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.

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.

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.

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 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.

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 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.

scholarly journals Convergence in Total Variation of Random Sums

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 194
Author(s):  
Luca Pratelli ◽  
Pietro Rigo
Keyword(s):  
Asymptotic Behavior ◽  
Total Variation ◽  
Random Variables ◽  
Total Variation Distance ◽  
Random Sums ◽  
Variation Distance ◽  
Random Indices

Let (Xn) be a sequence of real random variables, (Tn) a sequence of random indices, and (τn) a sequence of constants such that τn→∞. The asymptotic behavior of Ln=(1/τn)∑i=1TnXi, as n→∞, is investigated when (Xn) is exchangeable and independent of (Tn). We give conditions for Mn=τn(Ln−L)⟶M in distribution, where L and M are suitable random variables. Moreover, when (Xn) is i.i.d., we find constants an and bn such that supA∈B(R)|P(Ln∈A)−P(L∈A)|≤an and supA∈B(R)|P(Mn∈A)−P(M∈A)|≤bn for every n. In particular, Ln→L or Mn→M in total variation distance provided an→0 or bn→0, as it happens in some situations.

scholarly journals Detecting Different Road Infrastructural Elements Based on the Stochastic Characterization of Speed Patterns

2017 ◽  
Vol 2017 ◽  
pp. 1-11 ◽  
Author(s):  
Mario Muñoz-Organero ◽  
Ramona Ruiz-Blázquez
Keyword(s):  
Total Variation ◽  
Traffic Congestion ◽  
Automatic Detection ◽  
Mass Function ◽  
Total Variation Distance ◽  
Statistical Moments ◽  
Probability Mass Function ◽  
Traffic Lights ◽  
Traffic Conditions ◽  
Variation Distance

The automatic detection of road related information using data from sensors while driving has many potential applications such as traffic congestion detection or automatic routable map generation. This paper focuses on the automatic detection of road elements based on GPS data from on-vehicle systems. A new algorithm is developed that uses the total variation distance instead of the statistical moments to improve the classification accuracy. The algorithm is validated for detecting traffic lights, roundabouts, and street-crossings in a real scenario and the obtained accuracy (0.75) improves the best results using previous approaches based on statistical moments based features (0.71). Each road element to be detected is characterized as a vector of speeds measured when a driver goes through it. We first eliminate the speed samples in congested traffic conditions which are not comparable with clear traffic conditions and would contaminate the dataset. Then, we calculate the probability mass function for the speed (in 1 m/s intervals) at each point. The total variation distance is then used to find the similarity among different points of interest (which can contain a similar road element or a different one). Finally, a k-NN approach is used for assigning a class to each unlabelled element.

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