On perturbations of Stein operator

Abstract

In this paper, we obtain Stein operator for sum of n independent random variables (rvs) which is shown as perturbation of negative binomial (NB) operator. Comparing the operator with NB operator, we derive the error bounds for total variation distance by matching parameters. Also, three parameters approximation for such a sum is considered and is shown to improve the existing bounds in the literature.Finally, an application of our results to a function of waiting time for (k1, k2)-events is given.

Bounds on Negative Binomial Approximation to Call Function

“…[14], Wang and Xia [15], Kumar and Upadhye [9], among many others. Next, let X = N r,p and f = f z , defined in (1.1), then the Stein equation (2.1) leads to…”

Pseudo-binomial Approximation to $(k_1, k_2)$-runs

“…So, to obtain Stein operator, we use PGF approach with the recursive relation (iv) derived in Lemma 2.1. For more details and applications, we refer the reader to Upadhye et al [29] and Kumar and Upadhye [20].…”

“…Approximations to runs are widely studied in the literature, for example, negative binomial approximation to kruns (Wang and Xia [31]), Poisson approximation to (k 1 , k 2 )-events (Vellaisamy [30]) and Poisson approximations for the reliability of the system (Godbole [16]). Recently, Kumar and Upadhye [20] obtained bounds between negative binomial and a function of waiting time for (k 1 , k 2 )-events and Upadhye et al [29] derived a bound for binomial convoluted Poisson approximation to (1, 1)-runs. For more details and applications of runs, see Aki et al [2], Antzoulakos et al [3], Ankzoulakos and Chadjiconstantinidis [4], Balakrishnan and Koutras [5], Makri et al [22], Philippou et al [25], Philippou and Makri [26] and references therein.…”

“…A Stein operator can be obtained using different approaches available in the literature (see Stein [28], Barbour and Götze [6,18], Diaconis and Zabell [14], Ley et al [21] and Upadhye et al [29]). However, for identical trials, we focus on PGF approach given by Upadhye et al [29] and derive a Stein operator for B n k1,k2 as a perturbation of pseudo-binomial operator using the technique discussed by Kumar and Upadhye [20]. For non-identical trials, PGF for such distribution is generally complicated.…”

On Discrete Gibbs Measure Approximation to Runs

“…However, the distribution of the event of first type (B n k1,k2 ) is studied by Huang and Tsai [17] in 1991 where probability generating function (PGF), recursive relations for probability mass function (PMF), Poisson convergence and an extension of this distribution is given. Recently, approximation problem related B n k1,k2 is studied widely, for example, Poisson approximation to B n k1,k2 is given by Vellaisamy [29], binomial convoluted Poisson approximation to B n 1,1 is studied by Upadhye et al [28], negative binomial approximation to waiting time for B n k1,k2 and pseudo-binomial approximation to B n k1,k2 are given by Kumar and Upadhye [20,21].…”