scholarly journals On the asymptotic distribution of the discrete scan statistic

2006 ◽  
Vol 43 (04) ◽  
pp. 1137-1154 ◽  
Author(s):  
Michael V. Boutsikas ◽  
Markos V. Koutras
Keyword(s):  
Quality Control ◽  
Molecular Biology ◽  
Null Hypothesis ◽  
Success Probability ◽  
Poisson Approximation ◽  
Scan Statistic ◽  
Asymptotic Results ◽  
Compound Poisson Approximation ◽  
Extreme Value Theorem ◽  
Positive Integers

The discrete scan statistic in a binary (0-1) sequence of n trials is defined as the maximum number of successes within any k consecutive trials (n and k, n ≥ k, being two positive integers). It has been used in many areas of science (quality control, molecular biology, psychology, etc.) to test the null hypothesis of uniformity against a clustering alternative. In this article we provide a compound Poisson approximation and subsequently use it to establish asymptotic results for the distribution of the discrete scan statistic as n, k → ∞ and the success probability of the trials is kept fixed. An extreme value theorem is also provided for the celebrated Erdős-Rényi statistic.

scholarly journals On the asymptotic distribution of the discrete scan statistic

2006 ◽  
Vol 43 (4) ◽  
pp. 1137-1154 ◽  
Author(s):  
Michael V. Boutsikas ◽  
Markos V. Koutras
Keyword(s):  
Quality Control ◽  
Molecular Biology ◽  
Null Hypothesis ◽  
Success Probability ◽  
Poisson Approximation ◽  
Scan Statistic ◽  
Asymptotic Results ◽  
Compound Poisson Approximation ◽  
Extreme Value Theorem ◽  
Positive Integers

The discrete scan statistic in a binary (0-1) sequence of n trials is defined as the maximum number of successes within any k consecutive trials (n and k, n ≥ k, being two positive integers). It has been used in many areas of science (quality control, molecular biology, psychology, etc.) to test the null hypothesis of uniformity against a clustering alternative. In this article we provide a compound Poisson approximation and subsequently use it to establish asymptotic results for the distribution of the discrete scan statistic as n, k → ∞ and the success probability of the trials is kept fixed. An extreme value theorem is also provided for the celebrated Erdős-Rényi statistic.

Asymptotic results for the multiple scan statistic

2017 ◽  
Vol 54 (1) ◽  
pp. 320-330 ◽  
Author(s):  
M. V. Boutsikas ◽  
M. V. Koutras ◽  
F. S. Milienos
Keyword(s):  
Random Walk ◽  
Real Life ◽  
Poisson Approximation ◽  
Scan Statistics ◽  
Bernoulli Trials ◽  
Scan Statistic ◽  
Asymptotic Results ◽  
Compound Poisson Approximation ◽  
Multiple Scan ◽  
Random Walk Theory

AbstractThe contribution of the theory of scan statistics to the study of many real-life applications has been rapidly expanding during the last decades. The multiple scan statistic, defined on a sequence of n Bernoulli trials, enumerates the number of occurrences of k consecutive trials which contain at least r successes among them (r≤k≤n). In this paper we establish some asymptotic results for the distribution of the multiple scan statistic, as n,k,r→∞ and illustrate their accuracy through a simulation study. Our approach is based on an appropriate combination of compound Poisson approximation and random walk theory.

scholarly journals Testing the Validity of a Link Function Assumption in Repeated Type-II Censored General Step-Stress Experiments

Sankhya B ◽  
2021 ◽  
Author(s):  
Stefan Bedbur ◽  
Thomas Seiche
Keyword(s):  
Null Hypothesis ◽  
Statistical Tests ◽  
Operating Conditions ◽  
Link Function ◽  
Asymptotic Results ◽  
Failure Data ◽  
Stress Levels ◽  
Log Linear ◽  
Multiple Samples ◽  
General Step

AbstractIn step-stress experiments, test units are successively exposed to higher usually increasing levels of stress to cause earlier failures and to shorten the duration of the experiment. When parameters are associated with the stress levels, one problem is to estimate the parameter corresponding to normal operating conditions based on failure data obtained under higher stress levels. For this purpose, a link function connecting parameters and stress levels is usually assumed, the validity of which is often at the discretion of the experimenter. In a general step-stress model based on multiple samples of sequential order statistics, we provide exact statistical tests to decide whether the assumption of some link function is adequate. The null hypothesis of a proportional, linear, power or log-linear link function is considered in detail, and associated inferential results are stated. In any case, except for the linear link function, the test statistics derived are shown to have only one distribution under the null hypothesis, which simplifies the computation of (exact) critical values. Asymptotic results are addressed, and a power study is performed for testing on a log-linear link function. Some improvements of the tests in terms of power 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.

scholarly journals An Improved Error Bound for the Compound Poisson Approximation of a Nearly Homogeneous Portfolio

1987 ◽  
Vol 17 (2) ◽  
pp. 165-169 ◽  
Author(s):  
R. Michel
Keyword(s):  
Error Bound ◽  
Poisson Approximation ◽  
Claim Amount ◽  
Compound Poisson ◽  
Compound Poisson Approximation

AbstractFor the case of a portfolio with identical claim amount distributions, Gerber's error bound for the compound Poisson approximation is improved (in the case λ ⩾ 1). The result can also be applied to more general portfolios by partitioning them into homogeneous subportfolios.

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