On Negative Binomial Approximation
06 Mar 2013-Theory of Probability and Its Applications (Society for Industrial and Applied Mathematics)-Vol. 57, Iss: 1, pp 97-109
TL;DR: In this paper, negative binomial approximation to sums of independent Z +$-valued random variables using Stein's method is employed to obtain the error bounds Convolution of negative Binomial and Poisson distribution is used as a three-parametric approximation.
Abstract: This paper deals with negative binomial approximation to sums of independent ${\bf Z}_+$-valued random variables Stein's method is employed to obtain the error bounds Convolution of negative binomial and Poisson distribution is used as a three-parametric approximation
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TL;DR: 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.
19 citations
TL;DR: In this article, the authors developed Stein's method for conditional compound Poisson approximation, which is more appropriate in applications than a compound poisson distribution, since one can only start modelling the occurrence of rare events after such events have happened.
13 citations
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.
Abstract: In this article, we obtain a Stein operator for the sum of n independent random variables (rvs) which is shown as the perturbation of the negative binomial (NB) operator. Comparing the operator with NB operator, we derive the error bounds for total variation distance by matching parameters. Also, three-parameter 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.
11 citations
TL;DR: In this paper, a Stein operator for runs arising from identical and non-identical Bernoulli trials is derived via Stein method and the bounds obtained are new and their importance is demonstrated through an interesting application.
Abstract: A Stein operator for the runs is derived as a perturbation of an operator for discrete Gibbs measure. Due to this fact, using perturbation technique, the approximation results for runs arising from identical and non-identical Bernoulli trials are derived via Stein method. The bounds obtained are new and their importance is demonstrated through an interesting application.
7 citations
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.
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 $(k_1,k_2)$-events is given.
7 citations
References
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TL;DR: The methods introduced here apply to a very large class of approximating distributions on the non-negative integers, among which there is a natural class for higher-order approximations by probability distributions rather than signed measures (as previously).
Abstract: Barbour introduced a probabilistic view of Stein's method for estimating the error in probability approximations. However, in the case of approximations by general distributions on the integers, there have been no purely probabilistic proofs of Stein's bounds till this paper. Furthermore, the methods introduced here apply to a very large class of approximating distributions on the non-negative integers, among which there is a natural class for higher-order approximations by probability distributions rather than signed measures (as previously). The methods also produce Stein magic factors for process approximations which do not increase with the window of observation and which are simpler to apply than those in Brown, Weinberg and Xia.
106 citations
TL;DR: In this paper, the Poisson binomial distribution is approximated by a Binomial distribution and also by finite signed measures resulting from the corresponding Krawtchouk expansion, and bounds and asymptotic relations for the total variation distance and the point metric are given.
Abstract: The Poisson binomial distribution is approximated by a binomial distribution and also by finite signed measures resulting from the corresponding Krawtchouk expansion. Bounds and asymptotic relations for the total variation distance and the point metric are given.
89 citations
TL;DR: In this paper, an asymptotic expansion of the signed compound Poisson approximation was proposed, which is based on Stein's method for signed compound poisson approximation with uniform error.
Abstract: Let $W_n := \sum_{j=1}^n Z_j$ be a sum of independent integer-valued random variables. In this paper, we derive an asymptotic expansion for the probability $\mathbb{P}[W_n \in A]$ of an arbitrary subset $A \in \mathbb{Z}$. Our approximation improves upon the classical expansions by including an explicit, uniform error estimate, involving only easily computable properties of the distributions of the $Z_j:$ an appropriate number of moments and the total variation distance $d_{\mathrm{TV}}(\mathscr{L}(Z_j), \mathscr{L}(Z_j + 1))$. The proofs are based on Stein’s method for signed compound Poisson approximation.
85 citations
TL;DR: In this article, bounds on the rate of convergence to the negative binomial distribution are found, where this rate is measured by the total variation distance between probability laws, where the difference between the effect of particular indicator being one and the value of a geometrically distributed random variable is measured.
Abstract: Bounds on the rate of convergence to the negative binomial distribution are found, where this rate is measured by the total variation distance between probability laws. For an arbitrary discrete random variable written as a sum of indicators, an upper bound of coupling form is expressed as an average of terms each of which measures the difference between the effect of particular indicator being one and the value of a geometrically distributed random variable. When a monotone coupling exists a lower bound can also be shown. Application of these results is illustrated with the example of the Po´lya distribution for which the rate of approach to the negative binomial limit is found.
65 citations