A Convergence Theorem for Bivariate Exponential Dispersion Models

Corresponding Authors: Lila Ricci CEMIM, UNMdP, MdP, Argentina E-mail: lricci@mdp.edu.ar Abstract: Multivariate exponential dispersion models (MEDMs) were defined in 2013 by Jørgensen and Martínez. A particular case of MEDM is the bivariate Gamma model; in this article we prove that, under certain conditions, this is a limit distribution for MEDM generated by bivariate regularly varying measures, extending a previous result given by the aforementioned authors for the univariate case. As necessary tools for proving the main result, we use bivariate regularly varying functions and bivariate regularly varying measures; we also state a bivariate version of Tauberian Karamata’s theorems and a particular Karamata representation of bivariate slowly varying functions.


Introduction
The aim of this article is to extend to bi-dimensional space the result about Tauber type convergence of exponential dispersion models (EDM)s to a Gamma model, that has been proved by Jørgensen et al. (1994).
It was Tweedie (1947) pointed out the main properties of EDMs; but his ideas remained unknown for decades. On the other hand, Nelder and Wedderburn (1972) introduced a new class of statistical models named Natural Exponential Family, just when computational tools were being developed in such a way that it became possible to perform the required computations. Their importance comes from the fact that they can represent the error distribution in Generalized Linear Models (GLMs).
Later on, Jørgensen (1987) rescued Tweedie's ideas and defined an extended family of distributions named Exponential Dispersion Model. He published systematic studies of mathematical properties of EDMs in 1986 and 1987. In his own words, the main raison d'être of EDMs is to broaden the field of GLMs introduced by Nelder and Wedderburn (1972) allowing the researchers to choose between infinite probability distributions the one that optimally represents their data.
Under certain conditions, it has been proved that some EDMs converge to the Gamma distribution. An essential tool in the study of these domains of attraction has been the theory of regularly varying functions, they arise when the mean parameter goes to zero or infinity, while the dispersion parameter remains constant or asymptotically constant. That is why this convergence has been called of regular variation type. Using these resources Jørgensen et al. (1994) proved an important theorem, making it possible to asses Gamma convergence of some EDMs under weaker conditions than those required for asymptotic convergence of variance functions.
On the other hand, Jørgensen (2013;Jørgensen and Martínez, 2013) developed a unified methodology to build Multivariate Exponential Dispersion Models (MEDMs) with fixed known marginals and a flexible correlation structure. Based on previous univariate studies about EDMs convergence (Jørgensen et al., 1994;2009) they could conjecture that previous results might be extended to those MEDMs that they had just defined. Moreover, MEDMs are important for practitioners because they broaden the parametric distribution family covered by GLMs derived from Multivariate Natural Exponential Family.
In the next Section we give the definition of MEDMs; then we define bivariate regular variation and we state Karamata's theorems and we prove an asymptotic property of MEDMs, that is the main result of this article. Finally, some conclusions are developed. There are also three appendixes with some detailed calculations.

Multivariate Exponential Dispersion Models
A method to obtain MEDMs has been presented by Jørgensen (2013;Jørgensen and Martínez, 2013). It is based on an extended convolution method that ensures a k+k(k+1)/2 parameters distribution for k-dimensional models, with marginal distributions that belong to the same family; these new MEDMs have a flexible covariance structure. In this section the construction of the bivariate EDM will be detailed. Consider the probability density function: we will denote it by Y ∼ ED(µ, Σ). One slight disadvantage of the method is that only positive correlations are obtained. The parameter domain that ensures that K*(s 1 , s 2 ;θ) is a CGF could be broaden in order to admit negative correlations, but this issue remains to be investigated. With this in mind, Cuenin et al. (2015) give a variables-incommon method for constructing multivariate distributions admitting negative correlations, but it is restricted to Tweedie models.

A Particular Case: Bivariate Gamma
While passing from uni to multivariate distributions there is more than one direction to choose. In words of Letac: "While the names of distributions in ℝ are generally unambiguous, at the contrary in the jungle of distributions in k ℝ almost nothing is codified outside of the Wishart and Gaussian cases".
Let us consider Kibble and Moran bivariate Gamma distribution as given by Kotz et al. (2000) whose cumulant function is: The mean vector is:  , , in Appendix C (a detailed treatment can also be seen in Boggio's (2019)). The moment generating function (MGF) of the reproductive model in terms of µ is then: ; , 1 1 Note that when φ = 0, meaning independence, (1) becomes:

Bivariate Regular Variation and Karamata Theorems
In this section we present some definitions and results that will be needed in the next section. They include bivariate regular variation, Karamata Tauberian theorems about Laplace transforms and Karamata representation.

Bivariate Regular Variation
Regular variation functions were defined by J. Karamata (Goldie et al., 1987;de Haan, 1975), they behave asymptotically as their Laplace transforms. The next definition was given by Omey and Willekens (1989), extending the concept of regular variation to 2 + ⋅ ℝ

Defnition 1. A measurable function
y > 0 and t > 0, the limit: , , min t s u tx sy lim x y u t s α β →∞ = exists and is finite.
We will denote a regularly varying function at infinity with indexes α and β by u∈VR (α, β) ∞ ; analogously u(x, y) is regularly varying at 0 if is regularly varying at infinity and we will denote it by u∈ VR (α, β) 0 . If α = β = 0 the function is said to be slowly varying at infinity (zero) and we will denote it by L∈VL ∞(0) . The concept of regular variation can be extended to measures as follows.
Defnition 2: A measure ν on 2 + ℝ is said to vary regularly at infinity or zero with indexes , α β ∈ ℝ if the distribution

Bivariate Karamata's Theorems
Next we extend a theorem stated by Jørgensen et al. (1994), that relates regular variation of a measure with regular variation of its Laplace transform. Hereafter the notation "f(x)∼kg(x) when x→∞" means that Let ν be a measure on when min(t, s) → ∞, L∈VL ∞, α and β being non negative numbers and v the function given in Definition 2.
Proposition 1. The statement on the left is equivalent to affirm that Theorem 1 allows us to say that the MGF of the natural exponential family generated by such a measure takes the form: On the other hand, de Haan and Resnick (1987) proved an extension of Karamata representation to the multivariate regular variation case; we are interested in the particular case of bivariate slow variation. A slowly varying function 2 : L + → ℝ ℝ can be represented as:

The Main Result
In the univariate case several convergence theorems have been proved considering three types of convergence: central limit type, infinitely divisible type and regular variation type (Nielsen, 2000;Jørgensen, 1997: p149). Our main result concerns convergence of regular variation type, where the dispersion parameter remains constant while the mean tends to zero or infinity. In this section we extend a theorem that has been proved in the univariate context to the bivariate case. This theorem asses the asymptotic equivalence between some bivariate EDM and the bivariate Kibble and Moran's Gamma distribution, with weaker assumptions than previous theorems. Extension to k + ℝ is straightforward.
We will introduce now some notation in order to simplify next developments. Let the bivariate EDM (µ, Λ) be generated by a measure ν with support S ⊆ (0,∞) × (0,∞). If ν is regularly varying at zero or infinity with index α > 0, then given (3) its CGF is: where, l(x, y) = log L(x, y). The mean value vector takes the form: We define functions δ i (µ 1 , µ 2 ), i = 1, 2 as: , , , and the vector τ −1 (µ) can be expressed as: Now the main theorem can be stated.
Theorem 2: Let Y ∼ ED (µ, Σ) be a bivariate EDM generated by the measure ν with support S⊆(0,∞)×(0,∞). Suppose that ν is regularly varying at zero or infinity with the same index on both variables. Given (3) and if l(x, y) satisfies: The theorem will be proved for ν regularly varying at zero (L slowly varying in infinity); the proof for ν regularly varying at infinity is similar.
Proof: Let Z = (Z 1 , Z 2 ) T ∼ED * (θ, Λ) be the bivariate additive EDM generated by ν, constructed as described above with MGF:  , , and MGFs properties, the MGF for the perturbed reproductive model for µ i >0 fix, i = 1,2 and c small enough to ensure that cµ∈Ω:  . , Let us denote by h i (s; c, µ, Λ) the expressions with exponent λ i (i = 1,2) and by h 12 (s; c, µ, Λ) the one with exponent λ12 so the MGF can be written as follows: The following equalities will be proved: To prove (9a) note that from (6) and given condition (7), we have that: Before proving (9b) and in order to simplify the notation we define c µ Λ s we apply (4), in such a way that h 1 (s; c, µ, Λ), with τ −1 (cµ) in terms of ɶ θ can be expressed as: being ⋅ any norm and We also have that: Now, given that Putting together both results (11) and (12) and replacing in (10) and this is the expression for the MGF of the bivariate dispersion model Γ (µ, Σ α ) for independent variables, as was proved in (2). The matrix Σ α takes the following form: We present next an example of a bivariate EDM generated by a regularly varying measure that satisfies the conditions required by Theorem 2.