Sequential Ranks Research Articles

Signed sequential rank Shiryaev–Roberts schemes

AbstractWe develop Shiryaev–Roberts schemes based on signed sequential ranks to detect a persistent change in location of a continuous symmetric distribution with known median. The in‐control properties of these schemes are distribution‐free, hence they do not require a parametric specification of an underlying density function or the existence of any moments. Tables of control limits are provided. The out‐of‐control average run length properties of the schemes are gauged via theory‐based calculations and Monte Carlo simulation. Comparisons are made with two existing distribution‐free schemes. We conclude that the newly proposed scheme has much to recommend its use in practice. Implementation of the methodology is illustrated in an application to a data set from an industrial environment.

Control Limits for an Adaptive Self-Starting Distribution-Free CUSUM Based on Sequential Ranks

Since their introduction in 1954, cumulative sum (CUSUM) control charts have seen a widespread use beyond the conventional realm of statistical process control (SPC). While off-the-shelf implementations aimed at practitioners are available, their successful use is often hampered by inherent limitations which make them not easily reconcilable with real-world scenarios. Challenges commonly arise regarding a lack of robustness due to underlying parametric assumptions or requiring the availability of large representative training datasets. We evaluate an adaptive distribution-free CUSUM based on sequential ranks which is self-starting and provide detailed pseudo-code of a simple, yet effective calibration algorithm. The main contribution of this paper is in providing a set of ready-to-use tables of control limits suitable to a wide variety of applications where a departure from the underlying sampling distribution to a stochastically larger distribution is of interest. Performance of the proposed tabularized control limits is assessed and compared to competing approaches through extensive simulation experiments. The proposed control limits are shown to yield significantly increased agility (reduced detection delay) while maintaining good overall robustness.

Sequential rank CUSUMs for location and dispersion

We develop CUSUMs based on sequential ranks of the observations to detect changes over time in the location and dispersion of a distribution. The CUSUMs are distribution free in the sense that the appropriate control limits do not depend on the form or any parameters of the unknown underlying distribution. As such the CUSUMs are fully self starting. The inand out-of-control average run length properties of the CUSUMs are gauged qualitatively via theory-based calculations and quantitatively by Monte Carlo simulation. The CUSUMS are shown to perform very well when compared to some existing parametric and nonparametric CUSUMS. Implementation of the CUSUMs is illustrated in an application based on real data from an industrial environment.

Signed sequential rank CUSUMs

CUSUMs based on the signed sequential ranks of observations are developed for detecting location and scale changes in symmetric distributions. The CUSUMs are distribution-free and fully self-starting: given a specified in-control median and nominal in-control average run length, no parametric specification of the underlying distribution is required in order to find the correct control limits. If the underlying distribution is normal with unknown variance, a CUSUM based on the Van der Waerden signed rank score produces out-of-control average run lengths that are commensurate with those produced by the standard CUSUM for a normal distribution with known variance. For heavier tailed distributions, use of a CUSUM based on the Wilcoxon signed rank score is indicated. The methodology is illustrated by application to real data from an industrial environment.

The sequential normal scores transformation

ABSTRACTThe sequential analysis of series often requires nonparametric procedures, where the most powerful ones frequently use rank transformations. Reranking the data sequence after each new observation can become too intensive computationally. This led to the idea of sequential ranks, where only the most recent observation is ranked. However, difficulties finding, or approximating, the null distribution of the statistics may have contributed to the lack of popularity of these methods. In this article, we propose transforming the sequential ranks into sequential normal scores that are independent and asymptotically standard normal random variables. Thus, original methods based on the normality assumption may be used.A novel approach permits the inclusion of a priori information in the form of quantiles. It is developed as a strategy to increase the sensitivity of the scoring statistic. The result is a powerful convenient method to analyze non normal data sequences. Also, four variations of sequential normal scores are presented using examples from the literature. Researchers and practitioners might find this approach useful to develop nonparametric procedures to address new problems extending the use of parametric procedures when distributional assumptions are not met. These methods are especially useful with large data streams where efficient computational methods are required.

On locally most powerful sequential rank tests

ABSTRACTSequential ranks are defined as ranks of such observations, which have been observed so far in a sequential design. This article studies hypotheses tests based on sequential ranks for different situations. The locally most powerful sequential rank test is derived for the hypothesis of randomness against a general alternative, including the two-sample difference in location or regression in location as special cases for the alternative hypothesis. Further, the locally most powerful sequential rank tests are derived for the one-sample problem and for independence of two samples in an analogous spirit as the classical results of Hájek and Šidák (1967) for (classical) ranks. The locally most powerful tests are derived for a fixed sample size and the results bring arguments in favor of existing tests. In addition, we propose a sequential testing procedure based on these statistics of the locally most powerful tests. Principles of such sequential testing are explained on the two-sample Wilcoxon test based on sequential ranks.

Sequential rank CUSUM charts for angular data

A cumulative sum (CUSUM) control chart has desirable properties for checking whether a distribution has changed from an in-control to an out-of-control setting. Distribution-free CUSUMs based on sequential ranks to detect changes in the mean direction and dispersion of angular data are developed and some of their properties are illustrated by theoretical calculations and Monte Carlo simulation. Three applications to sequentially observed angular data from health science, industrial quality control and astrophysics are discussed.

Nonparametric Phase-II monitoring for detecting monotone trend based on inverse sampling

Recently, Mukherjee and Bandyopadhyay (J Stat Plan Inference, 2011, doi:10.1016/j.jspi.2011.02.017) introduced some partially sequential tests for detecting liner trend among the incoming series of observations when a training sample is available a-priori. Their work is very useful in econometric or environmental monitoring under certain situations. The present work is intended for generalization of their tests for any monotone trend. We develop two nonparametric tests for the identity of some unknown univariate continuous distribution functions against monotone or unidirectional trend in location. One of these two tests is based on usual ranks and the other is based on sequential ranks. These are typical nonparametric tests for monitoring structural changes. Performance of the two tests are compared using asymptotic studies as well as through some numerical results based on Monte-Carlo simulations. An illustration is offered using a real data on monthly production of certain beverage.

Some partially sequential nonparametric tests for detecting linear trend

In the present study, we develop two nonparametric partially sequential tests for detecting possible presence of linear trend among the incoming series of observations. We assume that a sample of fixed size is available a priori from some unknown univariate continuous population and there is no sign of trend among these historical observations. Our proposed tests can be viewed as the sequential type tests for monitoring structural changes. We use partial sequential sampling schemes based on usual ranks as well as on sequential ranks. We provide detailed discussion on asymptotic studies related to the proposed tests. We compare the two tests under various situations. We also present some numerical results based on simulation studies. Proposed tests are extremely important in profit making in volatile market through Margin Trading. We illustrate the mechanism with a detailed analysis of a stock price data.

Nonparametric sequential monitoring of longitudinal trials

This paper considers the sequential monitoring of multi-armed longitudinal clinical trials. We describe an approach that is relatively simple and accessible. Sequential ranks are used to form partial sum statistics, yielding processes that have independent increments, and hence can be approximated by Brownian motions. Three monitoring procedures are proposed. The first two are asymptotic, continuous analogues of the well-known Pocock and O'Brien-Fleming group sequential procedures, whereas the third procedure is exact. Performance of the procedures is assessed using Monte Carlo simulations. Data from an orthodontic clinical trial is used to illustrate the proposed methods, for the comparison of three treatment groups.

Some Rank-Based Two-Phase Procedures in Sequential Monitoring of Exchange Rate

Financial or economic crisis of 2008 and its severe negative impact on the human race is currently one of the focal themes of financial research. During this crisis period the Indian Rupee (INR) has significantly lost its ground compared to the United States Dollar (USD). In the present paper, we address the issues identifying warning and action points in a time sequential monitoring of the exchange rate records. We developed some approximate power 1 partially sequential nonparametric tests that are based on both usual and sequential ranks. Proposed procedures necessarily help us to capitalize the best features of the two different rank procedures. We use Wilcoxon score and the idea of curved stopping boundaries as in Bandyopadhyay et al. (2008). We vividly present our numerical findings obtained by sequential Monte-Carlo procedures. We aim at controlling the type I error rate in course of structural monitoring.

Robust kernel distance multivariate control chart using support vector principles

It is important to monitor manufacturing processes in order to improve product quality and reduce production cost. Statistical Process Control (SPC) is the most commonly used method for process monitoring, in particular making distinctions between variations attributed to normal process variability to those caused by ‘special causes’. Most SPC and multivariate SPC (MSPC) methods are parametric in that they make assumptions about the distributional properties and autocorrelation structure of in-control process parameters, and, if satisfied, are effective in managing false alarms/-positives and false-negatives. However, when processes do not satisfy these assumptions, the effectiveness of SPC methods is compromised. Several non-parametric control charts based on sequential ranks of data depth measures have been proposed in the literature, but their development and implementation have been rather slow in industrial process control. Several non-parametric control charts based on machine learning principles have also been proposed in the literature to overcome some of these limitations. However, unlike conventional SPC methods, these non-parametric methods require event data from each out-of-control process state for effective model building. The paper presents a new non-parametric multivariate control chart based on kernel distance that overcomes these limitations by employing the notion of one-class classification based on support vector principles. The chart is non-parametric in that it makes no assumptions regarding the data probability density and only requires ‘normal’ or in-control data for effective representation of an in-control process. It does, however, make an explicit provision to incorporate any available data from out-of-control process states. Experimental evaluation on a variety of benchmarking datasets suggests that the proposed chart is effective for process monitoring.

Sequential ranks and order statistics

A characterization of stationary sequences of random variables is given in terms of the sequential ranks and the order statistics. Bibliography: 3 titles.

Weak convergence of weighted empirical type processes under contiguous and changepoint alternatives

Let X 1, X 2,… be independent random variables. We study asymptotic behaviour of two-time parameter empirical type processes based on observations, ranks and sequential ranks. We introduce weight functions and derive the limiting distributions of these processes under the null hypothesis of X i being identically distributed, as well as under a class of continguous alternatives which can accommodate the possible occurrence of a changepoint in the series of measurements.

A new nonparametric quality control technique

Traditional control chart procedures process the information in the time sequence plot of successive values of the control variable. In this paper we consider a control chart procedure that uses the sequential ranks of the observations. The sequential rank of an observation is defined as its rank among the most recent g observations. A control chart technique that considers exponentially weighted moving averages (EWMA) of sequential ranks is proposed. This new nonparametric technique is outlier-resistant, and it performs well if one is concerned about slowly trending process levels.

A cusum procedure based on sequential ranks

A nonparametric quality-control statistic is proposed. The procedure is a cusum (cumulative sum) based on the sequential ranks of the observations which is designed to detect an abrupt change in the sampling distribution to a stochastically larger distribution. Such a procedure could be applied to establish that a process is in control or to detect assignable causes of loss of control. Meanwhile empirical data for constructing a parametric cusum procedure is being collected.

On the Asymptotic Theory of Statistics of Sequential Ranks

Previous article Next article On the Asymptotic Theory of Statistics of Sequential RanksA. M. Pardzhanadze and É. V. KhmaladzeA. M. Pardzhanadze and É. V. Khmaladzehttps://doi.org/10.1137/1131089PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] J. Oosterhoff and , W. Van Zwet, J. Jurechkova, A note on contiguity and Hellinger distanceContributions to Statistics, Reidel, Dordrecht, 1975, 157–166 Google Scholar[2] J. Hajek and , Z. Sidak, Theory of rank tests, Academic Press, New York, 1967, 297– 37:4925 0161.38102 Google Scholar[3] Ole Barndorff-Nielsen, On the limit behaviour of extreme order statistics, Ann. Math. Statist., 34 (1963), 992–1002 27:875 0119.15004 CrossrefGoogle Scholar[4] M. Reynolds, A sequential signed-rank test for symmetry, Ann. Statist., 3 (1975), 382–400 50:11665 0325.62050 CrossrefGoogle Scholar[5] E. V. Khmaladze and , A. M. Parjanadze, Functional limit theorems for linear statistics from sequential ranks, Probab. Theory Related Fields, 73 (1986), 585–595 88m:60098 0584.60044 CrossrefGoogle Scholar[6] P. Bickel and , M. Wichura, Convergence criteria for multiparameter stochastic processes and some applications, Ann. Math. Statist., 42 (1971), 1656–1670 52:4363 0265.60011 CrossrefGoogle Scholar[7] Georg Neuhaus, On weak convergence of stochastic processes with multidimensional time parameter, Ann. Math. Statist., 42 (1971), 1285–1295 45:2783 0222.60013 CrossrefGoogle Scholar[8] I. A. Ibragimov and , R. Z. Khas'minskii, Statistical estimation, Applications of Mathematics, Vol. 16, Springer-Verlag, New York, 1981vii+403, Asymptotic Theory 82g:62006 0467.62026 CrossrefGoogle Scholar[9] Patrick Billingsley, Convergence of probability measures, John Wiley & Sons Inc., New York, 1968xii+253 38:1718 0172.21201 Google Scholar[10] Peter Gaenssler and , Winfried Stute, Empirical processes: a survey of results for independent and identically distributed random variables, Ann. Probab., 7 (1979), 193–243 80d:60002 0402.60031 CrossrefGoogle Scholar[11] Ludger Rüschenforf, Asymptotic distributions of multivariate rank order statistics, Ann. Statist., 4 (1976), 912–923 54:8806 0359.62040 CrossrefGoogle Scholar[12] A. M. Pardzhanadze, Functional limit theorems for a random field constructed from sequential ranks, Summary of Reports of 19th Colloquium on Probability Theory and Mathematical Statistics, Gruz. NIINTI, Tbilisi, 1985, (In Russian.) Google Scholar[13] I. I. Gikhman and , A. V. Skorokhod, The theory of stochastic processes. I, Springer-Verlag, New York, 1974viii+570 49:11603 0291.60019 CrossrefGoogle Scholar[14] David M. Mason, On the use of a statistic based on sequential ranks to prove limit theorems for simple linear rank statistics, Ann. Statist., 9 (1981), 424–436 82h:62033 0459.62015 CrossrefGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails STATISTICAL SIMULATIONS METHOD IN APPLIED STATISTICSIndustrial laboratory. Diagnostics of materials, Vol. 85, No. 5 | 5 June 2019 Cross Ref 20 Weighted sequential empirical type processes with applications to change-point problemsOrder Statistics: Theory & Methods | 1 Jan 1998 Cross Ref The change-point problem for dependent observationsJournal of Statistical Planning and Inference, Vol. 53, No. 3 | 1 Aug 1996 Cross Ref Linear rank statistics, finite sampling, permutation tests and WinsorizingThe Annals of Statistics, Vol. 24, No. 3 | 1 Jun 1996 Cross Ref Weak convergence of weighted empirical type processes under contiguous and changepoint alternativesStochastic Processes and their Applications, Vol. 50, No. 2 | 1 Apr 1994 Cross Ref The Chibisov—O'Reilly theorem for empirical processes under contiguous measuresStatistics & Probability Letters, Vol. 19, No. 2 | 1 Jan 1994 Cross Ref Volume 31, Issue 4| 1987Theory of Probability & Its Applications563-742 History Submitted:04 November 1985Published online:17 July 2006 InformationCopyright © 1987 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1131089Article page range:pp. 669-682ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Functional limit theorems for linear statistics from sequential ranks

Let X 1 ,..., X n be a sequence of continuously distributed independent random variables. The normalized ranks R kn and sequential ranks S k , k=1,...,n, are defined by $${\text{R}}_{{\text{kn}}} = \frac{1}{{\text{n}}}\sum\limits_{{\text{j}} = 1}^{\text{n}} {{\text{I}}\{ {\text{X}}_{\text{j}} < {\text{X}}_{\text{k}} \} ,} {\text{ S}}_{\text{k}} = \frac{1}{{\text{k}}}\sum\limits_{{\text{j = }}1}^{\text{n}} {{\text{I}}\{ {\text{X}}_{\text{j}} < {\text{X}}_{\text{k}} \} .} $$ The subject of the present paper is the asymptotic behavior, as n→∞, of the process $$\frac{1}{{\sqrt {\text{n}} }}\sum\limits_{{\text{k}} \leqq {\text{nt}}} {{\text{a}}({\text{S}}_{\text{k}} ),} {\text{ }}0 \leqq {\text{t}} \leqq 1,$$ for a∈L 2 (0, 1), $$\int\limits_0^1 {{\text{adn}} = 0} $$ . For suitable a, the limiting law of that process is expressed as solution of a stochastic equation under the hypothesis of identically distributed X 1,..., X n as well as under a class of contiguous alternatives, which contains the occurrence of a change point in the series of measurements.

Limit theorems for generalized sequential rank statistics

A general class of statistics based on sequential ranks is introduced. Under suitable regularity conditions, an almost sure representation and invariance principle are established for this class. In particular, it is shown that these statistics can obey invariance principles that are radically different from those obeyed by the usual full rank statistics.

A Bahadur efficiency comparison between one and two sample rank statistics and their sequential rank statistic analogues

One and two sample rank statistics are shown in general to be more efficient in the Bahadur sense than their sequential rank statistic analogues as defined by Mason (1981, Ann. Statist. 9 424–436) and Lombard (1981, South African Statist. J. 15 129–152), even though the two families of statistics (those based on full ranks and those based on sequential ranks) have the same Pitman efficiency against local alternatives. In the process, general results on large deviation probabilities and laws of large numbers for statistics based on sequential ranks are obtained.