Runs and Scans with Applications

Balakrishnan, N.; Koutras, Markos V.

doi:10.1002/9781118150467

Cited by 192 publications

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“…binary variables. This latter distribution has been worked out in the literature (see, e.g., Balakrishnan and Koutras (2002) page 56; note a typo) and is given by …”

Section: Appendixmentioning

confidence: 99%

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A phase II nonparametric control chart based on precedence statistics with runs-type signaling rules

Chakraborti

Eryılmaz

Human

2009

Computational Statistics & Data Analysis

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a b s t r a c tNonparametric control charts do not require knowledge about the shape of the underlying distribution and can thus be attractive in certain situations. Two new Shewharttype nonparametric control charts are proposed for monitoring the unknown location parameter of a continuous population in Phase II (prospective) applications. The charts are based on control limits given by two specified order statistics from a reference sample, obtained from a Phase I (retrospective) analysis, and using some runs-type signaling rules. The plotting statistic can be any order statistic in a Phase II sample; the median is used here for simplicity and robustness. Exact run length distributions of the proposed charts are derived using conditioning and some results from the theory of runs. Tables are provided for practical implementation of the charts for a given in-control average run length (ARL 0 ) between 300 and 500. Comparisons of the average run length ARL, the standard deviation of run length (SDRL) and some run length percentiles show that the charts have robust in-control performance and are more efficient when the underlying distribution is t (symmetric with heavier tails than the normal) or gamma (1, 1) (right-skewed). Even for the normal distribution, the new charts are quite competitive. An illustrative numerical example is given. An added advantage of these charts is that they can be applied before all the data are collected which might lead to savings in time and resources in certain applications.

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“…binary variables. This latter distribution has been worked out in the literature (see, e.g., Balakrishnan and Koutras (2002) page 56; note a typo) and is given by …”

Section: Appendixmentioning

confidence: 99%

“…The required quantities are most conveniently derived by conditioning, noting that given X a:m and X b:m , the random variable T 2 has a geometric distribution of order 2 (see Balakrishnan and Koutras (2002)). The conditional expected value and the variance of T 2 are known to be…”

Section: Arl Var and Far Of 2-of-2 Dr Chartmentioning

confidence: 99%

A phase II nonparametric control chart based on precedence statistics with runs-type signaling rules

Chakraborti

Eryılmaz

Human

2009

Computational Statistics & Data Analysis

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“…Exact results for the distribution of the scan statistic were discussed in Fu (2001), Balakrishnan and Koutras (2002) and Fu and Lou (2003). Since the evaluation of the exact distribution is computationally intractable, especially for large values of the parameters n, k, several approximations and bounds have been developed during the last decades.…”

Section: Introductionmentioning

confidence: 99%

“…it has been an attractive subject of continuing research interest for the past few decades; see e.g. the monographs by Balakrishnan and Koutras (2002), Glaz, Naus and Wallenstein (2001) and the special issue edited by Glaz and Balakrishnan (1999).…”

Section: Introductionmentioning

confidence: 99%

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Extreme Value Results for Scan Statistics

2009

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Abstract:In the first part of this paper we focus on the classical scan and multiple scan statistic, defined on a sequence of independent and identically distributed (i.i.d.) binary trials and review a number of bounds and approximations for their distributions which have been developed by the aid of distance measures. Moreover, we discuss briefly a number of asymptotic results that have been established by setting up appropriate conditions guaranteeing the convergence (to zero) of the distance measures' upper bounds. In the second part, we study a multiple scan statistic enumerating variable by considering a general threshold-based framework, defined on i.i.d. continuous random variables. More specifically, we prove first a compound Poisson approximation for the total number of fixed length overlapping moving windows containing prespecified number of threshold exceedances. The classical scan and multiple scan statistic may be treated as a special case of this general model. Next we exploit the previous result to gain some new extreme value results for the scan enumerating statistic under the assumption that the continuous random variables belong to the maximum domain of attraction of one of the three extreme value distributions (Frechet, reversed Weibull, Gumbel). Finally, we elucidate how the general results can be applied in a number of classical continuous distributions (Pareto, Uniform, Exponential and Normal).

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