Identifying the time of change in the mean of a two-stage nested process

Statistical process control charts are used to distinguish between common cause and special cause sources of variability. Once a control chart signals, a search to find the special cause should be initiated. If process analysts had knowledge of the change point, the search to find the special cause could be easily facilitated. Relevant literature contains an array of solutions to the change-point problem; however, these solutions are most appropriate when the samples are assumed to be independent. Unfortunately, the assumption of independence is often violated in practice. This work considers one such case of non-independence that frequently occurs in practice as a result of multi-stage sampling. Due to its commonality in practice, we assume a two-stage nested random model as the underlying process model and derive and evaluate a maximum-likelihood estimator for the change point in the fixed-effects component of this model. The estimator is applied to electron microscopy data obtained following a genuine control chart signal and from a real machining process where the important quality characteristic is the size of the surface grains produced by the machining operation. We conduct a simulation study to compare relative performances between the proposed change-point estimator and a commonly used alternative developed under the assumption of independent observations. The results suggest that both estimators are approximately unbiased; however, the proposed estimator yields smaller variance. The implication is that the proposed estimator is more precise, and thus, the quality of the estimator is improved relative to the alternative.     

Identifying the time of a step change with multivariate single control charts

Change point estimation procedures simplify the efforts to search for and identify special causes in multivariate statistical process monitoring. After a signal is generated by the simultaneously used control charts or a single control chart, add-on change point procedure estimates the time of the change. In this study, multivariate joint change point estimation performance for simultaneous monitoring of both location and dispersion is compared under the assumption that various single charts are used to monitor the process. The change detection performance for several structural changes for the mean vector and covariance matrix is also discussed. It is concluded that choice of the control chart to obtain a signal may affect the change point detection performance.     

On the estimation of serial correlation in Markov-dependent production processes

In this paper, we present a study about the estimation of the serial correlation for Markov chain models which is used often in the quality control of autocorrelated processes. Two estimators, non-parametric and multinomial, for the correlation coefficient are discussed. They are compared with the maximum likelihood estimator [U.N. Bhat and R. Lal, Attribute control charts for Markov dependent production process, IIE Trans. 22 (2) (1990), pp. 181–188.] by using some theoretical facts and the Monte Carlo simulation under several scenarios that consider large and small correlations as well a range of fractions (p) of non-conforming items. The theoretical results show that for any value of p≠0.5 and processes with autocorrelation higher than 0.5, the multinomial is more precise than maximum likelihood. However, the maximum likelihood is better when the autocorrelation is smaller than 0.5. The estimators are similar for p=0.5. Considering the average of all simulated scenarios, the multinomial estimator presented lower mean error values and higher precision, being, therefore, an alternative to estimate the serial correlation. The performance of the non-parametric estimator was reasonable only for correlation higher than 0.5, with some improvement for p=0.5.     

On modelling data from degradation sample paths over time

This paper is mainly concerned with modelling data from degradation sample paths over time. It uses a general growth curve model with Box‐Cox transformation, random effects and ARMA(p, q) dependence to analyse a set of such data. A maximum likelihood estimation procedure for the proposed model is derived and future values are predicted, based on the best linear unbiased prediction. The paper compares the proposed model with a nonlinear degradation model from a prediction point of view. Forecasts of failure times with various data lengths in the sample are also compared.     

Monitoring correlated processes with binomial marginals

Few approaches for monitoring autocorrelated attribute data have been proposed in the literature. If the marginal process distribution is binomial, then the binomial AR(1) model as a realistic and well-interpretable process model may be adequate. Based on known and newly derived statistical properties of this model, we shall develop approaches to monitor a binomial AR(1) process, and investigate their performance in a simulation study. A case study demonstrates the applicability of the binomial AR(1) model and of the proposed control charts to problems from statistical process control.     

Monitoring the mean of autocorrelated observations with one generally weighted moving average control chart

Traditionally, using a control chart to monitor a process assumes that process observations are normally and independently distributed. In fact, for many processes, products are either connected or autocorrelated and, consequently, obtained observations are autocorrelative rather than independent. In this scenario, applying an independence assumption instead of autocorrelation for process monitoring is unsuitable. This study examines a generally weighted moving average (GWMA) with a time-varying control chart for monitoring the mean of a process based on autocorrelated observations from a first-order autoregressive process (AR(1)) with random error. Simulation is utilized to evaluate the average run length (ARL) of exponentially weighted moving average (EWMA) and GWMA control charts. Numerous comparisons of ARLs indicate that the GWMA control chart requires less time to detect various shifts at low levels of autocorrelation than those at high levels of autocorrelation. The GWMA control chart is more sensitive than the EWMA control chart for detecting small shifts in a process mean.     

Covariance analysis and associated spectra for classes of nonstationary processes

In 1951, Cramér introduced a class of nonstationary processes. This broad class of processes contains the important harmonizable and stationary classes of processes. The Cramér class can have additional structure imposed upon it through Cesàro summability considerations. These refined Cramér classes, termed (c,p)-summable Cramér, have recently been considered by Swift (in: M.M. Rao (Ed.), Real and Stochastic Analysis: Recent Advances, CRC Press, Boca Raton, FL, 1997, p. 303). In this paper, the relationship between the (c,p)-summable Cramér classes and the (KF,p) classes of processes introduced by Rao in 1985 is considered. The (KF,p) classes of processes are a generalization of the class of processes considered by Kampé de Feriet and Frenkiel. A continuity theorem for the (KF,p) classes is obtained. This result yields a spectral representation for the (KF,p) classes. Some (KF,p) class processes are shown to arise as the solution to a difference equation obtained from a linear model of a noisy communication channel.     

Process capability analysis for serially dependent processes of Poisson counts

Process capability indices evaluate the actual compliance of a process with given external specifications in a single number. For the case of a process of independent and identically distributed Poisson counts, two types of index have been proposed and investigated in the literature. The assumption of serial independence, however, is quite unrealistic for practice. We consider the case of an underlying Poisson INAR(1) process which has an AR(1)-like autocorrelation structure. We show that the performance of the estimated indices is degraded heavily if serial dependence is ignored. Therefore, we develop approaches for estimating the process capability (both for the observation and innovation process), which explicitly consider the observed degree of autocorrelation. For this purpose, we introduce a new unbiased estimator of the innovations’ mean of a Poisson INAR(1) process and derive its exact as well as asymptotic stochastic properties. In this context, we also present new explicit expressions for the third- and fourth-order moments of a Poisson INAR(1) process. Then the capability indices and the performance of their estimators are analysed and recommendations for practice are given.     

Euler(p,q) Processes and Their Application to Non Stationary Time Series with Time Varying Frequencies

We introduce Euler(p, q) processes as an extension of the Euler(p) processes for purposes of obtaining more parsimonious models for non stationary processes whose periodic behavior changes approximately linearly in time. The discrete Euler(p, q) models are a class of multiplicative stationary (M-stationary) processes and basic properties are derived. The relationship between continuous and discrete mixed Euler processes is shown. Fundamental to the theory and application of Euler(p, q) processes is a dual relationship between discrete Euler(p, q) processes and ARMA processes, which is established. The usefulness of Euler(p, q) processes is examined by comparing spectral estimation with that obtained by existing methods using both simulated and real data.     

Conditional maximum likelihood estimation in negative binomial INGARCH processes with known number of successes when the true parameter is at the boundary of the parameter space

The paper establishes the asymptotic distribution of the conditional maximum likelihood estimator for integer-valued generalized autoregressive conditional heteroskedastic (INGARCH) processes of conditional negative binomial distributions, with the number of successes in the definition of the negative binomial distribution being assumed to be known, when the true parameter is at the boundary of the parameter space. Based on the result, coefficient nullity tests are developed for model simplification. The proposed tests are investigated through a simulation study.     

Short run spc based upon the second order dynamic linear model for trend detection

Recently, several new applications of control chart procedures for short production runs have been introduced. Bothe (1989) and Burr (1989) proposed the use of control chart statistics which are obtained by scaling the quality characteristic by target values or process estimates of a location and scale parameter. The performance of these control charts can be significantly affected by the use of incorrect scaling parameters, resulting in either an excessive "false alarm rate," or insensitivity to the detection of moderate shifts in the process. To correct for these deficiencies, Quesenberry (1990, 1991) has developed the Q-Chart which is formed from running process estimates of the sample mean and variance. For the case where both the process mean and variance are unknown, the Q-chaxt statistic is formed from the standard inverse Z-transformation of a t-statistic. Q-charts do not perform correctly, however, in the presence of special cause disturbances at process startup. This has recently been supported by results published by Del Castillo and Montgomery (1992), who recommend the use of an alternative control chart procedure which is based upon a first-order adaptive Kalman filter model Consistent with the recommendations by Castillo and Montgomery, we propose an alternative short run control chart procedure which is based upon the second order dynamic linear model (DLM). The control chart is shown to be useful for the early detection of unwanted process trends. Model and control chart parameters are updated sequentially in a Bayesian estimation framework, providing the greatest degree of flexibility in the level of prior information which is incorporated into the model. The result is a weighted moving average control chart statistic which can be used to provide running estimates of process capability. The average run length performance of the control chart is compared to the optimal performance of the exponentially weighted moving average (EWMA) chart, as reported by Gan (1991). Using a simulation approach, the second order DLM control chart is shown to provide better overall performance than the EWMA for short production run applications     

Sample moments of the autocorrelations of moving average processes and a modification to bartlett'sasymptotic variance formula

In this paper we express the sample autocorrelations for a moving average process of order q as a function of its own theoretical autocorrelations and the sample autocorrelations for the generating white noise series. Approximate analytic expressions are then obtained forthe moments of the sample autocorrelations of the moving average process.

Using these expressions, together with numerical evidence, we show that Bartlett's asymptotic formula for the variance of the sample autocorrelations of moving average processes, which is used widely in identifying these processes, is a large overestimate when considering finitesample sizes.

Our approach is for motivational purposes and so is purely formal, the amount of mathematics presented being kept to a minimum.     

Investigation of the diesel consumption for trucks at an overburden stripping area by SPC study

This study examined whether diesel consumption used by trucks at a stripping area is controlled or not. The factors affecting diesel consumption were also investigated and some necessary solutions were presented. Diesel consumption was observed with the aid of control graphs. Abnormal situations in the diesel consumption were explored by means of Shewhart control graphs. The factors which are out of control were also presented in a cause–effect diagram, and suggestions for improvement were proposed. It has been determined that the main effect of the diesel consumption is the daily run number of the trucks. The main factors affecting the daily run number were also investigated.     

基于加权似然比检验的监控线性曲线的控制图

文章提出了一种基于加权似然比检验的阶段二监控线性曲线的控制图,称为WLRT图,并通过平均运行长度来衡量控制图的性能表现。模拟结果表明,WLRT图对于线性曲线的截距、斜率、标准差的变化及截距和斜率同时变化都具有很好的检测能力。通过与其他几种控制图的性能比较,得出WLRT图能较快地发现过程变化,而且设计简单、操作方便。     

A change point method for Phase II monitoring of generalized linear profiles

In this article, we adopt the change point approach to monitor the generalized linear profiles in phase II Statistical process control (SPC). Generalized linear profiles include a large class of profiles defined in one framework. In contrast to the conventional change point approach, we adopt the Rao score test rather than the likelihood ratio test. Simulated results show that our approach has a good performance over any possible single step change in process parameters for two special cases of generalized linear profiles, namely Poisson and binomial profiles. Some diagnostic aids are also given and a real example is introduced to shed light on the merits of our approach in real applications.     

中国股票市场的长期记忆性与趋势预测研究

本文首次在国内利用较新的精准局部似然函数法（Exact Local Whittle）,以上证指数为对象,估计了ARFIMA(p,d,q)模型的长期记忆参数d,并分析了上证指数的趋势性变化。估计结果和稳健性检验均表明,上证指数具有长期记忆性,以上证指数为代表的股票市场并非有效;模拟结果显示,当滚动窗口n=260,带宽m=[n0.65]时,长期记忆参数即估计量d既具备一致性,又具有渐进正态性。在2004年10月8日至2015年11月13日期间,模型给出了8次上涨或下跌的趋势转换信号,其中7次信号是正确的,仅有1次给出了错误信号;股票价格由下跌趋势转为上涨趋势、由上涨趋势转为下跌趋势两种情况相比,记忆参数d在前一种情况时下跌幅度更大,给出的信号更明显。     

Estimation Methods of the Long Memory Parameter: Monte Carlo Analysis and Application

Since the seminal paper of Granger & Joyeux (1980), the concept of a long memory has focused the attention of many statisticians and econometricians trying to model and measure the persistence of stationary processes. Many methods for estimating d, the long-range dependence parameter, have been suggested since the work of Hurst (1951). They can be summarized in three classes: the heuristic methods, the semi-parametric methods and the maximum likelihood methods. In this paper, we try by simulation, to verify the two main properties of [dcirc]: the consistency and the asymptotic normality. Hence, it is very important for practitioners to compare the performance of the various classes of estimators. The results indicate that only the semi-parametric and the maximum likelihood methods can give good estimators. They also suggest that the AR component of the ARFIMA (1, d, 0) process has an important impact on the properties of the different estimators and that the Whittle method is the best one, since it has the small mean squared error. We finally carry out an empirical application using the monthly seasonally adjusted US Inflation series, in order to illustrate the usefulness of the different estimation methods in the context of using real data.     

Evidence of ARCH(1) Errors in the Context of Spurious Regressions

The spurious regression phenomenon is related to first-order serially correlated errors. This study, using a Monte Carlo analysis, finds that this phenomenon is also related to ARCH(1) type errors.     

EVVMA and cusum control charts in the presence of correlation

The statistical properties of control charts are usually evaluated under the assumption that the observations from the process are independent. For many processes however, observations which are closely spaced in time will be correlated. This paper considers EWMA and CUSUM control charts for the process mean when the observations are from an AR(1) process with additional random error. This simple model may be a reasonable model for many processes encountered in practice. The ARL and steady state ARL of the EWMA and CUSUM charts are evaluated numerically using an integral equation approach and a Markov chain approach. The numerical results show that correlation can have a significant effect on the properties of these charts. Tables are given to aid in the design of these charts when the observations follow the assumed model.