On Stein operators for discrete approximations

Abstract

In this paper, a new method based on probability generating functions is used to obtain multiple Stein operators for various random variables closely related to Poisson, binomial and negative binomial distributions. Also, Stein operators for certain compound distributions, where the random summand satisfies Panjer's recurrence relation, are derived. A well-known perturbation approach for Stein's method is used to obtain total variation bounds for the distributions mentioned above. The importance of such approximations is illustrated, for example, by the binomial convoluted with Poisson approximation to sums of independent and dependent indicator random variables.

Stein’s method for comparison of univariate distributions

“…Other methods of constructing Stein operators are available. In [89] Stein operators for discrete compound distributions are derived by exploiting properties of the moment generating function. In [3], both Fourier and Malliavin-based aproaches are used to derive operators for targets which can be represented as linear combinations of independent chi-square random variables.…”

Approximations to Weighted Sums of Random Variables

“…Then, the distribution of W n is known as the distribution of (1, 1)-runs and it adopted our locally dependent structure with ω i = 1. For more details, see Huang and Tsai [10], Upadhye et al [19], Vellaisamy [20], and reference therein. Next, it can be easily verified that…”

“…are mean and variance of binomial and negative binomial distributions, respectively. For more details, we refer the reader to Brown and Xia [3], Eichelsbacher and Reinert [6], Kumar et al [11], Ley et al [12], Upadhye and Barman [18], Upadhye et al [19], and references therein. This paper is organized as follows.…”

Some new Stein operators for product distributions

“…As such, over the years, a number of techniques have been developed for finding Stein operators for a variety of distributions. These include the density method (Stein [26], Ley, Reinert and Swan [16], Ley and Swan [17], Mijoule, Reinert and Swan [19]), the generator method (Barbour [3], Götze [14]), the differential equation duality approach (Gaunt [10], Ley, Reinert and Swan [16]), and probability generating function and characteristic function based approaches of Upadhye,Čekanavičius and Vellaisamy [27] and Arras et al [2]. The corpus of literature concerning Stein operators and their applications is now vast, and it continues growing at a steady pace.…”