scholarly journals A Compound Poisson Approximation Inequality

2006 ◽  
Vol 43 (01) ◽  
pp. 282-288
Author(s):  
Erol A. Peköz
Keyword(s):  
System Reliability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Poisson Random Variable ◽  
Sequence Pattern ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Numerical Example

We give conditions under which the number of events which occur in a sequence ofm-dependent events is stochastically smaller than a suitably defined compound Poisson random variable. The results are applied to counts of sequence pattern appearances and to system reliability. We also provide a numerical example.

scholarly journals A Compound Poisson Approximation Inequality

2006 ◽  
Vol 43 (1) ◽  
pp. 282-288 ◽  
Author(s):  
Erol A. Peköz
Keyword(s):  
System Reliability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Poisson Random Variable ◽  
Sequence Pattern ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Numerical Example

We give conditions under which the number of events which occur in a sequence of m-dependent events is stochastically smaller than a suitably defined compound Poisson random variable. The results are applied to counts of sequence pattern appearances and to system reliability. We also provide a numerical example.

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.

scholarly journals Between Individual and Collective Model for the Total Claims

1988 ◽  
Vol 18 (2) ◽  
pp. 169-174 ◽  
Author(s):  
R. Kaas ◽  
A. E. van Heerwaarden ◽  
M. J. Goovaerts
Keyword(s):  
Life Insurance ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Collective Model ◽  
Individual Model ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
The Real ◽  
Stop Loss

AbstractThis article studies random variables whose stop-loss rank falls between a certain risk (assumed to be integer-valued and non-negative, but not necessarily of life-insurance type) and the compound Poisson approximation to this risk. They consist of a compound Poisson part to which some independent Bernoulli-type variables are added.Replacing each term in an individual model with such a random variable leads to an approximating model for the total claims on a portfolio of contracts that is computationally almost as attractive as the compound Poisson approximation used in the standard collective model. The resulting stop-loss premiums are much closer to the real values.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

scholarly journals Compound Poisson approximation for the distribution of extremes

2002 ◽  
Vol 34 (1) ◽  
pp. 223-240 ◽  
Author(s):  
A. D. Barbour ◽  
S. Y. Novak ◽  
A. Xia
Keyword(s):  
Point Processes ◽  
Poisson Approximation ◽  
Value Theory ◽  
Wasserstein Metric ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Limiting Behaviour ◽  
Explicit Bounds ◽  
Empirical Point ◽  
Poisson Cluster

Empirical point processes of exceedances play an important role in extreme value theory, and their limiting behaviour has been extensively studied. Here, we provide explicit bounds on the accuracy of approximating an exceedance process by a compound Poisson or Poisson cluster process, in terms of a Wasserstein metric that is generally more suitable for the purpose than the total variation metric. The bounds only involve properties of the finite, empirical sequence that is under consideration, and not of any limiting process. The argument uses Bernstein blocks and Lindeberg's method of compositions.

Poisson approximation for (k1, k2)-events via the Stein-Chen method

2004 ◽  
Vol 41 (4) ◽  
pp. 1081-1092 ◽  
Author(s):  
P. Vellaisamy
Keyword(s):  
Limit Theorem ◽  
Rate Of Convergence ◽  
Upper Bound ◽  
Success Probability ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Poisson Random Variable ◽  
Markov Dependent Trials ◽  
Special Case

Consider a sequence of independent Bernoulli trials with success probability p. Let N(n; k1, k2) denote the number of times that k1 failures are followed by k2 successes among the first n Bernoulli trials. We employ the Stein-Chen method to obtain a total variation upper bound for the rate of convergence of N(n; k1, k2) to a suitable Poisson random variable. As a special case, the corresponding limit theorem is established. Similar results are obtained for Nk3(n; k1, k2), the number of times that k1 failures followed by k2 successes occur k3 times successively in n Bernoulli trials. The bounds obtained are generally sharper than, and improve upon, some of the already known results. Finally, the technique is adapted to obtain Poisson approximation results for the occurrences of the above-mentioned events under Markov-dependent trials.

Poisson approximations for runs and patterns of rare events

1991 ◽  
Vol 23 (4) ◽  
pp. 851-865 ◽  
Author(s):  
Anant P. Godbole
Keyword(s):  
Success Probability ◽  
Standard Condition ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Poisson Random Variable ◽  
Runs And Patterns ◽  
Reliability Systems ◽  
Poisson Approximations

Consider a sequence of Bernoulli trials with success probability p, and let Nn,k denote the number of success runs of length among the first n trials. The Stein–Chen method is employed to obtain a total variation upper bound for the rate of convergence of Nn,k to a Poisson random variable under the standard condition npk→λ. This bound is of the same order, O(p), as the best known for the case k = 1, i.e. for the classical binomial-Poisson approximation. Analogous results are obtained for occurrences of word patterns, where, depending on the nature of the word, the corresponding rate is at most O(pk–m) for some m = 0, 2, ···, k – 1. The technique is adapted for use with two-state Markov chains. Applications to reliability systems and tests for randomness are discussed.

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