一种可用于检测过程尺度参数的非参数控制图
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摘要
传统的控制图多数是在已知过程分布的假设下构建的,这种控制图被称为参数控制图。然而,在实际应用中,大多数过程因为其数据的复杂性导致他们的精确分布往往难以确定。当预先指定的参数分布无效时,参数控制图的结果将不再可靠。为了解决这个问题,通常考虑非参数控制图,因为非参数控制图比参数控制图更加稳健。近年来对非参数控制图的研究越来越多,但大多数现有的控制图主要是用于检测位置参数的变化。本文提出一个新的非参数Shewhart控制图(称为LOG图),可用来检测未知连续过程分布的尺度参数。文中依据运行长度分布的均值,方差和分位数,分析了LOG图在过程受控和失控时的性能表现,并与其他非参数控制图进行比较。模拟结果表明,LOG图在不同过程分布下对检测尺度参数的漂移都具有很好的性能。最后用一个实例来说明LOG图在实际中的应用。
Traditional control charts are constructed under the assumption of known process distribution.Such control charts are known as the parametric control charts.In various applications,most of the data streams follow complex processes and their exact distributions are often untraceable.When a parametric distribution specified beforehand is invalid,some articles argue that results from such conventional parametric monitoring schemes would no longer be reliable.To address this problem,the nonparametric control chart is often considered,since a nonparametric control chart is more robust than a parametric one.Although research on nonparametric control charts has increased in recent years,the majority of existing charts focus only on detecting process location shifts.In this paper,we propose a new Shewhart-type distribution-free control chart(named as LOG chart)for monitoring of unknown scale parameter of continuous distributions.The in-control and out-of-control performance properties and a comparison with some other distribution-free control charts are presented in terms of the average,the standard deviation and some percentiles of the run length distribution.Numerical results based on Monte Carlo analysis show that the proposed LOG chart provides quite a satisfactory performance for a class of location-scale models.The application of our proposed LOG chart is illustrated by a real data example.
Traditional control charts are constructed under the assumption of known process distribution.Such control charts are known as the parametric control charts.In various applications,most of the data streams follow complex processes and their exact distributions are often untraceable.When a parametric distribution specified beforehand is invalid,some articles argue that results from such conventional parametric monitoring schemes would no longer be reliable.To address this problem,the nonparametric control chart is often considered,since a nonparametric control chart is more robust than a parametric one.Although research on nonparametric control charts has increased in recent years,the majority of existing charts focus only on detecting process location shifts.In this paper,we propose a new Shewhart-type distribution-free control chart(named as LOG chart)for monitoring of unknown scale parameter of continuous distributions.The in-control and out-of-control performance properties and a comparison with some other distribution-free control charts are presented in terms of the average,the standard deviation and some percentiles of the run length distribution.Numerical results based on Monte Carlo analysis show that the proposed LOG chart provides quite a satisfactory performance for a class of location-scale models.The application of our proposed LOG chart is illustrated by a real data example.
引文
[1]胡雪龙,周晓剑,黄卫东,蒋国平.CUSUM中位数控制图设计[J].数理统计与管理,2018,37(3):469-477
[2]张久军,李琦,杨瑞梅,何川.检测对数正态分布位置参数和尺度参数的控制图[J].数理统计与管理,201837(5):864-870.
[3]张久军,李二杰,张慧,卞宇婷,刘双.联合检测过程位置参数和尺度参数的非参数Cusum控制图[J].数理统计与管理,2017,36(5):833-842.
[4]生志荣,程龙生,顾玉萍.基于控制图的马田系统马氏空间生成机理研究[J].数理统计与管理,2017,36(6):1059-1068.
[5]Song Z,Liu Y,Li Z,Zhang J.A weighted likelihood ratio test-based chart for monitoring process mean and variability[J].Journal of Statistical Computation and Simulation,2018,88(2):1-22.
[6]Song Z,Liu Y,Li Z,Zhang J.A comparative study of memory-type control charts based on robust scale estimators[J].Quality and Reliability Engineering International,2018,34(6):1079-1102.
[7]李红梅,王青.基于GARCH型控制图的互联网金融预警实证分析-以余额宝为例[J].辽宁大学学报(哲学社会科学版),2018,46(2):38-44.
[8]Chakraborti S,Qiu P,Mukherjee A.Editorial to the special issue:Nonparametric statistical process control charts[J].Quality and Reliability Engineering International,2015,31(1):1-2.
[9]Nandini D.A note on the efficiency of nonparametric control chart for monitoring process variability[J].Economic Quality Control,2008,23(1):85-93.
[10]Nandini D.A non-parametric control chart for controlling variability based on squared rank test[J].Journal of Industrial and Systems Engineering,2008,2(2):114—125.
[11]Zhou M,Geng W.A robust control chart for monitoring dispersion[J].Journal of Applied Mathematics,2013,2013:1-5.
[12]Zhou M,Zhou Q,Geng W.A new nonparametric control chart for monitoring variability[J].Quality and Reliability Engineering International,2016,32(7):2471-2479.
[13]Villanueva-Guerra E,Tercerogomez V,Corderofranco A,Conover W.A control chart for variance based on squared ranks[J].Journal of Statistical Computation and Simulation,2017,87(4):1-26.
[14]Hajek,J,Sidak Z,Sen P.Theory of Rank Tests(2th Ed.)[M].San Diego,California:Academic Press,1999.
[15]Kossler W.Asymptotic power and efficiency of Lepage-type tests for the treatment of combined location-scale alternatives[R].Informatik-bericht nr.Humboldt-Universitat zu Berlin 200,2006.
[16]Song Z,Mukher.jee A,Liu Y,Zhang J.Optimizing joint location-scale monitoring—An adaptive distribution-free approach with minimal loss of information[J].European Journal of Operational Research,2019,274(3):1019-1036.
[17]王兆军,巩震,邹长亮.ARL计算方法综述[J].数理统计与管理,2011,30(3):467-497.
[18]Montgomery D C.Introduction to Statistical Quality Control(6th Ed.)[M].Hoboken,NJ:John Wiley&Sons,2009.
[2]张久军,李琦,杨瑞梅,何川.检测对数正态分布位置参数和尺度参数的控制图[J].数理统计与管理,201837(5):864-870.
[3]张久军,李二杰,张慧,卞宇婷,刘双.联合检测过程位置参数和尺度参数的非参数Cusum控制图[J].数理统计与管理,2017,36(5):833-842.
[4]生志荣,程龙生,顾玉萍.基于控制图的马田系统马氏空间生成机理研究[J].数理统计与管理,2017,36(6):1059-1068.
[5]Song Z,Liu Y,Li Z,Zhang J.A weighted likelihood ratio test-based chart for monitoring process mean and variability[J].Journal of Statistical Computation and Simulation,2018,88(2):1-22.
[6]Song Z,Liu Y,Li Z,Zhang J.A comparative study of memory-type control charts based on robust scale estimators[J].Quality and Reliability Engineering International,2018,34(6):1079-1102.
[7]李红梅,王青.基于GARCH型控制图的互联网金融预警实证分析-以余额宝为例[J].辽宁大学学报(哲学社会科学版),2018,46(2):38-44.
[8]Chakraborti S,Qiu P,Mukherjee A.Editorial to the special issue:Nonparametric statistical process control charts[J].Quality and Reliability Engineering International,2015,31(1):1-2.
[9]Nandini D.A note on the efficiency of nonparametric control chart for monitoring process variability[J].Economic Quality Control,2008,23(1):85-93.
[10]Nandini D.A non-parametric control chart for controlling variability based on squared rank test[J].Journal of Industrial and Systems Engineering,2008,2(2):114—125.
[11]Zhou M,Geng W.A robust control chart for monitoring dispersion[J].Journal of Applied Mathematics,2013,2013:1-5.
[12]Zhou M,Zhou Q,Geng W.A new nonparametric control chart for monitoring variability[J].Quality and Reliability Engineering International,2016,32(7):2471-2479.
[13]Villanueva-Guerra E,Tercerogomez V,Corderofranco A,Conover W.A control chart for variance based on squared ranks[J].Journal of Statistical Computation and Simulation,2017,87(4):1-26.
[14]Hajek,J,Sidak Z,Sen P.Theory of Rank Tests(2th Ed.)[M].San Diego,California:Academic Press,1999.
[15]Kossler W.Asymptotic power and efficiency of Lepage-type tests for the treatment of combined location-scale alternatives[R].Informatik-bericht nr.Humboldt-Universitat zu Berlin 200,2006.
[16]Song Z,Mukher.jee A,Liu Y,Zhang J.Optimizing joint location-scale monitoring—An adaptive distribution-free approach with minimal loss of information[J].European Journal of Operational Research,2019,274(3):1019-1036.
[17]王兆军,巩震,邹长亮.ARL计算方法综述[J].数理统计与管理,2011,30(3):467-497.
[18]Montgomery D C.Introduction to Statistical Quality Control(6th Ed.)[M].Hoboken,NJ:John Wiley&Sons,2009.
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