scholarly journals On magic factors in Stein’s method for compound Poisson approximation

2017 ◽  
Vol 22 (0) ◽  
Author(s):  
Fraser Daly
Keyword(s):  
Poisson Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Compound Poisson ◽  
Compound Poisson Approximation

scholarly journals Solving the Stein Equation in compound poisson approximation

1998 ◽  
Vol 30 (02) ◽  
pp. 449-475 ◽  
Author(s):  
A. D. Barbour ◽  
Sergey Utev
Keyword(s):  
Poisson Approximation ◽  
Stein’S Method ◽  
Restricted Range ◽  
Stein's Method ◽  
Kolmogorov Distance ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Equation ◽  
The Mean ◽  
Accuracy Of Approximation

The accuracy of compound Poisson approximation can be estimated using Stein's method in terms of quantities similar to those which must be calculated for Poisson approximation. However, the solutions of the relevant Stein equation may, in general, grow exponentially fast with the mean number of ‘clumps’, leading to many applications in which the bounds are of little use. In this paper, we introduce a method for circumventing this difficulty. We establish good bounds for those solutions of the Stein equation which are needed to measure the accuracy of approximation with respect to Kolmogorov distance, but only in a restricted range of the argument. The restriction on the range is then compensated by a truncation argument. Examples are given to show that the method clearly outperforms its competitors, as soon as the mean number of clumps is even moderately large.

Compound Poisson approximation and the clustering of random points

2000 ◽  
Vol 32 (1) ◽  
pp. 19-38 ◽  
Author(s):  
A. D. Barbour ◽  
Marianne Månsson
Keyword(s):  
Poisson Distribution ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Compound Poisson Distribution ◽  
Stein's Method ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Unit Square

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are covered by some translate of a small square C. If n|C| is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W can be much more closely approximated by an appropriate compound Poisson distribution CP(λ1, λ2,…). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that iλi be decreasing.

Compound Poisson approximation and the clustering of random points

2000 ◽  
Vol 32 (01) ◽  
pp. 19-38 ◽  
Author(s):  
A. D. Barbour ◽  
Marianne Månsson
Keyword(s):  
Poisson Distribution ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Compound Poisson Distribution ◽  
Stein's Method ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Unit Square

Let n random points be uniformly and independently distributed in the unit square, and count the number W of subsets of k of the points which are covered by some translate of a small square C. If n|C| is small, the number of such clusters is approximately Poisson distributed, but the quality of the approximation is poor. In this paper, we show that the distribution of W can be much more closely approximated by an appropriate compound Poisson distribution CP(λ1, λ2,…). The argument is based on Stein's method, and is far from routine, largely because the approximating distribution does not satisfy the simplifying condition that iλ i be decreasing.

scholarly journals Solving the Stein Equation in compound poisson approximation

1998 ◽  
Vol 30 (2) ◽  
pp. 449-475 ◽  
Author(s):  
A. D. Barbour ◽  
Sergey Utev
Keyword(s):  
Poisson Approximation ◽  
Stein’S Method ◽  
Restricted Range ◽  
Stein's Method ◽  
Kolmogorov Distance ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Equation ◽  
The Mean ◽  
Accuracy Of Approximation

The accuracy of compound Poisson approximation can be estimated using Stein's method in terms of quantities similar to those which must be calculated for Poisson approximation. However, the solutions of the relevant Stein equation may, in general, grow exponentially fast with the mean number of ‘clumps’, leading to many applications in which the bounds are of little use. In this paper, we introduce a method for circumventing this difficulty. We establish good bounds for those solutions of the Stein equation which are needed to measure the accuracy of approximation with respect to Kolmogorov distance, but only in a restricted range of the argument. The restriction on the range is then compensated by a truncation argument. Examples are given to show that the method clearly outperforms its competitors, as soon as the mean number of clumps is even moderately large.

scholarly journals Compound Poisson Approximation for Dissociated Random Variables via Stein's Method

1999 ◽  
Vol 8 (4) ◽  
pp. 335-346 ◽  
Author(s):  
PETER EICHELSBACHER ◽  
MAŁGORZATA ROOS
Keyword(s):  
System Reliability ◽  
Random Variables ◽  
Poisson Approximation ◽  
Stein’S Method ◽  
Stein's Method ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Equation

In the present paper we consider compound Poisson approximation by Stein's method for dissociated random variables. We present some applications to problems in system reliability. In particular, our examples have the structure of an incomplete U-statistics. We mainly apply techniques from Barbour and Utev, who gave new bounds for the solutions of the Stein equation in compound Poisson approximation in two recent papers.

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