Score tests for independence in parametric competing risks models

A popular model for competing risks postulates the existence of a latent unobserved failure time for each risk. Assuming that these underlying failure times are independent is attractive since it allows standard statistical tools for right-censored lifetime data to be used in the analysis. This paper proposes simple independence score tests for the validity of this assumption when parametric regression models are used to model the individual risks. The score tests are derived for the alternatives that specify that copulas are responsible for a possible dependency between competing risks. The test statistics are functions of the Cox and Snell residuals. A variance estimator is derived by writing the score function and the Fisher information matrix for the marginal models as stochastic integrals. A simulation study and a numerical example illustrate the methodology proposed in this paper.      

Algorithms for digital measurement of random-signal rolling mean and variance

Recurrent algorithms have been derived for calculating the rolling mean and variance, which reduce the memory volume required, improve the speed, and raise the reliability.Translated from Izmeritel'naya Tekhnika, No. 12, pp. 11–12, December, 1993.     

Self-starting control chart for simultaneously monitoring process mean and variance

A self-starting control chart, based on the likelihood ratio test and the exponentially weighted moving average procedure, is proposed for monitoring the process mean and variance simultaneously when the process parameters are unknown. A table is presented to assist in the design of the control chart with different parameters. Its in-control average run length can be evaluated by a two-dimensional Markov chain model. Moreover, the diagnostic aids of the proposed chart are given. Monte Carlo simulation results compared with some competing methods in the literature show that the proposed approach has quite satisfactory charting performance across a range of possible shifts when the process parameters are unknown, even including the detection of a decrease in variability. A real data example from industrial manufacturing is used to demonstrate its implementation.     

Robust CUSUM charts for monitoring the process mean and variance

This paper considers the problem of obtaining robust control charts for detecting changes in the mean µ and standard deviation σ of process observations that have a continuous distribution. The standard control charts for monitoring µ and σ are based on the assumption that the process distribution is normal. However, the process distribution may not be normal in many situations, and using these control charts can lead to very misleading conclusions. Although some control charts for µ can be tuned to be robust to non‐normal distributions, the most critical problem with non‐robustness is with the control chart for σ. This paper investigates the performance of two CUSUM chart combinations that can be made to be robust to non‐normality. One combination consists of the standard CUSUM chart for µ and a CUSUM chart of absolute deviations from target for σ, where these CUSUM charts are tuned to detect relatively small parameter shifts. The other combination is based on using winsorized observations in the standard CUSUM chart for µ and a CUSUM chart of squared deviations from target for σ. Guidance is given for selecting the design parameters and control limits of these robust CUSUM chart combinations. When the observations are actually normal, using one of these robust CUSUM chart combination will result in some reduction in the ability to detect moderate and large changes in µ and σ, compared with using a CUSUM chart combination that is designed specifically for the normal distribution. Copyright © 2009 John Wiley & Sons, Ltd.     

An adaptive chart for monitoring the process mean and variance

Traditionally, an chart is used to control the process mean and an R chart is used to control the process variance. However, these charts are not sensitive to small changes in the process parameters. The adaptive and R charts might be considered if the aim is to detect small disturbances. Due to the statistical character of the joint and R charts with fixed or adaptive parameters, they are not reliable in identifying the nature of the disturbance, whether it is one that shifts the process mean, increases the process variance, or leads to a combination of both effects. In practice, the speed with which the control charts detect process changes may be more important than their ability in identifying the nature of the change. Under these circumstances, it seems to be advantageous to consider a single chart, based on only one statistic, to simultaneously monitor the process mean and variance. In this paper, we propose the adaptive non‐central chi‐square statistic chart. This new chart is more effective than the adaptive and R charts in detecting disturbances that shift the process mean, increase the process variance, or lead to a combination of both effects. Copyright © 2006 John Wiley & Sons, Ltd.     

Bivariate semiparametric control charts for simultaneous monitoring of process mean and variance

In the present article, two semiparametric bivariate control charts are presented, which use order statistics and are effective in jointly monitoring of possible shifts in the process mean and/or variance. To achieve that both the median location (or more generally the location of a specific order statistic) and the number of specific observations of the test sample lying between the control limits are taken into account. The false alarm rate and the in-control average run length are not affected by the marginal distributions, while the effect of the dependence structure on them is negligible; therefore, they can be used as fully nonparametric charts. A performance-comparison study is carried out, and an illustrative example is provided using a real-world data set.     

Some theorems on conditional mean convergence and conditional almost sure convergence for randomly weighted sums of dependent random variables

In (Ordó?ez Cabrera and Volodin, J. Math. Anal. Appl. 305:644?C658, 2005), the authors introduce the notion of h-integrability of an array of random variables with respect to an array of constants, and obtained some mean convergence theorems for weighted sums of random variables subject to some special kinds of dependence. In view of the important role played by conditioning and dependence in the models used to describe many situations in the applied sciences, the concepts and results in the aforementioned paper are extended herein to the case of randomly weighted sums of dependent random variables when a sequence of conditioning sigma-algebras is given. The dependence conditions imposed on the random variables (conditional negative quadrant dependence and conditional strong mixing) as well as the convergence results obtained are conditional relative to the conditioning sequence of sigma-algebras. In the last section, a strong conditional convergence theorem is also established by using a strong notion of conditional h-integrability.     

Optimal coordinate sensor placements for estimating mean and variance components of variation sources

In-process optical coordinate measuring machines offer the potential to diagnose the sources of the variations that are responsible for product quality defects. Such a sensor system can thus help manufacturers to improve product quality and reduce process downtime. The effective use of sensor data in the diagnosis of the sources of variations depends on the optimal design of the sensor system, which is often also called the problem of sensor placement. This paper addresses coordinate sensor placement for the diagnosis of dimensional variation sources in assembly processes. Sensitivity indices for the detection of the process mean and variance components are defined as the design criteria and are derived in terms of process layout and sensor deployment information. Exchange algorithms, originally developed for optimal experimental design, are revised and then used to maximize the detection sensitivity. A sort-and-cut procedure is proposed, which is able to significantly improve the algorithm efficiency of the current exchange routine. The resulting optimal sensor layout and its implications are illustrated in the specific context of a panel assembly process.     

U-tests for variance components in linear mixed models

We propose a U-statistics-based test for null variance components in linear mixed models and obtain its asymptotic distribution (for increasing number of units) under mild regularity conditions that include only the existence of the second moment for the random effects and of the fourth moment for the conditional errors. We employ contiguity arguments to derive the distribution of the test under local alternatives assuming additionally the existence of the fourth moment of the random effects. Our proposal is easy to implement and may be applied to a wide class of linear mixed models. We also consider a simulation study to evaluate the behaviour of the U-test in small and moderate samples and compare its performance with that of exact F-tests and of generalized likelihood ratio tests obtained under the assumption of normality. A practical example in which the normality assumption is not reasonable is included as illustration.     

Control charts for monitoring the mean and percentiles of Weibull processes with variance components

In this article, we study Shewhart and exponentially weighted moving average control charts for monitoring the mean or, equivalently, the percentiles of a Weibull process when additional sources of variation, also known as variance components, are present. We adopt a frailty model to describe the monitored process. We derive analytical properties for this model and use them to develop control charts. We consider charts for the sample mean and exponentially weighted moving averages. We compare their average run length performances to their traditional counterparts when they do not account for variance components.     

Goodness-of-fit tests for Log-GARCH and EGARCH models

This paper studies goodness-of-fit tests and specification tests for an extension of the Log-GARCH model, which is both asymmetric and stable by scaling. A Lagrange-multiplier test is derived for testing the extended Log-GARCH against more general formulations taking the form of combinations of Log-GARCH and exponential GARCH (EGARCH). The null assumption of an EGARCH is also tested. Portmanteau goodness-of-fit tests are developed for the extended Log-GARCH. An application to real financial data is proposed.     

Multivariate EWMA control charts using individual observations for process mean and variance monitoring and diagnosis

Most multivariate control charts in the literature are designed to detect either mean or variation shifts rather than both. A simultaneous use of the Hotelling T ² and |S| control charts has been proposed but the Hotelling T ² reacts to mean shifts, dispersion changes, and changes of correlations among responses. The combination of two multivariate control charts into one chart sometimes loses the ability to provide detailed diagnostic information when a process is out-of-control. In this research a new multivariate control chart procedure based on exponentially weighted moving average (EWMA) statistics is proposed to monitor process mean and variance simultaneously to identify proper sources of variations. Two multivariate EWMA control charts using individual observations are proposed to achieve a quick detection of mean or variance shifts or both. Simulation studies show that the proposed charts are capable of identifying appropriate types of shifts in terms of correct detection percentages. A manufacturing example is used to demonstrate how the proposed charts can be properly set-up based on average run length values via simulations. In addition, correct detection rates of the proposed charts are explored.     

X charts with supplementary samples to control the mean and variance

A standard X chart for controlling a process takes regular individual observations, for instance every half hour. This article proposes a modification of the X chart that allows one to take supplementary samples. The supplementary sample is taken (and the X and R values computed) when the current value of X falls outside the control limits. With the supplementary sample, the signal of out-of-control is given by an X value outside the X chart's control limits or an R value outside the R chart's control limit. The proposed chart is designed to hold the supplementary sample frequency, during the in-control period, as low as 5% or less. In this context, the practitioner might prefer to verify an out-of-control condition by simply comparing the X and R values with the control limits. In other words, without plotting the X and R points. The X chart with supplementary samples has two major advantages when compared with the standard X and R charts: (a) the user will be plotting X values instead of X and R values; (b) the shifts in the process mean and/or changes in the process variance are detected faster.     

On correct expression of variance of double progressive mean statistic for monitoring Poisson observations

Control charts play a vital role in tracing unfavorable changes in the ongoing process parameters to enhance the quality of the output. Attribute charting schemes are commonly utilized for tracing shifts in fraction nonconforming and number of nonconformities per unit in any production process. A double progressive mean charting structure for monitoring Poisson observations (PDPM) is recently proposed by Alevizakos and Koukouvinos¹ for efficient monitoring of the average number of nonconformities in the production unit. Unfortunately, the variance expression used in the charting statistic of the said article needs correction. The focus of this study is to provide correct derivation and expression of the variance of the PDPM statistic.     

A comparison study on effectiveness and robustness of control charts for monitoring process mean and variance

This article compares the effectiveness and robustness of nine typical control charts for monitoring both process mean and variance, including the most effective optimal and adaptive sequential probability ratio test (SPRT) charts. The nine charts are categorized into three types (the type, CUSUM type and SPRT type) and three versions (the basic version, optimal version and adaptive version). While the charting parameters of the basic charts are determined by common wisdoms, the parameters of the optimal and adaptive charts are designed optimally in order to minimize an index average extra quadratic loss for the best overall performance. Moreover, the probability distributions of the mean shift δ_µ and standard deviation shift δ_σ are studied explicitly as the influential factors in a factorial experiment. The main findings obtained in this study include: (1) From an overall viewpoint, the SPRT‐type chart is more effective than the CUSUM‐type chart and type chart by 15 and 73%, respectively; (2) in general, the adaptive chart outperforms the optimal chart and basic chart by 16 and 97%, respectively; (3) the optimal CUSUM chart is the most effective fixed sample size and sampling interval chart and the optimal SPRT chart is the best choice among the adaptive charts; and (4) the optimal sample sizes of both the charts and the CUSUM charts are always equal to one. Furthermore, this article provides several design tables which contain the optimal parameter values and performance indices of 54 charts under different specifications. Copyright © 2011 John Wiley & Sons, Ltd.     

The Max EWMAMS control chart for joint monitoring of process mean and variance with individual observations

A traditional approach to monitor both the location and the scale parameters of a quality characteristic is to use two separate control charts. These schemes have some difficulties in concurrent tracking and interpretation. To overcome these difficulties, some researchers have proposed schemes consisting of only one chart. However, none of these schemes is designed to work with individual observations. In this research, an exponentially weighted moving average (EWMA)‐based control chart that plots only one statistic at a time is proposed to simultaneously monitor the mean and variability with individual observations. The performance of the proposed scheme is compared with one of the two other existing combination charts by simulation. The results show that in general the proposed chart has a significantly better performance than the other combination charts. Copyright © 2010 John Wiley & Sons, Ltd.