Robustness of the EWMA and the combined X¯–EWMA control charts with variable sampling intervals to non-normality

The exponentially weighted moving average (EWMA) control charts with variable sampling intervals (VSIs) have been shown to be substantially quicker than the fixed sampling intervals (FSI) EWMA control charts in detecting process mean shifts. The usual assumption for designing a control chart is that the data or measurements are normally distributed. However, this assumption may not be true for some processes. In the present paper, the performances of the EWMA and combined X¯–EWMA control charts with VSIs are evaluated under non-normality. It is shown that adding the VSI feature to the EWMA control charts results in very substantial decreases in the expected time to detect shifts in process mean under both normality and non-normality. However, the combined X¯–EWMA chart has its false alarm rate and its detection ability is affected if the process data are not normally distributed.     

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.     

A weibull process failure mechanism for the economic design of mewma control charts

The design parameters of the multivariate exponentially weighted moving average (MEWMA) control chart may be chosen according to economic and/or statistical considerations. The economic model proposed for the design of the'MEWMA chart assumes a Markovian process failure mechanism following an exponential distribution. We'assess the sensitivity of the resulting economic design for the MEWMA to deviations from this assumption. In particular, the generalization, from an exponential to a Weibull distribution of process failure, is used to study the selection of MEWMA chart parameters given process cost and time information. We conclude that the quality of the resulting design (in terms of expected cost) is not substantially affected by mis-specification of the distribution of process failure.     

Exact inference for a simple step-stress model with competing risks for failure from exponential distribution under Type-II censoring

In reliability analysis, accelerated life-testing allows for gradual increment of stress levels on test units during an experiment. In a special class of accelerated life tests known as step-stress tests, the stress levels increase discretely at pre-fixed time points, and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. In this article, we consider the simple step-stress model under Type-II censoring when the lifetime distributions of the different risk factors are independently exponentially distributed. Under this setup, we derive the maximum likelihood estimators (MLEs) of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. The exact distributions of the MLEs of the parameters are then derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions and the parametric bootstrap method, we discuss the construction of confidence intervals for the parameters and assess their performance through Monte Carlo simulations. Finally, we illustrate the methods of inference discussed here with an example.     

A new nonparametric synthetic EWMA control chart for monitoring process mean

The exponentially weighted moving average (EWMA) control charts are widely used in chemical and process industries because of their excellent speed in catching small to moderate shifts in the process target. In usual practice, many data come from a process where the monitoring statistic is non-normally distributed or it follows an unknown probability distribution. This necessitates the use of distribution-free/nonparametric control charts for monitoring the deviations from the process target. In this paper, we integrate the existing EWMA sign chart with the conforming run length chart to propose a new synthetic EWMA (SynEWMA) sign chart for monitoring the process mean. The SynEWMA sign chart encompasses the synthetic sign and EWMA sign charts. Monte Carlo simulations are used to compute the run length profiles of the SynEWMA sign chart. Based on a comprehensive comparison, it turns out that the SynEWMA sign chart is able to perform substantially better than the existing EWMA sign chart. Both real and simulated data sets are used to explain the working and implementation of existing and proposed control charts.     

Order restricted Bayesian inference for exponential simple step-stress model

A step-stress model has received a considerable amount of attention in recent years. In the usual step-stress experiment, a stress level is allowed to increase at each step to get rapid failure of the experimental units. The expected lifetime of the experimental unit is shortened as the stress level increases. Although extensive amount of work has been done on step-stress models, not enough attention has been paid to analyze step-stress models incorporating this information. We consider a simple step-stress model and provide Bayesian inference of the unknown parameters under cumulative exposure model assumption. It is assumed that the lifetime of the experimental units are exponentially distributed with different scale parameters at different stress levels. It is further assumed that the stress level increases at each step, hence the expected lifetime decreases. We try to incorporate this restriction using the prior assumptions. It is observed that different censoring schemes can be incorporated very easily under a general setup. Monte Carlo simulations have been performed to see the effectiveness of the proposed method, and two datasets have been analyzed for illustrative purposes.     

Independent or dependent competing risks: does it make a difference

This article investigates the consequences of departures from independence when the component lifetimes in a series system are exponentially distributed. Such departures are studied when the joint distribution is assumed to follow either one of the three Gumbel bivariate exponential models, the Downton bivariate exponential model, or the Oakes bivariate exponential model. Two distinct situations are considered. First, in theoretical modeling of series systems, when the distribution of the component lifetimes is assumed, one wishes to compute system reliability and mean system life. Second, errors in parametric and nonparametric estimation of component reliability and component mean life are studied based on life-test data collected on series systems when the assumption of independence is made     

A minimum cost exponential cusum test

A minimum cost CUSUM test for an event rate increase when inter-event times are exponentially distributed is presented. Optimal values of the test decision parameters, h and k, are developed from a renewal reward model of the event cycle by combining a non-linear optimization technique with an exact method for determining exponential average run lengths. Test robustness for event cycle parameter estimates and departures from the assumption of exponentially distributed inter-event times are discussed in the context of an injury monitoring scenario. Robustness to positively serially correlated observations emanating from EAR(1) and EMA(1) processes is also examined.     

Monitoring Multivariate Simple Linear Profiles in the Presence of between Profile Autocorrelation

In recent years, statistical profile monitoring has emerged as a relatively new and potentially useful subarea of statistical process control and has attracted attention of many researchers and practitioners. A profile, waveform, or signature is a function that relates a dependent or a response variable to one or more independent variables. Different statistical methods have been proposed by researchers to monitor profiles where each method requires its own assumptions. One of the common and implicit assumptions in most of the proposed procedures is the assumption of independent residuals. Violation of this assumption can affect the performance of control procedures and ultimately leading to misleading results. In this article, we study phase II analysis of monitoring multivariate simple linear profiles when the independency assumption is violated. Three time series based methods are proposed to eliminate the effect of correlation that exists between multivariate profiles. Performances of the proposed methods are evaluated using average run length (ARL) criterion. Numerical results indicate satisfactory performance for the proposed methods. A simulated example is also used to show the application of the proposed methods.     

Bayesian X̄ control limits for exponentially distributed measurements

This article deals with the construction of an X? control chart using the Bayesian perspective. We obtain new control limits for the X? chart for exponentially distributed data-generating processes through the sequential use of Bayes’ theorem and credible intervals. Construction of the control chart is illustrated using a simulated data example. The performance of the proposed, standard, tolerance interval, exponential cumulative sum (CUSUM) and exponential exponentially weighted moving average (EWMA) control limits are examined and compared via a Monte Carlo simulation study. The proposed Bayesian control limits are found to perform better than standard, tolerance interval, exponential EWMA and exponential CUSUM control limits for exponentially distributed processes.     

A multiple warm standby δ-shock system with a repairman having multiple vacations

In this article, a warm standby n-unit system is studied. The system is operational as long as there is one unit normal. The unit online, which has a lifetime distribution governed by a phase-type distribution, is also attacked by a shock from some external causes. Assume that shocks arrive according to a Poisson process. Whenever an interarrival time of shock is less than a threshold, the unit online fails. The lifetimes of the units in warm standby is exponentially distributed. A repairman who can take multiple vacations repairs the failed units based on the “first-in-first-out” rule. The repair times and the vacation times of repairman are governed by different phase-type distributions. For this system, the Markov process governing the system is constructed. The system is studied in a transient and stationary regime; the availability, the reliability, the rates of occurrence of the different types of failures, and the working probability of the repairman are calculated. A numerical application is performed to illustrate the calculations.     

Maximum likelihood and bayes prediction of current system lifetime

The Power Law Process is often used to analyse failure data of repairable systems undergoing development testing where the system failure intensity decreases as a result of repeated application of corrective actions. At the end of the development program, the system failure intensity is assumed to remain constant and the current system lifetime is assumed to be exponentially distributed. In this paper, prediction limits on the current system lifetime have been derived both in the maximum likelihood and Bayesian context. Exact values and a closed form approximation of percentage points of the pivotal quantity used in the classical approach are given in the case of failure truncated testing. For both failure and time truncated testing, the Bayesian approach is developed both when no prior knowledge is available and when information on the reliability growth rate can be given. A numerical example is also given.     

One-sided Continuous Tolerance Limits and their Accompanying Sample Size Problem

Tolerance limits are those limits that contain a certain proportion of the distribution of a characteristic with a given probability. 'They are used to make sure that the production will not be outside of specifications' (Amin & Lee, 1999). Usually, tolerance limits are constructed at the beginning of the monitoring of the process. Since they are calculated just one time, these tolerance limits cannot reflect changes of tolerance level over the lifetime of the process. This research proposes an algorithm to construct tolerance limits continuously over time for any given distribution. This algorithm makes use of the exponentially weighted moving average (EWMA) technique. It can be observed that the sample size required by this method is reduced over time.     

Likelihood inference for the component lifetime distribution based on progressively censored parallel systems data

Point and interval estimators for the scale parameter of the component lifetime distribution of a k-component parallel system are obtained when the component lifetimes are assumed to be independently and identically exponentially distributed. We prove that the maximum likelihood estimator of the scale parameter based on progressively Type-II censored system lifetimes is unique and can be obtained by a fixed-point iteration procedure. In particular, we illustrate that the Newton–Raphson method does not converge for any initial value. Furthermore, exact confidence intervals are constructed by a transformation using normalized spacings and other component lifetime distributions including Weibull distribution are discussed.     

The extended GWMA control chart

This study extends the generally weighted moving average (GWMA) control chart by imitating the double exponentially weighted moving average (DEWMA) technique. The proposed chart is called the double generally weighted moving average (DGWMA) control chart. Simulation is employed to evaluate the average run length characteristics of the GWMA, DEWMA and DGWMA control charts. An extensive comparison of these control charts reveals that the DGWMA control chart with time-varying control limits is more sensitive than the GWMA and the DEWMA control charts for detecting medium shifts in the mean of a process when the shifts are between 0.5 and 1.5 standard deviations. Additionally, the GWMA control chart performs better when the mean shifts are below the 0.5 standard deviation, and the DEWMA control performs better when the mean shifts are above the 1.5 standard deviation. The design of the DGWMA control chart is also discussed.     

Robust Hotelling's T2 control charts under non-normality: the case of t-Student distribution

Traditional multivariate control charts are based upon the assumption that the observations follow a multivariate normal distribution. In many practical applications, however, this supposition may be difficult to verify. In this paper, we use control charts based on robust estimators of location and scale to improve the capability of detection observations out of control under non-normality in the presence of multiple outliers. Concretely, we use a simulation process to analyse the behaviour of the robust alternatives to Hotelling's T ², which use minimum volume ellipsoidal (MVE) and minimum covariance determinant (MCD) in the presence of observations with a Student's t-distribution. The results show that these robust control charts are good alternatives for small deviations from normality due to the fact that the percentage of out-of-control observations detected for these charts in the Phase II are higher.     

Designing historical control studies with survival endpoints using exact statistical inference

Historical control trials compare an experimental treatment with a previously conducted control treatment. By assigning all recruited samples to the experimental arm, historical control trials can better identify promising treatments in early phase trials compared with randomized control trials. Existing designs of historical control trials with survival endpoints are based on asymptotic normal distribution. However, it remains unclear whether the asymptotic distribution of the test statistic is close enough to the true distribution given relatively small sample sizes in early phase trials. In this article, we address this question by introducing an exact design approach for exponentially distributed survival endpoints, and compare it with an asymptotic design in both real examples and simulation examples. Simulation results show that the asymptotic test could lead to bias in the sample size estimation. We conclude the proposed exact design should be used in the design of historical control trials.     

Sample size calculations for noninferiority trials for time-to-event data using the concept of proportional time

Noninferiority trials intend to show that a new treatment is ‘not worse'' than a standard-of-care active control and can be used as an alternative when it is likely to cause fewer side effects compared to the active control. In the case of time-to-event endpoints, existing methods of sample size calculation are done either assuming proportional hazards between the two study arms, or assuming exponentially distributed lifetimes. In scenarios where these assumptions are not true, there are few reliable methods for calculating the sample sizes for a time-to-event noninferiority trial. Additionally, the choice of the non-inferiority margin is obtained either from a meta-analysis of prior studies, or strongly justifiable ‘expert opinion'', or from a ‘well conducted'' definitive large-sample study. Thus, when historical data do not support the traditional assumptions, it would not be appropriate to use these methods to design a noninferiority trial. For such scenarios, an alternate method of sample size calculation based on the assumption of Proportional Time is proposed. This method utilizes the generalized gamma ratio distribution to perform the sample size calculations. A practical example is discussed, followed by insights on choice of the non-inferiority margin, and the indirect testing of superiority of treatment compared to placebo.KEYWORDS: Generalized gamma, noninferiority, non-proportional hazards, proportional time, relative time, sample size     

A weighted likelihood ratio test-based chart for monitoring process mean and variability

In this paper, a new single exponentially weighted moving average (EWMA) control chart based on the weighted likelihood ratio test, referred to as the WLRT chart, is proposed for the problem of monitoring the mean and variance of a normally distributed process variable. It is easy to design, fast to compute, and quite effective for diverse cases including the detection of the decrease in variability and individual observation case. The optimal parameters that can be used as a design aid in selecting specific parameter values based on the average run length (ARL) and the sample size are provided. The in-control (IC) and out-of-control (OC) performance properties of the new chart are compared with some other existing EWMA-type charts. Our simulation results show that the IC run length distribution of the proposed chart is similar to that of a geometric distribution, and it provides quite a robust and satisfactory overall performance for detecting a wide range of shifts in the process mean and/or variability.     

Economic and economic statistical designs for three-level control charts

In some situations, an appropriate quality measure uses three or more discrete levels to classify a product characteristic. For these situations, some control charts have been developed based on statistical criteria regardless of economic considerations. In this paper, we develop economic and economic statistical designs (ESD) for 3-level control charts. We apply the cost model proposed by Costa and Rahim.[Economic design of X charts with variable parameters: the Markov chain approach, J Appl Stat 28 (2001), 875–885] Furthermore, we assume that the length of time that the process remains in control is exponentially distributed which allows us to apply the Markov chain approach for developing the cost model. We apply a genetic algorithm to determine the optimal values of model parameters by minimizing the cost function. A numerical example is provided to illustrate the performance of the proposed models and to compare the cost of the pure economic and ESD for three-level control charts. A sensitivity analysis is also conducted in this numerical example.