Distribution Theory of Runs and Patterns and Its Applications - A Finite Markov Chain Imbedding Approach

Fu, James C.; Lou, Wendy Wen Yi

doi:10.1142/9789812779205

Cited by 90 publications

(96 citation statements)

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“…respectively (see Fu and Lou (2003); Theorems 5.2 and 7.4 pages 68 and 143) where r + 1 denotes the total number of states (i.e. there are r non-absorbing states and one absorbing state which is entered when the chart signals), = × is the identity matrix, = × is called the essential transition probability sub-matrix which contains all the probabilities of going from one non-absorbing state to another, 1 =1 × is a column vector with all elements equal to one…”

Section: Computation Of the Run-length Distributionmentioning

confidence: 99%

A nonparametric exponentially weighted moving average signed-rank chart for monitoring location

Graham

Chakraborti

Human

2011

Computational Statistics & Data Analysis

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Nonparametric control charts can provide a robust alternative in practice to the data analyst when there is a lack of knowledge about the underlying distribution. A nonparametric exponentially weighted moving average (NPEWMA) control chart combines the advantages of a nonparametric control chart with the better shift detection properties of a traditional EWMA chart. A NPEWMA chart for the median of a symmetric continuous distribution was introduced by Amin and Searcy (1991) using the Wilcoxon signed-rank statistic (see Gibbons and Chakraborti, 2003). This is called the nonparametric exponentially weighted moving average Signed-Rank (NPEWMA-SR) chart. However, important questions remained unanswered regarding the practical implementation as well as the performance of this chart. In this paper we address these issues with a more indepth study of the two-sided NPEWMA-SR chart. A Markov chain approach is used to compute the run-length distribution and the associated performance characteristics. Detailed guidelines and recommendations for selecting the chart's design parameters for practical implementation are provided along with illustrative examples. An extensive simulation study is done on the performance of the chart including a detailed comparison with a number of existing control charts, including the traditional EWMA chart for subgroup averages and some nonparametric charts i.e. runs-rules enhanced Shewhart-type SR charts and the NPEWMA chart based on signs. Results show that the NPEWMA-SR chart performs just as well as and in some cases better than the competitors. A summary and some concluding remarks are given.

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Section: Computation Of the Run-length Distributionmentioning

confidence: 99%

A nonparametric exponentially weighted moving average signed-rank chart for monitoring location

Graham

Chakraborti

Human

2011

Computational Statistics & Data Analysis

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“…Все размещаемые элементы предварительно упорядочены, поэто-му на каждом шаге можно выразить переходные вероятности цепи через со-отношения в спецификации. В работе [10] этот метод назван Finitе Markov Chain Imbedding Approach, а в книге [14] можно найти его подробное изло-жение. Ранее указанный метод позволил получить формулы для вычисления распределений ряда комбинаторных статистик, заданных на последователь-ностях с фиксированным составом: числа возрастаний [11], числа отрезков убывания [13], числа пар соседних элементов последовательности, имею-щих вид (i, i + 1) [15], вектора, состоящего из чисел возрастаний, убываний и уровней [12].…”

Section: д в шуваевunclassified

Computing the distribution of a combinatorial statistics defined on permutations of a given multiset

Шуваев¹,

Shuvaev²

2011

Матем. вопр. криптогр.

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Для конечной целочисленной последовательности α рассматривается число X(α) чтений α слева направо, необходимое для обращения ко всем элементам последовательности в порядке неубывания. Получены форму-лы для вычисления точного распределения X(α), когда α равновероятно распределена на множестве всех перестановок n заданных чисел. Ключевые слова: число чтений, цепь МарковаComputing the distribution of a combinatorial statistics defined on permutations of a given multiset D. V. Shuvaev TVP Laboratory, MoscowAbstract. For a finite integer sequence α we consider the number X(α) of readings of α from left to right necessary to count all elements of the sequence in a nondecreasing order. We describe a method to calculate the distribution of X(α) exactly when α is uniformly distributed on the set of all permutations of n given numbers.

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“…the total number of successes in all the success runs) of length greater than or equal to a prespecified length, and the waiting time for the first time that the abovementioned statistic equals or exceeds a predetermined level (Antzoulakos et al (2003b)). These authors studied sequences of trials ordered on a line by exploiting a Markov chain embedding technique; see Fu and Koutras (1994), and Fu and Lou (2003).…”

Section: F S Makri Et Almentioning

confidence: 99%

Success run statistics defined on an urn model

2007

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Statistics denoting the numbers of success runs of length exactly equal and at least equal to a fixed length, as well as the sum of the lengths of success runs of length greater than or equal to a specific length, are considered. They are defined on both linearly and circularly ordered binary sequences, derived according to the Pólya-Eggenberger urn model. A waiting time associated with the sum of lengths statistic in linear sequences is also examined. Exact marginal and joint probability distribution functions are obtained in terms of binomial coefficients by a simple unified combinatorial approach. Mean values are also derived in closed form. Computationally tractable formulae for conditional distributions, given the number of successes in the sequence, useful in nonparametric tests of randomness, are provided. The distribution of the length of the longest success run and the reliability of certain consecutive systems are deduced using specific probabilities of the studied statistics. Numerical examples are given to illustrate the theoretical results.

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Distribution Theory of Runs and Patterns and Its Applications - A Finite Markov Chain Imbedding Approach

Cited by 90 publications

References 0 publications

A nonparametric exponentially weighted moving average signed-rank chart for monitoring location

A nonparametric exponentially weighted moving average signed-rank chart for monitoring location

Computing the distribution of a combinatorial statistics defined on permutations of a given multiset

Success run statistics defined on an urn model

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