Discrete, Continuous and Conditional Multiple Window Scan Statistics

2013 ◽  
Vol 50 (04) ◽  
pp. 1089-1101 ◽  
Author(s):  
Tung-Lung Wu ◽  
Joseph Glaz ◽  
James C. Fu
Keyword(s):  
Scan Statistics ◽  
Quality Monitoring ◽  
Blood Component ◽  
Bernoulli Trials ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Theoretical Results ◽  
Markov Chain Imbedding Technique ◽  
Multiple Window

The distributions of discrete, continuous and conditional multiple window scan statistics are studied. The finite Markov chain imbedding technique has been applied to obtain the distributions of fixed window scan statistics defined from a sequence of Bernoulli trials. In this manuscript the technique is extended to compute the distributions of multiple window scan statistics and the exact powers for multiple pulse and Markov dependent alternatives. An application in blood component quality monitoring is provided. Numerical results are also given to illustrate our theoretical results.

scholarly journals Discrete, Continuous and Conditional Multiple Window Scan Statistics

2013 ◽  
Vol 50 (4) ◽  
pp. 1089-1101 ◽  
Author(s):  
Tung-Lung Wu ◽  
Joseph Glaz ◽  
James C. Fu
Keyword(s):  
Scan Statistics ◽  
Quality Monitoring ◽  
Blood Component ◽  
Bernoulli Trials ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Theoretical Results ◽  
Markov Chain Imbedding Technique ◽  
Multiple Window

The distributions of discrete, continuous and conditional multiple window scan statistics are studied. The finite Markov chain imbedding technique has been applied to obtain the distributions of fixed window scan statistics defined from a sequence of Bernoulli trials. In this manuscript the technique is extended to compute the distributions of multiple window scan statistics and the exact powers for multiple pulse and Markov dependent alternatives. An application in blood component quality monitoring is provided. Numerical results are also given to illustrate our theoretical results.

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 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.

On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials

2003 ◽  
Vol 40 (3) ◽  
pp. 623-642 ◽  
Author(s):  
James C. Fu ◽  
Yung-Ming Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Markov Chain Imbedding ◽  
Exact Distributions ◽  
Markov Chain Imbedding Technique ◽  
Comprehensive Study

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials

2003 ◽  
Vol 40 (02) ◽  
pp. 346-360 ◽  
Author(s):  
James C. Fu ◽  
Liqun Wang ◽  
W. Y. Wendy Lou
Keyword(s):  
Large Deviation ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Longest Run ◽  
Exact Distributions ◽  
Markov Dependent Trials ◽  
Success Run ◽  
Longest Success Run ◽  
Theoretical Results

Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.

scholarly journals Approximate probabilities for runs and patterns in i.i.d. and Markov-dependent multistate trials

2009 ◽  
Vol 41 (01) ◽  
pp. 292-308 ◽  
Author(s):  
James C. Fu ◽  
Brad C. Johnson
Keyword(s):  
Matrix Theory ◽  
Tail Probability ◽  
Simple Pattern ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Normal Approximations ◽  
Runs And Patterns ◽  
Markov Chain Imbedding Technique ◽  
Better Than

LetXn(Λ) be the number of nonoverlapping occurrences of a simple pattern Λ in a sequence of independent and identically distributed (i.i.d.) multistate trials. For fixedk, the exact tail probability P{Xn(∧) < k} is difficult to compute and tends to 0 exponentially asn→ ∞. In this paper we use the finite Markov chain imbedding technique and standard matrix theory results to obtain an approximation for this tail probability. The result is extended to compound patterns, Markov-dependent multistate trials, and overlapping occurrences of Λ. Numerical comparisons with Poisson and normal approximations are provided. Results indicate that the proposed approximations perform very well and do significantly better than the Poisson and normal approximations in many cases.

On ordered series and later waiting time distributions in a sequence of Markov dependent multistate trials

2003 ◽  
Vol 40 (03) ◽  
pp. 623-642 ◽  
Author(s):  
James C. Fu ◽  
Yung-Ming Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Markov Chain Imbedding ◽  
Exact Distributions ◽  
Markov Chain Imbedding Technique ◽  
Comprehensive Study

The sooner and later waiting time problems have been extensively studied and applied in various areas of statistics and applied probability. In this paper, we give a comprehensive study of ordered series and later waiting time distributions of a number of simple patterns with respect to nonoverlapping and overlapping counting schemes in a sequence of Markov dependent multistate trials. Exact distributions and probability generating functions are derived by using the finite Markov chain imbedding technique. Examples are given to illustrate our results.

On exact and large deviation approximation for the distribution of the longest run in a sequence of two-state Markov dependent trials

2003 ◽  
Vol 40 (2) ◽  
pp. 346-360 ◽  
Author(s):  
James C. Fu ◽  
Liqun Wang ◽  
W. Y. Wendy Lou
Keyword(s):  
Large Deviation ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Longest Run ◽  
Exact Distributions ◽  
Markov Dependent Trials ◽  
Success Run ◽  
Longest Success Run ◽  
Theoretical Results

Consider a sequence of outcomes from Markov dependent two-state (success-failure) trials. In this paper, the exact distributions are derived for three longest-run statistics: the longest failure run, longest success run, and the maximum of the two. The method of finite Markov chain imbedding is used to obtain these exact distributions, and their bounds and large deviation approximation are also studied. Numerical comparisons among the exact distributions, bounds, and approximations are provided to illustrate the theoretical results. With some modifications, we show that the results can be easily extended to Markov dependent multistate trials.

scholarly journals Approximate probabilities for runs and patterns in i.i.d. and Markov-dependent multistate trials

2009 ◽  
Vol 41 (1) ◽  
pp. 292-308 ◽  
Author(s):  
James C. Fu ◽  
Brad C. Johnson
Keyword(s):  
Matrix Theory ◽  
Tail Probability ◽  
Simple Pattern ◽  
Finite Markov Chain ◽  
Finite Markov Chain Imbedding ◽  
Markov Chain Imbedding ◽  
Normal Approximations ◽  
Runs And Patterns ◽  
Markov Chain Imbedding Technique ◽  
Better Than

Let Xn(Λ) be the number of nonoverlapping occurrences of a simple pattern Λ in a sequence of independent and identically distributed (i.i.d.) multistate trials. For fixed k, the exact tail probability P{Xn (∧) < k} is difficult to compute and tends to 0 exponentially as n → ∞. In this paper we use the finite Markov chain imbedding technique and standard matrix theory results to obtain an approximation for this tail probability. The result is extended to compound patterns, Markov-dependent multistate trials, and overlapping occurrences of Λ. Numerical comparisons with Poisson and normal approximations are provided. Results indicate that the proposed approximations perform very well and do significantly better than the Poisson and normal approximations in many cases.

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