On Stein operators for discrete approximations
TLDR
In this article, 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.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, the 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.read more
Citations
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Stein's method for comparison of univariate distributions
TL;DR: In this article, a new general version of Stein's method for univariate distributions is proposed, which is based on a linear difference or differential-type operator, and the resulting Stein identity highlights the unifying theme behind the literature on Stein's methods both for continuous and discrete distributions.
Journal ArticleDOI
A state-dependent queueing system with asymptotic logarithmic distribution
TL;DR: In this paper, a Markovian single-server queueing model with Poisson arrivals and state-dependent service rates, characterized by a logarithmic steady-state distribution, is considered.
Journal ArticleDOI
Pseudo-binomial approximation to (k1,k2)-runs
N. S. Upadhye,Amit Kumar +1 more
TL;DR: In this article, a pseudo-binomial approximation to (k 1, k 2 )-runs under non-i.i.d. setup is proposed. But the approximation results are of optimal order and improve the existing results.
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On perturbations of Stein operator
Amit Kumar,N. S. Upadhye +1 more
TL;DR: In this paper, a Stein operator for the sum of n independent random variables (rvs) which is shown as the perturbation of the negative binomial (NB) operator is derived.
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On Perturbations of Stein Operator
Amit Kumar,N. S. Upadhye +1 more
TL;DR: In this article, the Stein operator for sum of independent random variables (rvs) is shown as perturbation of negative binomial (NB) operator, and the error bounds for total variation distance by matching parameters are derived.
References
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Book
Normal Approximations with Malliavin Calculus
Ivan Nourdin,Giovanni Peccati +1 more
TL;DR: In this article, the authors provide an ideal introduction both to Stein's method and Malliavin calculus, from the standpoint of normal approximations on a Gaussian space, and explain the connections between Stein's methods and Mallian calculus of variations.
Book
Normal Approximation by Stein's Method
TL;DR: In this paper, Stein's method is used for non-linear statistics and multivariate normal approximations for independent random variables with moderate deviations, and a non-normal approximation for nonlinear statistics.
Journal ArticleDOI
Fundamentals of Stein's method
TL;DR: The main concepts and techniques of Stein's method for distributional approximation by the normal, Poisson, exponential, and geometric distributions, and its relation to concentration of measure inequalities are discussed in this paper.
Journal ArticleDOI
Compound Poisson Approximation for Nonnegative Random Variables Via Stein's Method
TL;DR: In this paper, the authors extend Stein's method to a compound Poisson distribution setting, where the distribution of random variables is a finite positive measure on $(0, ∞).