Abstract
We build upon recent advances on the distributional aspect of Stein's method to propose a novel and flexible technique for computing Stein operators for random variables that can be written as products of independent random variables. We show that our results are valid for a wide class of distributions including normal, beta, variance-gamma, generalized gamma and many more. Our operators are kth degree differential operators with polynomial coefficients; they are straightforward to obtain even when the target density bears no explicit handle. As an application, we derive a new formula for the density of the product of k independent symmetric variance-gamma distributed random variables.
Highlights
In 1972, Charles Stein (1920–2016) [41] introduced a powerful method for estimating the error in normal approximations
We provide an answer for functionals of the form F (x1, . . . , xd) = xα1 1 · · · xαd d with αi ∈ R and Xi’s with polynomial Stein operator satisfying a specific commutativity assumption (Assumption 3 below)
We stress that Stein operators are of use in applications beyond proving approximation theorems; for example, in obtaining distributional properties [16,19] and other surprising applications include the derivation of formulas for definite integrals of special functions [18]
Summary
In 1972, Charles Stein (1920–2016) [41] introduced a powerful method for estimating the error in normal approximations. For instance the p.d.f. γ(x) = (2π)−1/2e−x2/2 of the standard normal distribution satisfies the first order ODE γ (x)+xγ(x) = 0 leading, by integration by parts, to the already mentioned operator Af (x) = f (x) − xf (x) This is useful for densities defined implicitly via ODEs. This is useful for densities defined implicitly via ODEs Such are by no means the only methods for deriving differential Stein operators and, for any given X, one can determine an entire ecosystem of Stein operators, leading to the natural question of which operator to choose. We stress that Stein operators are of use in applications beyond proving approximation theorems; for example, in obtaining distributional properties [16,19] and other surprising applications include the derivation of formulas for definite integrals of special functions [18].
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