scholarly journals A Pólya Approximation to the Poisson-Binomial Law

2012 ◽  
Vol 49 (3) ◽  
pp. 745-757 ◽  
Author(s):  
Max Skipper
Keyword(s):  
Total Variation ◽  
Upper Bounds ◽  
Numerical Comparison ◽  
Bernoulli Random Variables ◽  
Polya Distribution ◽  
Variation Distance ◽  
Mean And Variance ◽  
Probability Mass ◽  
Distributional Approximations ◽  
Mass Functions

Using Stein's method, we derive explicit upper bounds on the total variation distance between a Poisson-binomial law (the distribution of a sum of independent but not necessarily identically distributed Bernoulli random variables) and a Pólya distribution with the same support, mean, and variance; a nonuniform bound on the pointwise distance between the probability mass functions is also given. A numerical comparison of alternative distributional approximations on a somewhat representative collection of case studies is also exhibited. The evidence proves that no single one is uniformly most accurate, though it suggests that the Pólya approximation might be preferred in several parameter domains encountered in practice.

scholarly journals A Pólya Approximation to the Poisson-Binomial Law

2012 ◽  
Vol 49 (03) ◽  
pp. 745-757 ◽  
Author(s):  
Max Skipper
Keyword(s):  
Total Variation ◽  
Upper Bounds ◽  
Numerical Comparison ◽  
Bernoulli Random Variables ◽  
Polya Distribution ◽  
Variation Distance ◽  
Mean And Variance ◽  
Probability Mass ◽  
Distributional Approximations ◽  
Mass Functions

Using Stein's method, we derive explicit upper bounds on the total variation distance between a Poisson-binomial law (the distribution of a sum of independent but not necessarily identically distributed Bernoulli random variables) and a Pólya distribution with the same support, mean, and variance; a nonuniform bound on the pointwise distance between the probability mass functions is also given. A numerical comparison of alternative distributional approximations on a somewhat representative collection of case studies is also exhibited. The evidence proves that no single one is uniformly most accurate, though it suggests that the Pólya approximation might be preferred in several parameter domains encountered in practice.

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.

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.

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.

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.

Poisson and compound Poisson approximations for random sums of random variables

1996 ◽  
Vol 33 (01) ◽  
pp. 127-137 ◽  
Author(s):  
P. Vellaisamy ◽  
B. Chaudhuri
Keyword(s):  
Lower Bound ◽  
Total Variation ◽  
Random Variables ◽  
Upper Bounds ◽  
Random Sums ◽  
Compound Poisson ◽  
Sums Of Random Variables ◽  
Variation Distance ◽  
Poisson Approximations ◽  
Better Than

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

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.

Poisson and compound Poisson approximations for random sums of random variables

1996 ◽  
Vol 33 (1) ◽  
pp. 127-137 ◽  
Author(s):  
P. Vellaisamy ◽  
B. Chaudhuri
Keyword(s):  
Lower Bound ◽  
Total Variation ◽  
Random Variables ◽  
Upper Bounds ◽  
Random Sums ◽  
Compound Poisson ◽  
Sums Of Random Variables ◽  
Variation Distance ◽  
Poisson Approximations ◽  
Better Than

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

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