Distributions related to events

Dafnis, Spiros D.; Antzoulakos, Demetrios L.; Philippou, Andreas N.

doi:10.1016/j.jspi.2009.12.019

Cited by 19 publications

(24 citation statements)

References 13 publications

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“…(k − 1) + ν 1 (k)][10ν 1 (k − 1) + 13ν 1 (k)] 16a4 remains to apply (26) to complete the proof. Let M be a finite variation measure concentrated on integers andk |k||M {k}| < ∞.…”

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confidence: 99%

A Compound Poisson Convergence Theorem for Sums of $$m$$ m -Dependent Variables

Čekanavičius

Vellaisamy

2014

J Theor Probab

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We prove the Simons-Johnson theorem for the sums S n of m-dependent random variables, with exponential weights and limiting compound Poisson distribution CP(s, λ). More precisely, we give sufficient conditions for ∞ k=0 e hk |P (S n = k)−CP(s, λ){k}| → 0 and provide an estimate on the rate of convergence. It is shown that the Simons-Johnson theorem holds for weighted Wasserstein norm as well. The results are then illustrated for N (n; k 1 , k 2 ) and k-runs statistics.

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“…(k − 1) + ν 1 (k)][10ν 1 (k − 1) + 13ν 1 (k)] 16a4 remains to apply (26) to complete the proof. Let M be a finite variation measure concentrated on integers andk |k||M {k}| < ∞.…”

mentioning

confidence: 99%

A Compound Poisson Convergence Theorem for Sums of $$m$$ m -Dependent Variables

Čekanavičius

Vellaisamy

2014

J Theor Probab

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“…A Y is a Stein operator for sum of n independent rvs by matching mean with negative binomial rv. Now, for g ∈ G Z ∩ G Y , taking the expectation of U Y with respect to Y and using (10), we get required result.…”

Section: One-parameter Approximationmentioning

confidence: 96%

“…Taking the expectation w.r.t. T , we have Now, g ∈ G Z ∩ G T , taking supremum and using (10), we get required result.…”

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confidence: 91%

On perturbations of Stein operator

Kumar

Upadhye

2017

Communications in Statistics - Theory and Methods

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In this paper, we obtain Stein operator for sum of n independent random variables (rvs) which is shown as perturbation of negative binomial (NB) operator. Comparing the operator with NB operator, we derive the error bounds for total variation distance by matching parameters. Also, three parameters approximation for such a sum is considered and is shown to improve the existing bounds in the literature.Finally, an application of our results to a function of waiting time for (k1, k2)-events is given.

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“…. In a recent paper, Dafnis et al (2010) studied distributions related to the (k 1 , k 2 ) events: at least (exactly, at most) k 1 consecutive 0's are followed by at least (exactly, at most) k 2 consecutive 1's. Presently, for k ≥ 1, we define the pattern E 0 : there are k consecutive successes and for k ≥ 2, we define the patterns E 1 : two successes are separated by at most k − 2 failures, E 2 : two successes are separated by exactly k − 2 failures, E 3 : two successes are separated by at least k − 2 failures.…”

Section: Introductionmentioning

confidence: 99%

Distributions of patterns of two successes separated by a string of k-2 failures

2010

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Let Z 1 , Z 2 , . . . be a sequence of independent Bernoulli trials with constant success and failure probabilities p = Pr(Z t = 1) and q = Pr(Z t = 0) = 1 − p, respectively, t = 1, 2, . . . . For any given integer k ≥ 2 we consider the patterns E 1 : two successes are separated by at most k − 2 failures, E 2 : two successes are separated by exactly k − 2 failures, and E 3 : two successes are separated by at least k − 2 failures. Denote by Nn,k ) the number of occurrences of the pattern E i , i = 1, 2, 3, in Z 1 , Z 2 , . . . , Z n when the non-overlapping (respectively overlapping) counting scheme for runs and patterns is employed. Also, let T (i) r,k (resp. W (i) r,k ) be the waiting time for the r − th occurrence of the pattern E i , i = 1, 2, 3, in Z 1 , Z 2 , . . . according to the non-overlapping (resp. overlapping) counting scheme. In this article we conduct a systematic study of N (i) n,k , M (i) n,k , T (i) r,k and W (i)r,k (i = 1, 2, 3) obtaining exact formulae, explicit or recursive, for their probability generating functions, probability mass functions and moments. An application is given.

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Distributions related to events

Cited by 19 publications

References 13 publications

A Compound Poisson Convergence Theorem for Sums of $$m$$ m -Dependent Variables

A Compound Poisson Convergence Theorem for Sums of $$m$$ m -Dependent Variables

On perturbations of Stein operator

Distributions of patterns of two successes separated by a string of k-2 failures

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