇=𝜇_𝑁(𝑝,𝜃_𝑁)⋅𝜇_𝑌⟂(𝜃_𝑁,𝛼,𝜃), and, in the coordinates (𝑝,𝜃_𝑁,𝜇_𝑌,𝛼), the constant 𝜃 is equal to 𝜃=𝜃𝑝,𝜃_𝑁,𝜇_𝑌,𝛼=𝑝⋅𝜇^−𝛼=𝑝⋅𝜇_𝑌⋅^𝑟_𝑘=1𝑘⋅𝑐_𝑘𝜃_𝑁⋅𝑝^𝑘−𝛼. (29)

Proof. The result follows by combining Theorems 5 and 14 making the observation that a multiparameter Hermite distribution can always be put in the form of Lemma 15 (generalization of Example 16). The assertion about the orthogonal parameters to the means 𝜇_𝑁,𝜇_𝑌,𝜇 follows along the same arguments as in Example 9 using (27).