About the Stein equation for the generalized inverse Gaussian and Kummer distributions

We propose a Stein characterization of the Kummer distribution on (0, $\infty$). This result follows from our observation that the density of the Kummer distribution satisfies a certain differential equation, leading to a solution of the related Stein equation. A bound is derived for the solution, under a condition on the parameters. The derivation of this bound is carried out using the same framework as in Gaunt 2017 [A Stein characterisation of the generalized hyper-bolic distribution. ESAIM: Probability and Statistics, 21, 303--316] in the case of the generalized inverse Gaussian distribution, which we revisit by correcting a minor error in the latter paper.


Introduction
For a > 0, b ∈ R, c > 0, the Kummer distribution with parameters a, b, c has density k a,b,c (x) = 1 Γ(a)ψ(a, a − b + 1; c) where ψ is the confluent hypergeometric function of the second kind. The generalized inverse Gaussian (hereafter GIG) distribution with parameters p ∈ R, a > 0, b > 0 has density g p,a,b (x) = (a/b) p/2 2K p ( √ ab) x p−1 e − 1 2 (ax+b/x) , x > 0, where K p is the modified Bessel function of the third kind. For details on GIG and Kummer distributions see [4,5,11] and references therein, where one can see for instance that these distributions are involved in some characterization problems related to the so-called Matsumoto-Yor property.
In this paper, these two distributions are considered in the context of Stein's method. This method introduced in [9] is a technique used to bound the error in the approximation of the distribution of a random variable of interest by another probability (for instance the normal) distribution. For an overview of Stein's method see [1,7]. The first steps of this method consist in finding an operator called Stein operator characterizing the targeted distribution, then solving the corresponding socalled Stein equation.
One finds in [9] a seminal instance of the method, where Stein showed that a random variable X has a standard normal distribution if and only if for all real-valued absolutely continuous function f such that E |f ′ (Z)| < ∞ for Z ∼ N(0, 1), where h is a bounded function and Z a random variable following the standard nor- If a function f h is a solution of the previous equation, then for any random variable Thus, in order to bound |E(h(U)) − E(h(Z))| given h, its enough to find a solution f h of the Stein equation and to bound the left-hand side of the previous equation. The problem of solving the Stein equation for other distributions than the standard normal distribution and bounding the solution and its derivatives has been widely studied in the literature (see [3] among many others).
The aim of this paper is to solve the Stein equation and derive a bound of the solution for the Kummer distribution (which is new) and for the generalized inverse Gaussian distribution (which has been done in [2], but there was a little mistake in the bound of the solution).
The idea of this paper emerged by reading the remarkable work by [2] about a Stein characterization of the generalized hyperbolic distribution of which the generalized inverse Gaussian distribution (GIG) is a limiting case. Among many other results, [2] solved the GIG Stein equation and bounded the solution by using a general result obtained in [8] when the targeted distribution has a density g satisfying for some polynomial functions s and τ . Also a bound was obtained for the solution under the condition that the function τ be a decreasing linear function. But since this linearity condition does not hold in the GIG case, the bound given by [2] has to be slightly corrected. This is done in Theorem 3.1 after recalling the general framework of Schoutens [8] and adapting it to the cases where τ is decreasing but not necessarily linear. Indeed, we realized that the procedure adopted in [8] still works, via a slight change, even if τ is not linear.
Observing that the Kummer density also satisfies (1.1), we can use the same methodology (Theorem 4.2) for this distribution. We have to put the restrictions p ≤ −1 for the GIG density and 1 − b − c ≤ 0 for the Kummer density in order for the corresponding function τ to be decreasing on (0, ∞).
In Section 1 we recall the general framework established by [8] for densities g satisfying (1.1) without the assumption of linearity of τ . We retrieve the Stein operator given in [8] by using the density approach initiated in [10] and further developed in [6].
In Section 2 we show the application of this method to the GIG distribution as mentioned in [2] by giving the right bound for the solution of the Stein equation. Section 3 is devoted to the Stein characterization and the Stein equation related to the Kummer distribution.
2 Stein characterization in the Schoutens framework Theorem 1 in [8] addressed the problem of establishing a Stein characterization for probability distributions with density g satisfying (1.1) for some polynomial functions s et τ , and proved that a Stein operator in this case is f → sf ′ + τ f . We realized (see the following theorem) that the same Stein operator can be arrived at by using the density approach designed in [10] and [6]. The support of the density may be any interval, but here we take this support to be (0, ∞) in the purpose of the application to the GIG and Kummer distributions. Theorem 2.1 Consider a density g on (0, ∞) such that (1.1) holds for some polynomial functions s and τ . Then a positive random variable X has density g if and only if for any differentiable function f such that lim Proof: We use Corollary 2.1 of [6]. According to this corollary, a Stein operator related to the density function g is Applyng this operator to sf , we have Theorem 2.1 shows that the Stein equation related to any density g satisfying (1.1) enjoys the tractable form where W is random variable with density g. Schoutens [8] found a solution to the Stein equation (2.1) and established a bound for the solution, under the condition that the function τ be a decreasing linear function (which is the case for the so-called Pearson and Ord classes of distributions considered in [8]).
The following result comes from Proposition 1 in [8]. We again take the support of the density function to be (0, ∞).
where C is constant.
• Suppose lim x→0 s(x)g(x) = 0. For the solution to be bounded, it is necessary that

Proof:
Multiplying both sides of (2.1) by g(x) we have which, by (1.1), can be written As a consequence, there exists a constant C such that The second expression for f h follows from the fact that, since W has density g, The following proposition proves that the solution given by (2.2) is bounded indeed if h is bounded, and thus is the unique bounded solution to the Stein equation associated to the density g. A bound is provided.  [8] without the assumption that τ is linear. With this assumption, [8] established the same bound with α = E(X) (for a random variable X with density g), which is not true if τ is not linear. The proof given below follows the lines of that of [8] where we observed that the assumption of linearity of τ was used nowhere except to state that its only zero is α = E(X).

The proof of Proposition 2.3 uses the following lemma :
Lemma 2.1 Under the assumptions of Proposition 2.3, Proof: Suppose x < α. Since τ is positive and decreasing on (0, α), we have because of (1.1) and as lim t→0 s(t)g(t) = 0.
In the two next sections we apply the previous results to the GIG and Kummer distributions.

About the Stein equation of the generalized inverse Gaussian distribution
Recall that the density of the GIG distribution with parameters p ∈ R, a > 0, b > 0 is where K p is the modified Bessel function of the third kind. Let Then, as observed by [2], the GIG density g p,a,b satisfies This enables us to apply Theorem 2.1 to retrieve the following Stein characterization of the GIG distribution given in [4] and [2]: The corresponding Stein equation is where h is a bounded function and W a random variable following the GIG distribution with parameters p, a, b.
We apply Proposition 2.2 and Proposition 2.3 to solve Equation (3.3) and bound the solution. Let us check that the assumptions of these propositions are true in the GIG case.
where W follows the GIG distribution with parameters p, a, b.
If h is a bounded continuous function and p ≤ −1, then the function defined by (3.4) is the unique bounded solution of (3.3) and Remark 3.1 This result was claimed by Gaunt (see [2]) with α = E(X) by applying Proposition 1 of [8]. The only slight mistake is that τ is not a polynomial function of degree one as in [8].

About the Stein equation related to the Kummer distribution
Recall that for a > 0, b ∈ R, c > 0, the Kummer distribution K(a, b, c) has density where ψ is the confluent hypergeometric function of second kind. Let The corresponding Stein equation is where W has density k a,b,c . We have  Remark 4.1 These results could be used in future work to provide rates of convergence in limit problems related to the GIG and Kummer distributions.