A note on the asymptotic distribution of the process capability index C pmk

Process capability indices have been widely used to evaluate the process performance to the continuous improvement of quality and productivity. The distribution of the estimator of the process capability index C _pmk is very complicated and the asymptotic distribution is proposed by Chen and Hsu [The asymptotic distribution of the processes capability index C _pmk , Comm. Statist. Theory Methods 24(5) (1995), pp. 1279–1291]. However, we found a critical error for the asymptotic distribution when the population mean is not equal to the midpoint of the specification limits. In this paper, a correct version of the asymptotic distribution is given. An asymptotic confidence interval of C _pmk by using the correct version of asymptotic distribution is proposed and the lower bound can be used to test if the process is capable. A simulation study of the coverage probability of the proposed confidence interval is shown to be satisfactory. The relation of six sigma technique and the index C _pmk is also discussed in this paper. An asymptotic testing procedure to determine if a process is capable based on the index of C _pmk is also given in this paper.     

A Bayesian procedure for assessing process performance based on the third-generation capability index

Capability indices that qualify process potential and process performance are practical tools for successful quality improvement activities and quality program implementation. Most existing methods to assess process capability were derived on the basis of the traditional frequentist point of view. This paper considers the problem of estimating and testing process capability based on the third-generation capability index C _pmk from the Bayesian point of view. We first derive the posterior probability p for the process under investigation is capable. The one-sided credible interval, a Bayesian analog of the classical lower confidence interval, can be obtained to assess process performance. To investigate the effectiveness of the derived results, a series of simulation was undertaken. The results indicate that the performance of the proposed Bayesian approach depends strongly on the value of ξ=(μ?T)/σ. It performs very well with the accurate coverage rate when μ is sufficiently far from T. In those cases, they have the same acceptable performance even though the sample size n is as small as 25.     

The asymptotic distribution of the process capability index Cpmk

The main result of this paper is that under some regularity conditions, the distribution of an estimator of the process capability index C_pmk is asymptotically normal.     

An asymptotic distribution of wright’s process capability index sensitive to skewness

In a recent paper (J. Statist. Comput. Simul., 1995, Vol.53, pp. 195–203) P. A. Wright proposed a new process capability index C_s which generalizes the Pearn-Kotz-Johnson’s index C_pmk by taking into account the skewness (in addition to deviation of the mean from tliCrntarget already incorporated in C_pmk ). The purpose of this article is to study the consistency and asymptotics of an estimate ?_s of C_s The asymptotic distribution provides an insight into some desirable properties of the estimate which are not apparent from its original definition     

A general class of capability indices in the case of asymmetric tolerances

Vannman has earlier studied a class of capability indices, containing the indices C _p , C _pk , C _pm and C _pmk , when the tolerances are symmetric. We study the properties of this class when the tolerances are asymmetric and suggest a new enlargened class of indices. Under the assumption of normality an explicit form of the distribution of the new class of the estimated indices is provided. Numerical investigations are made to explore the behavior of the estimators of the indices for different values of the parameters. Based on the estimator a decision rule that can be used to determine whether the process can be considered capable or not is provided and suitable criteria for choosing an index from the family are suggested.     

Reliable confidence intervals for assessing normal process incapability

This study aims to provide a reliable confidence interval for assessing the process incapability index [C_pp]. The concept of the generalized pivotal quantities is utilized for constructing the generalized confidence interval for [C_pp]. And, simulations are performed for demonstrating our proposed method and one existent method. The results show that the empirical confidences of these two methods are significantly affected by the degree of process departure. Therefore, we suggest the practitioners to select proper one for capability testing purpose based on the information of degree of process departure.     

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.     

A General Class of Test Procedures for Left-Truncated and Right-Censored Data

When the distribution of one of the characteristics of a process is non normal, methods based on empirical percentiles suggest the use of several process capability indices (PCIs) which are similar to the usual C _p , C _pk , C _pm , and C _pmk indices. However most of these PCIs apply only to the case of symmetrical tolerances. To take into account the asymmetry of the tolerances as well as the asymmetry of the process distribution, new PCIs which improve the previous ones are proposed. In the end and in order to validate the method proposed here, we apply it to a real production case.     

Estimating process capability index Cpk: classical approach versus Bayesian approach

Process capability index C_pk has been the most popular one used in the manufacturing industry dealing with problems of measuring reproduction capability of processes to enhance product development with very low fraction of defectives. In the manufacturing industry, lower confidence bound (LCB) estimates the minimum process capability providing pivotal information for quality engineers to monitoring the process and assessing process performance for quality assurance. The main objective of this paper is to compare and contrast the LCBs on C_pk using two approaches, Classical method and Bayesian method.     

A robust process capability index

The existing process capability indices (PCI's) assume that the distribution of the process being investigated is normal. For non-normal distributions, PCI's become unreliable in that PCI's may indicate the process is capable when in fact it is not. In this paper, we propose a new index which can be applied to any distribution. The proposed indexC_f:, is directly related to the probability of non-conformance of the process. For a given random sample, the estimation of C_f boils down to estimating non-parametrically the tail probabilities of an unknown distribution. The approach discussed in this paper is based on the works by Pickands (1975) and Smith (1987). We also discuss the construction of bootstrap confidence intervals of C_f: based on the so-called accelerated bias correction method (BC _a:). Several simulations are carried out to demonstrate the flexibility and applicability of C_f:. Two real life data sets are analyzed using the proposed index.     

Developing and designing of an efficient variables sampling system based on the process capability index

Process capability indices (PCIs) are extensively used in the manufacturing industries in order to confirm whether the manufactured products meet their specifications or not. PCIs can be used to judge the process precision, process accuracy, and the process performance. So developing of sampling plans based on PCIs is inevitable and those plans will be very much useful for maintaining and improving the product quality in the manufacturing industries. In view of this, we propose a variables sampling system based on the process capability index C_pmk, which takes into account of process yield and process loss, when the quality characteristic under study will have double specification limits. The proposed sampling system will be effective in compliance testing. The advantages of this system over the existing sampling plans are also discussed. In order to determine the optimal parameters, tables are also constructed by formulating the problem as a nonlinear programming in which the average sample number is minimized by satisfying the producer and consumer risks.     

An asymptotic confidence interval for the process capability index Cpm

Abstract

The use of indices as an estimation tool of process capability is long-established among the statistical quality professionals. Numerous capability indices have been proposed in last few years. C_pm constitutes one of the most widely used capability indices and its estimation has attracted much interest. In this paper, we propose a new method for constructing an approximate confidence interval for the index C_pm. The proposed method is based on the asymptotic distribution of the index C_pm obtained by the Delta Method. Under some regularity conditions, the distribution of an estimator of the process capability index C_pm is asymptotically normal.     

The probability density function of process capability index Cpmk

We derive an explicit and computationally convenient form for the probability density function of the estimator of the process capability index C_pmk ( Pearn, Kotz and Johnson ), when sampling from a normal distribution.     

On a Process Capability Index for Asymmetric Specifications

Abstract

Among the process capability indices considered in the literature C _pm is one of the most widely used, despite the fact that its performance often becomes unsatisfactory in the case of processes with asymmetric specifications, i.e., processes whose target value is not located at the midpoint of the specification area. In this article, a new index that is a variant of C _pm , is introduced and shown to overcome this deficiency. In particular, the proposed index performs satisfactorily for processes with symmetric or asymmetric specifications. Moreover, the article compares the suggested index with the existing indices for asymmetric specifications, investigates the distributional properties of its estimator under the assumption of normality and deals with the assessment of confidence intervals using three bootstrap methods. The coverage achieved by each of these methods is investigated via simulation.     

Estimating Process Capability Index C PM Using a Bootstrap Sequential Sampling Procedure

Construction of a confidence interval for process capability index C _PM is often based on a normal approximation with fixed sample size. In this article, we describe a different approach in constructing a fixed-width confidence interval for process capability index C _PM with a preassigned accuracy by using a combination of bootstrap and sequential sampling schemes. The optimal sample size required to achieve a preassigned confidence level is obtained using both two-stage and modified two-stage sequential procedures. The procedure developed is also validated using an extensive simulation study.     

On Confidence Intervals for Process Capability Indices in a One-Way Random Model

In this article, we investigated the bootstrap calibrated generalized confidence limits for process capability indices C _pk for the one-way random effect model. Also, we derived Bissell's approximation formula for the lower confidence limit using Satterthwaite's method and calculated its coverage probabilities and expected values. Then we compared it with standard bootstrap (SB) method and generalized confidence interval method. The simulation results indicate that the confidence limit obtained offers satisfactory coverage probabilities. The proposed method is illustrated with the help of simulation studies and data sets.     

Construction of closed interval for process capability indices Cpu,Cpl, and Spk based on Boole’s inequality and de Morgan’s laws

ABSTRACT

Considerable effort has been spent on the development of confidence intervals for process capability indices (PCIs) based on the sampling distribution of the PCI or the transferred PCI. However, there is still no definitive way to construct a closed interval for a PCI. The aim of this study is to develop closed intervals for the PCIs C_pu, C_pl, and S_pk based on Boole's inequality and de Morgan's laws. The relationships between different sample sizes, the significance levels, and the confidence intervals of the PCIs C_pu, C_pl, and S_pk are investigated. Then, a testing model for interval estimation for the PCIs C_pu, C_pl, and S_pk is built as a powerful tool for measuring the quality performance of a product. Finally, an applied example is given to demonstrate the effectiveness and applicability of the proposed method and the testing model.     

Letter to the editor the asymptotic distribution of the process capability index cpmk

In theorem 3.1 of Chen and Hsu (1995) an expressin for an upper confidence bound for C_pmk in the case when µ = M and so C_pmk = C_pm, is given as a multiple of C_pm. However, if the value µ = M known then it would be more reasonable to use the estimator obtained by replacing X_n by M.     

Optimal designing of a new mixed variable lot-size chain sampling plan based on the process capability index

In this paper, a new mixed sampling plan based on the process capability index (PCI) C_pk is proposed and the resultant plan is called mixed variable lot-size chain sampling plan (ChSP). The proposed mixed plan comprises of both attribute and variables inspections. The variable lot-size sampling plan can be used for inspection of attribute quality characteristics and for the inspection of measurable quality characteristics, the variables ChSP based on PCI will be used. We have considered both symmetric and asymmetric fraction non conforming cases for the variables ChSP. Tables are developed for determining the optimal parameters of the proposed mixed plan based on two points on the operating characteristic (OC) approach. In order to construct the tables, the problem is formulated as a non linear programming where the average sample number function is considered as an objective function to be minimized and the lot acceptance probabilities at acceptable quality level and limiting quality level under the OC curve are considered as constraints. The practical implementation of the proposed mixed sampling plan is explained with an illustrative real time example. Advantages of the proposed sampling plan are also discussed in terms of comparison with other existing sampling plans.