An iterative technique for bounding derivatives of solutions of Stein equations

Abstract

We introduce a simple iterative technique for bounding derivatives of solutions of Stein equations Lf = h − Eh(Z), where L is a linear differential operator and Z is the limit random variable. Given bounds on just the solutions or certain lower order derivatives of the solution, the technique allows one to deduce bounds for derivatives of any order, in terms of supremum norms of derivatives of the test function h. This approach can be readily applied to many Stein equations from the literature. We consider a number of applications; in particular, we derive new bounds for derivatives of any order of the solution of the general variancegamma Stein equation. Finally, we present a connection between Stein equations and Poisson equations, from which we first recognised the importance of the iterative technique to Stein's method.1 for all functions f belonging to some measure determining class. For continuous random variables, L is a differential operator; for discrete random variables, L is a difference operator. Such characterisations have often been derived via Stein's density approach [53], [54] or the generator approach of Barbour and Götze [3], [32]. The scope of the density approach has recently been extended by [38], and other techniques for obtaining Stein characterisations are discussed in that work.The characterisation (1.1) leads to the so-called Stein equation:where the test function h is real-valued. The second step of Stein's method, which will be the focus of this paper, concerns the problem of obtaining a solution f to the Stein equation (1.2) and then establishing estimates for f and some of its lower order derivatives (for continuous distributions). Evaluating (1.2) at a random variable W and taking expectations gives3) and thus the problem of bounding the quantity Eh(W ) − Eh(Z) reduces to solving (1.2) and bounding the right-hand side of (1.3). The third step of Stein's method concerns the problem of bounding the expectation E[Lf (W )]. For continuous limit distributions, such bounds are usually obtained via Taylor expansions and coupling techniques. For many classical distributions, the problem of obtaining the first two necessary ingredients is relatively tractable. As a result, over the years, Stein's method has been adapted to many standard distributions, including the beta [11], [30], gamma [28], [39], exponential [5], [19], [45], Laplace [48] and, more generally, the class of variance-gamma distributions [21]. The method has also been adapted to distributions arising from specific problems, such as preferential attachment graphs [46], the Curie-Weiss model [6] and statistical mechanics [13], [14]. For a comprehensive overview of the literature, see [38]. Estimates for solutions of Stein equationsThe Stein equations of many classical distributions, such as the normal, beta and gamma, are linear first order ODEs with simple coefficients. As a result, the problem of solving the Stein equation and bounding the derivatives of the solution is reasonably tractable.In fact, for Stein characterisation...

A Stein characterisation of the generalized hyperbolic distribution

“…Finally, applying a shift by µ gives a Stein operator the scaled and shifted Student's t-distribution with density (2.6): 15) which is in agreement with the Stein operator of [15] for the scaled and shifted Student's tdistribution. In the special case δ = √ ν, µ = 0, we obtain the Stein operator for Student's t-distribution that was obtained by [42].…”

“…In the special case δ = √ ν, µ = 0, we obtain the Stein operator for Student's t-distribution that was obtained by [42]. It is interesting to note that the solution of the Stein equation corresponding to (3.15) is bounded if the test function h is bounded, and that its n-th derivative is bounded if ν − 2(n − 1) > 0 and the derivatives of h up to (n − 1)-th order are bounded (see [15]). …”

“…To determine explicit bounds for f , f ′ and higher order derivatives of f , we would need to bound a number of expressions involving integrals of modified Bessel functions. Bounds of expressions of a similar form to those we would encounter are given in [23], and were used to bound the derivatives of the solution of the variance-gamma Stein equation (see [21] and [15]). However, since we do not use the GH Stein equation to prove any approximation results in this paper, we omit this analysis.…”

“…The solution of this Stein equation and its first derivative are bounded if h is bounded (see [21], Lemma 3.3), and it was shown by [15] that the n-th derivative of the solution is bounded if all derivatives of h up to (n − 1)-th order are bounded.…”

“…To date, no bounds have been obtained for solution of the general GIG(λ, a, b) Stein equation (3.18). However, the limiting case a → 0, which is the inverse-gamma distribution, was treated by [15]. They showed that the solution of the Stein equation (with a = 0) is bounded if λ < −1 and the test function h is bounded, and that the n-th derivative of the solution is bounded if λ < −(2n + 1) and the n-th derivative of h exists and is bounded.…”