Jyh-Jen Horng Shiau

A two-stage spline smoothing method for partially linear models

Abstract Rice (1986) showed that the partial spline estimate of the parametric component in a semiparametric regression model is generally biased and it is necessary to undersmooth the nonparametric component to force the bias to be negligible with respect to the standard error. We propose a two-stage spline smoothing method for estimating the parametric and nonparametric components in a semiparametric model. By appropriately choosing rates for smoothing parameters, we show that the parametric component can be estimated at the parametric rate with the new estimate without undersmoothing the nonparametric component. We also show that the same result holds for the partial regression estimate proposed independently by Denby (1986) and Speckman (1988). Asymptotic normality results for the parametric component are also shown for both estimates. Furthermore, we associate these estimates with Wellners (1986) efficient scores methods.

A Bayesian procedure for process capability assessment

The usual practice of judging process capability by evaluating point estimates of some process capability indices has a flaw that there is no assessment on the error distributions of these estimates. However, the distributions of these estimates are usually so complicated that it is very difficult to obtain good interval estimates. In this paper we adopt a Bayesian approach to obtain an interval estimation, particularly for the index Cpm. The posterior probability p that the process under investigation is capable is derived; then the credible interval, a Bayesian analogue of the classical confidence interval, can be obtained. We claim that the process is capable if all the points in the credible interval are greater than the pre-specified capability level ω, say 1.33. To make this Bayesian procedure very easy for practitioners to implement on manufacturing floors, we tabulate the minimum values of Ĉpm/ω, for which the posterior probability p reaches the desirable level, say 95%. For the special cases where the process mean equals the target value for Cpm and equals the midpoint of the two specification limits for Cpk, the procedure is even simpler; only chi-square tables are needed. Copyright

Capability assessment for processes with multiple characteristics: A generalization of the popular index Cpk

PU. We not only provided some tables but also presented an application example. Copyright

Monitoring Nonlinear Profiles with Random Effects by Nonparametric Regression

The monitoring of process/product profiles is presently a growing and promising area of research in statistical process control. This study is aimed at developing monitoring schemes for nonlinear profiles with random effects. We utilize the technique of principal components analysis to analyze the covariance structure of the profiles and propose monitoring schemes based on principal component (PC) scores. The number of the PC scores used in constructing control charts is crucial to the detecting power. In the Phase I analysis of historical data, due to the dependency of the PC-scores, we adopt the usual Hotelling T 2 chart to check the stability. For Phase II monitoring, we study individual PC-score control charts, a combined chart scheme that combines all the PC-score charts, and a T 2 chart. Although an individual PC-score chart may be perfect for monitoring a particular mode of variation, a chart that can detect general shifts, such as the T 2 chart and the combined chart scheme, is more feasible in practice. The performances of the schemes under study are evaluated in terms of the average run length.

A note on Bayesian estimation of process capability indices

Process capability indices are useful for assessing the capability of manufacturing processes. Most traditional methods are obtained from the frequentist point of view. We view the problem from the Bayes and empirical Bayes approaches by using non-informative and conjugate priors, respectively.

Effective Control Charts for Monitoring Multivariate Process Dispersion

When monitoring process dispersion, it is common to pay more attention to dispersion increases than to decreases for practical reasons. Nonetheless, it is also important to detect dispersion decreases for two reasons: (i) it deserves further investigations as to why the process has improved; and (ii) if the process has changed, the settings of the control chart would need to be adjusted for effective future monitoring. In this paper, we first propose an effective control chart for detecting multivariate dispersion decreases in phase II process monitoring, which is constructed using the same approach as that of the one-sided likelihood-ratio-test-based multivariate chart proposed recently in the literature for detecting dispersion increases. We then discuss a combined charting scheme by combining these two one-sided charts for detecting either dispersion increases or decreases. Comparative simulation studies show that the proposed combined control charting scheme outperforms several existing two-sided control charts in terms of the average run length when the process dispersion indeed increases or decreases. Two real-life examples are presented to demonstrate the applicability of the proposed charts. Copyright

Yield‐Related Process Capability Indices for Processes of Multiple Quality Characteristics

Process capability indices (PCIs) have been widely used in industries for assessing the capability of manufacturing processes. Castagliola and Castellanos (Quality Technology and Quantitative Management 2005, 2(2):201–220), viewing that there were no clear links between the definition of the existing multivariate PCIs and theoretical proportion of nonconforming product items, defined a bivariate Cpk and Cp (denoted by BCpk and BCp, respectively) based on the proportions of nonconforming product items over four convex polygons for bivariate normal processes with a rectangular specification region. In this paper, we extend their definitions to MCpk and MCp for multivariate normal processes with flexible specification regions. To link the index to the yield, we establish a ‘reachable’ lower bound for the process yield as a function of MCpk. An algorithm suitable for such processes is developed to compute the natural estimate of MCpk from process data. Furthermore, we construct via the bootstrap approach the lower confidence bound of MCpk, a measure often used by producers for quality assurance to consumers. As for BCp, we first modify the original definition with a simple preprocessing step to make BCp scale-invariant. A very efficient algorithm is developed for computing a natural estimator of BCp. This new approach of BCp can be easily extended to MCp for multivariate processes. For BCp, we further derive an approximate normal distribution for , which enables us to construct procedures for making statistical inferences about process capability based on data, including the hypothesis testing, confidence interval, and lower confidence bound. Finally, the proposed procedures are demonstrated with three real data sets. Copyright

A note on mse coverage intervals in a partial spline model

A partial spline model is used to estimate an unknown function which is smooth except for some break points. Assuming the break points are known, a Generalized Cross-Validated smoothing spline estimation method is proposed. Some interval estimation methods for the magnitude of the discontinuities based on the mean square error are introduced and investigated.

Robustness of the EWMA Control Chart to Non-normality for Autocorrelated Processes

Abstract Most commonly used control charts for monitoring quality characteristics of the processes were developed under the assumption that the observations are randomly sampled from a normal population. It is well known that these control charts have more false alarms than usual when processes are positively autocorrelated. One remedy is to adjust the control limits such that the modified control charts can achieve an about right false-alarm rate. In this paper, we investigate the robustness of such modified individuals Shewhart control chart and modified exponentially weighted moving average (EWMA) control chart to the usual normality assumption of the white noise term in an AR(1) process with positive autocorrelation. The performances of the control charts under study are evaluated on the basis of the average run length (ARL) curves. It is found that the modified EWMA control chart is more robust to the normality assumption than the modified individuals Shewhart control chart in terms of the in-control ARL for some heavy-tailed symmetric distributions and some skewed distributions. Results also show that the choice of the EWMA smoothing parameter A is very crucial to the ARL performance. However, choosing an appropriate value for λ is not easy and many practitioners may simply choose a value of 0.1 or 0.2, which are values commonly suggested for the standard EWMA charts designed for independent normal data. Unfortunately, the modified EWMA control chart with these popular values of λ does not perform well enough for some of the positively autocorrelated non-normal data in our study. In a preliminary study for improving the robustness, we consider two control charts with data averaging schemes called the moving-average EWMA chart and subgroup-average EWMA chart, respectively. A small simulation study shows that the subgroup-average EWMA control chart with the same naïve choice of λ = 0.1 or 0.2 indeed outperforms the modified EWMA control chart with a tradeoff of slight inefficiency on the detecting power for the case under study.

A Distribution-Free Multivariate Control Chart for Phase I Applications

The purpose of this paper is to provide a novel distribution-free control chart for monitoring the location parameter vector of a multivariate process in phase I analysis. To be robust to the process distribution, the spatial sign statistic that defines the multivariate direction of an observation is used to construct a Shewhart-type control chart for detecting out-of-control observations in historical phase I data. The proposed control chart is distribution free in the sense that the false-positive rate (or false alarm rate), the proportion of wrongly classified in-control samples, can be controlled at the specified value for elliptical-direction distributions. In addition, we demonstrate through simulation studies that the false-positive rate of the proposed chart is robust to the shift size of the out-of-control condition if we only delete the most extreme out-of-control observation at each iteration of phase I analysis. Compared with the traditional Hotellings T2 control chart and some of its robust versions, the proposed chart is generally more powerful in detecting out-of-control observations and more robust to the normality assumption. Copyright