Multivariate Capability Indices: Distributional and Inferential Properties

Process capability indices have been widely used in the manufacturing industry for measuring process reproduction capability according to manufacturing specifications. Properties of the univariate processes have been investigated extensively, but are comparatively neglected for multivariate processes where multiple dependent characteristics are involved in quality measurement. In this paper, we consider two commonly used multivariate capability indices MC_p and MC_pm, to evaluate multivariate process capability. We investigate the statistical properties of the estimated MC_p and obtain the lower confidence bound for MC_p. We also consider testing MC_p, and provide critical values for testing if a multivariate process meets the preset capability requirement. In addition, an approximate confidence interval for MC_pm is derived. A simulation study is conducted to ascertain the accuracy of the approximation. Three examples are presented to illustrate the applicability of the obtained results.     

Multivariate process capability using principal component analysis in the presence of measurement errors

Various different definitions of multivariate process capability indices have been proposed in the literature. Most of the research works related to multivariate process capability indices assume no gauge measurement errors. However, in industrial applications, despite the use of highly advanced measuring instruments, account needs to be taken of gauge imprecision. In this paper we are going to examine the effects of measurement errors on multivariate process capability indices computed using the principal components analysis. We show that measurement errors alter the results of a multivariate process capability analysis, resulting in either a decrease or an increase in the capability of the process. In order to achieve accurate process capability assessments, we propose a method useful for overcoming the effects of gauge measurement errors.     

Lower confidence bounds as precision measure for truncated processes

Process capability index C_p has been the most popular one used in the manufacturing industry to provide numerical measures on process precision. For normally distributed processes with automatic fully inspections, the inspected processes follow truncated normal distributions. In this article, we provide the formulae of moments used for the Edgeworth approximation on the precision measurement C_p for truncated normally distributed processes. Based on the developed moments, lower confidence bounds with various sample sizes and confidence levels are provided and tabulated. Consequently, practitioners can use lower confidence bounds to determine whether their manufacturing processes are capable of preset precision requirements.     

A comparison of methods forsing loss-based capability index

The process capability index C _pm, sometimes called the loss-based index, has been proposed to the manufacturing industry for measuring process reproduction capability. This index incorporates the variation of production items with respect to the target value and the specification limits preset in the factory. To estimate the loss-based index properly and accurately, certain frequentist and Bayesian perspectives have been proposed to obtain lower confidence bounds (LCBs) for providing minimum process capability. The LCBs not only provide critical information regarding process performance but are also used to determine whether an improvement was made in a capability index and by extension in reducing the fraction of non-conforming items. In this paper, under the assumption of normality, based on frequentist and Bayesian senses, several existing approaches for constructing LCBs of C _pm are presented. Depending on the statistical methods used, we then classify these existing approaches into three categories and compared them in terms of the coverage rates and the mean values of the LCBs via simulations. The relative advantages and disadvantages of these approaches are summarized with some highlights of the relevant findings.     

Estimating total and specific process capability indices in three-stage processes with measurement errors

In multistage processes, two kinds of process capability indices (specific and total process capability indices) are defined in each stage. The total process capability index calculates the capability of each stage when it is affected by the previous stages and the specific process capability index calculates the capability of the stage when the effects of the previous stages are eliminated. Process capability indices in multistage processes are proposed under the assumption of no measurement errors. However, sometimes this assumption may be violated and leads to misleading interpretations. In this paper, the effects of measurement errors on the specific and total process capability indices in the second and third stages of the three-stage processes are statistically analysed. In addition, the effects of the measurement errors on the bias and the mean square error value of the total and specific process capability indices in the second and third stages are studied. Finally, the effects of the measurement errors on the specific and total process capability indices are shown through a numerical example.     

Comparing two measures of creatinine clearance: an application of errors-in-variables and bootstrap techniques

Two indices of creatinine clearance (an index of kidney function) are compared on a group of cancer patients who underwent chemotherapy with a potentially nephrotoxic drug. The standard index, measured creatinine clearance MCC, is cumbersome to use, whereas the more convenient alternative, estimated creatinine clearance ECC, has not yet been conclusively evaluated on cancer patients. We conclude that under certain clinical conditions ECC and MCC are identically calibrated for males, but not for females, and we obtain estimated true and false positive rates for assessing the use of ECC instead of MCC as a diagnostic tool. We use a model that is formally equivalent to an errors-in-variables model with (unbalanced) repeated observations and correlated measurement errors. The bootstrap is used to obtain standard errors and confidence limits.     

Bounds on the effect of heteroscedasticity on the chow test for structural change

This paper considers the effect of heteroscedastic regression errors on the size of the Chow test for structural stability. We show that bounds can be placed on the true size of this test in the light of such misspecification, and on the true critical value needed to achieve any desired significance level when using the test under various degrees of heteroscedasticity. These bounds are data-independent, and some cases are tabulated. Examples are given to illustrate the practical application of the critical value bounds.     

Process Capability Assessment for Asymmetric Tolerances with Consideration of Gauge Measurement Errors

In recent years, the issue of process capability assessment in the presence of gauge measurement errors (GME) for cases with symmetric tolerances was investigated enthusiastically. However, even processes with symmetric tolerances are very common in practical situations, cases of asymmetric tolerances also occur in manufacturing industries. In this article, a novel approach, called the generalized confidence interval (GCI) approach, is applied to assess the capabilities of processes with asymmetric tolerances in the presence of the GME. To examine the performance of the proposed approach, an exhaustive simulation was conducted. The conclusion is that the proposed approach appears quite satisfactorily for assessing process performance with asymmetric tolerances in the presence of GME in terms of the coverage rate (CR) and the average value of the generalized lower confidence limits.     

Micro-Level Estimation of Optimal Consumption Choice With Intertemporal Nonseparability in Preferences and Measurement Errors

This article investigates the presence of habit formation in household consumption, using data from the Panel Study of Income Dynamics. We develop an econometric model of internal habit formation of the multiplicative specification. The restrictions of the model allow for classical measurement errors in consumption without parametric assumptions on the distribution of measurement errors. We estimate the parameters by nonlinear generalized method of moments and find that habit formation is an important determinant of household food-consumption patterns. Using the parameter estimates, we develop bounds for the expectation of the implied heterogenous intertemporal elasticity of substitution and relative risk aversion that account for measurement errors, and compute confidence intervals for these bounds. Supplementary materials for this article are available online.     

Distributional and Inferential Properties of the Process Loss Indices

Johnson (1992) developed the process loss index Le, which is defined as the ratio of the expected quadratic loss to the square of half specification width. Tsui (1997) expressed the index Le as Le=Lpe+Lot, which provides an uncontaminated separation between information concerning the potential relative expected loss (Lpe) and the relative off-target squared (Lot), as the ratio of the process variance and the square of the half specification width, and the square of the ratio of the deviation of mean from the target and the half specification width, respectively. In this paper, we consider these three loss function indices, and investigate the statistical properties of their natural estimators. For the three indices, we obtain their UMVUEs and MLEs, and compare the reliability of the two estimators based on the relative mean squared errors. In addition, we construct 90%, 95%, and 99% upper confidence limits, and the maximum values of L^e for which the process is capable, 90%, 95%, and 99% of the time. The results obtained in this paper are useful to the practitioners in choosing good estimators and making reliable decisions on judging process capability.     

Lower bounds for confidence coefficients for confidence intervals for finite population quantiles

Assuming stratified simple random sampling, a confidence interval for a finite population quantile may be desired. Using a confidence interval with endpoints given by order statistics from the combined stratified sample, several procedures to obtain lower bounds (and approximations for the lower bounds) for the confidence coefficients are presented. The procedures differ with respect to the amount of prior information assumed about the var-iate values in the finite population, and the extent to which sample data is used to estimate the lower bounds.     

Impact of measurement errors on the performance and distributional properties of the multivariate capability index $$\mathbf{NMC }_\mathbf{pm }$$

Current industrial processes are sophisticated enough to be tied to only one quality variable to describe the process result. Instead, many process variables need to be analyze together to assess the process performance. In particular, multivariate process capability analysis (MPCIs) has been the focus of study during the last few decades, during which many authors proposed alternatives to build the indices. These measures are extremely attractive to people in charge of industrial processes, because they provide a single measure that summarizes the whole process performance regarding its specifications. In most practical applications, these indices are estimated from sampling information collected by measuring the variables of interest on the process outcome. This activity introduces an additional source of variation to data, that needs to be considered, regarding its effect on the properties of the indices. Unfortunately, this problem has received scarce attention, at least in the multivariate domain. In this paper, we study how the presence of measurement errors affects the properties of one of the MPCIs recommended in previous researches. The results indicate that even little measurement errors can induce distortions on the index value, leading to wrong conclusions about the process performance.     

On C pk Assessment in the Presence of Tool Wear

Process capability indices are routinely used in manufacturing industries for process monitoring. A basic assumption while using process capability indices is that there are no assignable causes of variation present. However, when variation due to an assignable cause is present and is tolerated, the conventional methods of capability measurement become inaccurate. In this article, we suggest an estimate of C_pk assuming that the process capability changes dynamically. We obtain an exact form of the sampling distribution in the presence of a systematic assignable cause. We discuss the problem of testing whether a given process is capable. The critical values for different sample sizes are obtained based on the sampling distribution. An example involving tool wear problem is presented.     

Confidence Intervals for Variance Components in Measurement System Capability Studies

In Measurement System Analysis a relevant issue is how to find confidence intervals for the parameters used to evaluate the capability of a gauge. In literature approximate solutions are available but they produce so wide intervals that they are often not effective in the decision process. In this article we introduce a new approach and, with particular reference to the parameter γ_R, i.e., the ratio of the variance due to the process and the variance due to the instrument, we show that, under quite realistic assumptions, we obtain confidence intervals narrower than other methods. An application to a real microelectronic case study is reported.     

Counting by weighing: construction of two-sided confidence intervals

Counting by weighing is widely used in industry and often more efficient than counting manually which is time consuming and prone to human errors especially when the number of items is large. Lower confidence bounds on the numbers of items in infinitely many future bags based on the weights of the bags have been proposed recently in Liu et al. [Counting by weighing: Know your numbers with confidence, J. Roy. Statist. Soc. Ser. C 65(4) (2016), pp. 641–648]. These confidence bounds are constructed using the data from one calibration experiment and for different parameters (or numbers), but have the frequency interpretation similar to a usual confidence set for one parameter only. In this paper, the more challenging problem of constructing two-sided confidence intervals is studied. A simulation-based method for computing the critical constant is proposed. This method is proven to give the required critical constant when the number of simulations goes to infinity, and shown to be easily implemented on an ordinary computer to compute the critical constant accurately and quickly. The methodology is illustrated with a real data example.     

Epidemics in heterogeneous communities: estimation of R0 and secure vaccination coverage

A stochastic multitype model for the spread of an infectious disease in a community of heterogeneous individuals is analysed. In particular, estimates of R ₀ (the basic reproduction number) and the critical vaccination coverage are derived, where estimation is based on final size data of an outbreak in the community. It is shown that these key parameters cannot be estimated consistently from data; only upper and lower bounds can be estimated. Confidence regions for the upper bounds are derived, thus giving conservative estimates of R ₀ and the fractions necessary to vaccinate.     

A process capability index that is based on the proportion of conformance

In this paper a new process capability index is proposed, which is based on the proportion of conformance of the process and has several appealing features. This index is simple in its assessment and interpretation and is applicable to normally or non-normally distributed processes. Likewise, its value can be assessed for continuous or discrete processes, it can be used under either unilateral or bilateral tolerances and the assessment of confidence limits for its true value is not very involved, under specific distributional assumptions. Point estimators and confidence limits for this index are investigated, assuming two very common continuous distributions (normal and exponential).     

Skewed zero-bound distributions and process capability indices for upper specifications

A common practical situation in process capability analysis, which is not well developed theoretically, is when the quality characteristic of interest has a skewed distribution with a long tail towards relatively large values and an upper specification limit only exists. In such situations, it is not uncommon that the smallest possible value of the characteristic is 0 and this is also the best value to obtain. Hence a target value 0 is assumed to exist. We investigate a new class of process capability indices for this situation. Two estimators of the proposed index are studied and the asymptotic distributions of these estimators are derived. Furthermore, we suggest a decision procedure useful when drawing conclusions about the capability at a given significance level, based on the estimated indices and their asymptotic distributions. A simulation study is also performed, assuming that the quality characteristic is Weibull-distributed, to investigate the true significance level when the sample size is finite.     

A process capability index with reliable confidence intervals

A well-know process capability index is slightly modified in this article to provide a new measure of process capability which takes account of the process location and variability, and for which point estimator and confidence intervals do exist that are insensitive to departures from the assumption of normal variability. Two examples of applications based on real data are presented.     

Inequalities for Expected Current Record Statistics

In this article, we obtain sharp distribution-free bounds for the expected value of the gap between the current records and record values as well as upper sharp bounds for the spacings between any two upper current records. We also present two-sided bounds on the errors in approximating the means of current records by inverse hazard functions.