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 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.

scholarly journals Continuous, Discrete, and Conditional Scan Statistics

2012 ◽  
Vol 49 (01) ◽  
pp. 199-209 ◽  
Author(s):  
James C. Fu ◽  
Tung-Lung Wu ◽  
W.Y. Wendy Lou
Keyword(s):  
Rates Of Convergence ◽  
Scan Statistics ◽  
Random Permutations ◽  
Bernoulli Trials ◽  
Finite Markov Chain ◽  
Scan Statistic ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Runs And Patterns ◽  
Theoretical Results

The distributions for continuous, discrete, and conditional discrete scan statistics are studied. The approach of finite Markov chain imbedding, which has been applied to random permutations as well as to runs and patterns, is extended to compute the distribution of the conditional discrete scan statistic, defined from a sequence of Bernoulli trials. It is shown that the distribution of the continuous scan statistic induced by a Poisson process defined on (0, 1] is a limiting distribution of weighted distributions of conditional discrete scan statistics. Comparisons of rates of convergence as well as numerical comparisons of various bounds and approximations are provided to illustrate the theoretical results.

scholarly journals Discrete Scan Statistics Generated by Exchangeable Binary Trials

2010 ◽  
Vol 47 (4) ◽  
pp. 1084-1092 ◽  
Author(s):  
Serkan Eryilmaz
Keyword(s):  
Closed Form ◽  
Random Variables ◽  
Random Variable ◽  
Scan Statistics ◽  
Bernoulli Trials ◽  
Scan Statistic ◽  
Binary Trials ◽  
Special Case ◽  
Independent Identically Distributed

Let {Xi}i=1n be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable Sn,m to be the maximum number of 1s within any m consecutive trials in {Xi}i=1n. The random variable Sn,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {Xi}i=1n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of Sn,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

scholarly journals Continuous, Discrete, and Conditional Scan Statistics

2012 ◽  
Vol 49 (1) ◽  
pp. 199-209 ◽  
Author(s):  
James C. Fu ◽  
Tung-Lung Wu ◽  
W.Y. Wendy Lou
Keyword(s):  
Rates Of Convergence ◽  
Scan Statistics ◽  
Random Permutations ◽  
Bernoulli Trials ◽  
Finite Markov Chain ◽  
Scan Statistic ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Runs And Patterns ◽  
Theoretical Results

The distributions for continuous, discrete, and conditional discrete scan statistics are studied. The approach of finite Markov chain imbedding, which has been applied to random permutations as well as to runs and patterns, is extended to compute the distribution of the conditional discrete scan statistic, defined from a sequence of Bernoulli trials. It is shown that the distribution of the continuous scan statistic induced by a Poisson process defined on (0, 1] is a limiting distribution of weighted distributions of conditional discrete scan statistics. Comparisons of rates of convergence as well as numerical comparisons of various bounds and approximations are provided to illustrate the theoretical results.

scholarly journals Discrete Scan Statistics Generated by Exchangeable Binary Trials

2010 ◽  
Vol 47 (04) ◽  
pp. 1084-1092 ◽  
Author(s):  
Serkan Eryilmaz
Keyword(s):  
Closed Form ◽  
Random Variables ◽  
Random Variable ◽  
Scan Statistics ◽  
Bernoulli Trials ◽  
Scan Statistic ◽  
Binary Trials ◽  
Special Case ◽  
Independent Identically Distributed

Let {X i } i=1 n be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable S n,m to be the maximum number of 1s within any m consecutive trials in {X i } i=1 n . The random variable S n,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {X i } i=1 n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of S n,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

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