Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities

1999 ◽  
Vol 36 (1) ◽  
pp. 97-104 ◽  
Author(s):  
Michael Weba
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Upper Bounds ◽  
Risk Theory ◽  
Total Variation Distance ◽  
Medium Size ◽  
Bernoulli Random Variables ◽  
Variation Distance ◽  
Approximation Problems

In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p1,p2,…, pn is often approximated by a Poisson distribution with parameter λ = p1 + p2 + pn. Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p1,p2,…,pn.Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.

Bounds for the total variation distance between the binomial and the Poisson distribution in case of medium-sized success probabilities

1999 ◽  
Vol 36 (01) ◽  
pp. 97-104 ◽  
Author(s):  
Michael Weba
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Upper Bounds ◽  
Risk Theory ◽  
Total Variation Distance ◽  
Medium Size ◽  
Bernoulli Random Variables ◽  
Variation Distance ◽  
Approximation Problems

In applied probability, the distribution of a sum of n independent Bernoulli random variables with success probabilities p 1,p 2,…, p n is often approximated by a Poisson distribution with parameter λ = p 1 + p 2 + p n . Popular bounds for the approximation error are excellent for small values, but less efficient for moderate values of p 1,p 2,…,p n . Upper bounds for the total variation distance are established, improving conventional estimates if the success probabilities are of medium size. The results may be applied directly, e.g. to approximation problems in risk theory.

Poisson approximation of multivariate Poisson mixtures

2003 ◽  
Vol 40 (02) ◽  
pp. 376-390 ◽  
Author(s):  
Bero Roos
Keyword(s):  
Total Variation ◽  
Poisson Approximation ◽  
Upper Bounds ◽  
Risk Theory ◽  
Total Variation Distance ◽  
Poisson Mixtures ◽  
Poisson Distributions ◽  
Variation Distance

We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson approximation of generalized multinomial distributions.

Compound Poisson approximation for the Johnson-Mehl model

2000 ◽  
Vol 37 (01) ◽  
pp. 101-117
Author(s):  
Torkel Erhardsson
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Random Intervals ◽  
Variation Distance ◽  
The One

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

Compound Poisson approximation for the Johnson-Mehl model

2000 ◽  
Vol 37 (1) ◽  
pp. 101-117 ◽  
Author(s):  
Torkel Erhardsson
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Random Intervals ◽  
Variation Distance ◽  
The One

We consider the uncovered set (i.e. the complement of the union of growing random intervals) in the one-dimensional Johnson-Mehl model. Let S(z,L) be the number of components of this set at time z > 0 which intersect (0, L]. An explicit bound is known for the total variation distance between the distribution of S(z,L) and a Poisson distribution, but due to clumping of the components the bound can be rather large. We here give a bound for the total variation distance between the distribution of S(z,L) and a simple compound Poisson distribution (a Pólya-Aeppli distribution). The bound is derived by interpreting S(z,L) as the number of visits to a ‘rare’ set by a Markov chain, and applying results on compound Poisson approximation for Markov chains by Erhardsson. It is shown that under a mild condition, if z→∞ and L→∞ in a proper fashion, then both the Pólya-Aeppli and the Poisson approximation error bounds converge to 0, but the convergence of the former is much faster.

Poisson approximation of multivariate Poisson mixtures

2003 ◽  
Vol 40 (2) ◽  
pp. 376-390 ◽  
Author(s):  
Bero Roos
Keyword(s):  
Total Variation ◽  
Poisson Approximation ◽  
Upper Bounds ◽  
Risk Theory ◽  
Total Variation Distance ◽  
Poisson Mixtures ◽  
Poisson Distributions ◽  
Variation Distance

We show how good multivariate Poisson mixtures can be approximated by multivariate Poisson distributions and related finite signed measures. Upper bounds for the total variation distance with applications to risk theory and generalized negative multinomial distributions are given. Furthermore, it turns out that the ideas used in this paper also lead to improvements in the Poisson approximation of generalized multinomial distributions.

On the distance between the distributions of random sums

2003 ◽  
Vol 40 (01) ◽  
pp. 87-106 ◽  
Author(s):  
Bero Roos ◽  
Dietmar Pfeifer
Keyword(s):  
Total Variation ◽  
Approximation Problem ◽  
Upper Bounds ◽  
Total Variation Distance ◽  
Random Sum ◽  
Random Sums ◽  
Random Summation ◽  
Variation Distance ◽  
Stop Loss

In this paper, we consider the total variation distance between the distributions of two random sums S M and S N with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum S N if M and N have the same first moments.

On the distance between the distributions of random sums

2003 ◽  
Vol 40 (1) ◽  
pp. 87-106 ◽  
Author(s):  
Bero Roos ◽  
Dietmar Pfeifer
Keyword(s):  
Total Variation ◽  
Approximation Problem ◽  
Upper Bounds ◽  
Total Variation Distance ◽  
Random Sum ◽  
Random Sums ◽  
Random Summation ◽  
Variation Distance ◽  
Stop Loss

In this paper, we consider the total variation distance between the distributions of two random sums SM and SN with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum SN if M and N have the same first moments.

scholarly journals Poisson Approximation of the Number of Cliques in Random Intersection Graphs

2010 ◽  
Vol 47 (3) ◽  
pp. 826-840 ◽  
Author(s):  
Katarzyna Rybarczyk ◽  
Dudley Stark
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Upper Bound ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Intersection Graphs ◽  
Fixed Integer ◽  
Random Subset ◽  
Random Intersection Graph ◽  
Variation Distance

A random intersection graphG(n,m,p) is defined on a setVofnvertices. There is an auxiliary setWconsisting ofmobjects, and each vertexv∈Vis assigned a random subset of objectsWv⊆Wsuch thatw∈Wvwith probabilityp, independently for allv∈Vand allw∈W. Given two verticesv1,v2∈V, we setv1∼v2if and only ifWv1∩Wv2≠ ∅. We use Stein's method to obtain an upper bound on the total variation distance between the distribution of the number ofh-cliques inG(n,m,p) and a related Poisson distribution for any fixed integerh.

scholarly journals Poisson Approximation of the Number of Cliques in Random Intersection Graphs

2010 ◽  
Vol 47 (03) ◽  
pp. 826-840 ◽  
Author(s):  
Katarzyna Rybarczyk ◽  
Dudley Stark
Keyword(s):  
Total Variation ◽  
Poisson Distribution ◽  
Upper Bound ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Intersection Graphs ◽  
Fixed Integer ◽  
Random Subset ◽  
Random Intersection Graph ◽  
Variation Distance

A random intersection graph G(n, m, p) is defined on a set V of n vertices. There is an auxiliary set W consisting of m objects, and each vertex v ∈ V is assigned a random subset of objects W v ⊆ W such that w ∈ W v with probability p, independently for all v ∈ V and all w ∈ W . Given two vertices v 1, v 2 ∈ V , we set v 1 ∼ v 2 if and only if W v 1 ∩ W v 2 ≠ ∅. We use Stein's method to obtain an upper bound on the total variation distance between the distribution of the number of h-cliques in G(n, m, p) and a related Poisson distribution for any fixed integer h.

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