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Journal of Probability and Statistics
Volume 2012 (2012), Article ID 819730, 15 pages
http://dx.doi.org/10.1155/2012/819730
Research Article

Bayesian Estimation of Generalized Process Capability Indices

Department of Statistics, Visva-Bharati University, Santiniketan 731 235, India

Received 14 May 2011; Revised 6 December 2011; Accepted 13 December 2011

Academic Editor: Shein-chung Chow

Copyright © 2012 Sudhansu S. Maiti and Mahendra Saha. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Process capability indices (PCIs) aim to quantify the capability of a process of quality characteristic (X) to meet some specifications that are related to a measurable characteristic of its produced items. One such quality characteristic is life time of items. The specifications are determined through the lower specification limit (L), the upper specification limit (U), and the target value (T). Maiti et al. (2010) have proposed a generalized process capability index that is the ratio of proportion of specification conformance to proportion of desired conformance. Bayesian estimation of the index has been considered under squared error loss function. Normal, exponential (nonnormal), and Poisson (discrete) processes have been taken into account. Bayes estimates of the index have been compared with the frequentist counterparts. Data sets have been analyzed.

1. Introduction

The purpose of process capability index (PCI) is to provide a numerical measure on whether a production process is capable of producing items within the specification limits or not. It becomes very popular in assessing the capability of manufacturing process in practice during the past decade. More and more efforts have been devoted to studies and applications of each PCIs. For example, the 𝐶𝑝 and 𝐶𝑝𝑘 indices have been used in Japan and in the US automotive industry. For more information on PCIs, see Hsiang and Taguchi [1], Choi and Owen [2], Pearn et al. [3], Pearn and Kotz [4], Pearn and Chen [5], Mukherjee [6], Yeh and Bhattacharya [7], Borges and Ho [8], Perakis and Xekalaki [9, 10], and Maiti et al. [11].

The usual practice is to estimate these PCIs from data and then judge the capability of the process by these estimates. Most studies on PCIs are based on the traditional frequentist point of view. The main objective of this note is to provide both point and interval estimators of the PCIs given by Maiti et al. [11] from the Bayesian point of view. We believe this effort is well justified since the Bayesian estimation has become one of popular approaches in estimation. In addition, the Bayesian approach has one great advantage over the traditional frequentist approach: the posterior distribution is sometimes very easy to derive, and credible intervals, which are the Bayesian analogue of the classical confidence interval, can be easily obtained either by theoretical derivation or Monte Carlo methods. Lower credible limits (lcls) are constructed. Upper credible limits can also be obtained in a similar manner. However, only the case of lcls is considered as these are of greater interest (due to the fact that large values of PCIs are desirable).

The paper is organized as follows. We give a brief review on the PCIs, 𝐶𝑝, 𝐶𝑝𝑘, C𝑝𝑚, and 𝐶𝑝𝑦 in Section 2. In Sections 3, 4, and 5, we derive the Bayes estimators for 𝐶𝑝𝑦 (with process median being the process center) with respect to some chosen priors under the assumption of normal, exponential (nonnormal), and Poisson (discrete) distribution, respectively. Simulation results have been reported and discussed in Section 6. In Section 7, data sets have been analyzed to demonstrate the application of the proposed Bayesian procedure. Section 8 concludes.

2. Review of Some Process Capability Indices

The most popular PCIs are 𝐶𝑝, 𝐶𝑝𝑘, and 𝐶𝑝𝑚. The 𝐶𝑝 index is defined as𝐶𝑝=𝑈𝐿6𝜎,(2.1) where 𝐿 and 𝑈 are the lower and upper specification limits, respectively, and 𝜎 is the process standard deviation. Note that 𝐶𝑝 does not depend on the process mean. The 𝐶𝑝𝑘 is then introduced to reflect the impact of 𝜇 on the process capability indices. The 𝐶𝑝𝑘 index is defined as 𝐶𝑝𝑘=min𝜇𝐿,3𝜎𝑈𝜇3𝜎.(2.2) The 𝐶𝑝𝑚 index was introduced by Chan et al. [12]. This index takes into account the influence of the departure of the process mean 𝜇 from the process target 𝑇. The 𝐶𝑝𝑚 index is defined as 𝐶𝑝𝑚=𝑈𝐿6𝜎2+(𝜇𝑇)2.(2.3) Maiti et al. [11] suggested a more generalized measure which is directly or indirectly associated with all the previously defined capability indices. The measure is as follows: 𝐶𝑝𝑦=𝑝𝑝0,(2.4) where 𝑝 is the process yield that is, 𝑝=𝐹(𝑈)𝐹(𝐿), 𝐹(𝑡)=𝑃(𝑋𝑡) is the cumulative distribution function of X, and 𝑝0 is the desirable yield that is, 𝑝0=𝐹(𝑈𝐷𝐿)𝐹(𝐿𝐷𝐿), 𝐿𝐷𝐿 and 𝑈𝐷𝐿 be the lower and upper desirable limit, respectively. When the process is off centered, then 𝐹(𝐿)+𝐹(𝑈)1 but the proportion of desired conformance is achieved. In that case, the index is as follows: 𝐶𝑝𝑦𝑘𝐶=min𝑝𝑦𝑢,𝐶𝑝𝑦𝑙,(2.5) where 𝐶𝑝𝑦𝑢=𝐹𝜇(𝑈)𝐹𝑒(1/2)𝛼2,𝐶𝑝𝑦𝑙=𝐹𝜇𝑒𝐹(𝐿)(1/2)𝛼1,(2.6) with 𝜇𝑒 being the median of the distribution and the process center is to be located such that 𝐹(𝜇𝑒)=(𝐹(𝐿)+𝐹(𝑈))/2 that is, 𝐹(𝐿)+𝐹(𝑈)=1, 𝛼1=𝑃(𝑋<𝐿𝐷𝐿), and 𝛼2=𝑃(𝑋>𝑈𝐷𝐿). It generally happens that process target 𝑇 is such that 𝐹(𝑇)=(𝐹(𝐿)+𝐹(𝑈))/2; if 𝐹(𝑇)(𝐹(𝐿)+𝐹(𝑈))/2, the situation may be described as “generalized asymmetric tolerances” have been described by the term “asymmetric tolerances” when 𝑇(𝐿+𝑈)/2. Under this circumstance, the index is defined as follows:𝐶𝑝𝑇𝑘=min𝐹(𝑈)𝐹(𝑇)(1/2)𝛼2,𝐹(𝑇)𝐹(𝐿)(1/2)𝛼1.(2.7)

3. Bayes Estimate of 𝐶𝑝𝑦 for Normal Process

Let 𝑥1, 𝑥2, 𝑥3, …, 𝑥𝑛 be 𝑛 observations from normal distribution with parameter 𝜇 and 𝜎2. Then, the joint distribution of 𝑥1, 𝑥2, 𝑥3, …, 𝑥𝑛 is1𝐿(𝑥𝜇,𝜎)=(2𝜋)𝑛/2𝜎2𝑛/2𝑒(1/2𝜎2)(𝑥𝑖𝜇)2.(3.1) Regarding selection of the the prior distributions, it is advisable to choose conjugate prior, since in this situation, even if prior parameters are unknown in practice, these may be estimated approximately from the likelihood functions as discussed in subsequent sections. When there is no information about the parameter(s) of the distribution, noninformative prior choice is good one.

3.1. Conjugate Prior Distributions

Here we assume that the prior distribution of (𝜇,𝜎2) is of the following form 𝑔𝜇,𝜎2=𝑔1𝜇𝜎2𝑔2𝜎2,(3.2) where the 𝜇 given 𝜎2 follows normal distribution with mean 𝜇0 and variance 𝜎2 and 𝜎2 follows an inverted gamma distribution of the form: 𝑓𝜎2=𝛽𝛼Γ(𝛼)𝜎2(𝛼+1)𝑒(𝛽/𝜎2);𝜎2>0,𝛼,𝛽>0.(3.3) Hence the posterior distribution of (𝜇,𝜎2) is given by𝑔𝜇,𝜎2=𝐿𝑥𝑥/𝜇,𝜎2𝑔𝜇,𝜎20𝐿𝑥/𝜇,𝜎2𝑔𝜇,𝜎2𝑑𝜇𝑑𝜎2=2(𝑛+1)𝜎(2𝜋)𝑒((𝑛+1)/2𝜎2)(𝜇(𝑥0/(𝑛+1)))2𝑊1((𝑛+1)/2)+𝛼Γ(((𝑛+1)/2)+𝛼)𝜎2((𝑛/2)+𝛼+1)𝑒(𝛽/𝜎2),(3.4) where 𝑥0=𝑛𝑥+𝜇0 and 𝑊1=𝑥2𝑖+2𝛽+𝜇20(𝑥20/(𝑛+1)).

If the process quality characteristic follows normal distribution with mean 𝜇 and variance 𝜎2, then the generalized process capability index is given by 𝐶𝑝𝑦=𝑝𝑝0=Φ((𝑈𝜇)/𝜎)Φ((𝐿𝜇)/𝜎)𝑝0.(3.5) Then, the Bayes estimate of 𝐶𝑝𝑦 under squared error loss is given by𝐶𝑝𝑦𝐶=𝐸𝑝𝑦=1𝑥𝑝00Φ𝑈𝜇𝜎Φ𝐿𝜇𝜎𝑔𝜇,𝜎2𝑥𝑑𝜇𝑑𝜎2.(3.6) Now, 𝐸𝐶2𝑝𝑦=1𝑥𝑝200Φ𝑈𝜇𝜎Φ𝐿𝜇𝜎2𝑔𝜇,𝜎2𝑥𝑑𝜇𝑑𝜎2(3.7) and hence𝐶Var𝑝𝑦𝐶𝑥=𝐸2𝑝𝑦𝑥𝐸2𝐶𝑝𝑦.𝑥(3.8)

It is to be noted that the Bayes estimate of 𝐶𝑝𝑦 and the variance depend on the parameters of the prior distribution of 𝜇 and 𝜎2. These parameters could be estimated by means of an empirical Bayes procedure, see Lindley [13] and Awad and Gharraf [14]. Given the random samples (𝑋1,𝑋2,,𝑋𝑛), the likelihood function of 𝜇 given 𝜎2 is normal density with mean (𝑋) and the likelihood function of 𝜎2 is inverted gamma with 𝛽=𝑛𝑖=1(𝑋𝑖𝑋)2/2 and 𝛼=(𝑛3)/2. Hence it is proposed to estimate the prior parameters 𝜇0, and 𝛽 and 𝛼 from the samples by 𝑋, and 𝑛𝑖=1(𝑋𝑖𝑋)2/2 and (𝑛3)/2, respectively. The variances of these estimators are 𝜎2/𝑛, (𝑛1)𝜎4/2 and 0, respectively. The expressions of 𝑥0 and 𝑊1 will be (𝑛+1)𝑋 and 2𝑛𝑖=1(𝑋𝑖𝑋)2, respectively.

3.2. Noninformative Prior Distributions

Here we assume that the prior distribution of (𝜇,𝜎2) is noninformative of the form𝑔𝜇,𝜎21𝜎2.(3.9) Hence the posterior distribution of (𝜇,𝜎2) is of the form𝑔𝜇,𝜎2=1𝑥𝜎𝑒(2𝜋)(1/2𝜎2)(𝜇𝑥)22𝑥𝑖𝑥2/2𝑛/2𝜎Γ(𝑛/2)2((𝑛+1)/2)𝑒((𝑥𝑖𝑥)2)/2𝜎2.(3.10) Estimates are to be found out in the same way as in Section 3.1.

4. Bayes Estimate of 𝐶𝑃𝑦 for Exponential Process

Let 𝑥1, 𝑥2, 𝑥3, …,𝑥𝑛 be 𝑛 observations from exponential distribution with parameter 𝜆. Then, the joint distribution of 𝑥1, 𝑥2, 𝑥3, …,𝑥𝑛 is𝐿(𝑥𝜆)=𝜆𝑛𝑒𝑥𝜆𝑖.(4.1)

4.1. Conjugate Prior Distributions

Here we assume that the prior distribution of 𝜆 is gamma with parameter (𝑚,𝑎) that is, the distribution of 𝜆 is given as 𝑎𝑔(𝜆)=𝑚Γ𝑒(𝑚)𝑎𝜆𝜆𝑚1,𝜆>0.(4.2) Hence the posterior distribution of 𝜆 is given as𝑔(𝜆𝑥)=𝐿(𝑥/𝜆)𝑔(𝜆)0=𝑥𝐿(𝑥/𝜆)𝑔(𝜆)𝑑𝜆𝑖+𝑎𝑚+𝑛𝜆Γ(𝑚+𝑛)𝑚+𝑛1𝑒𝑥𝜆(𝑎+𝑖),𝜆>0.(4.3) As a process whose distribution can be regarded to be the exponential distribution, the generalized process capability index is given by 𝐶𝑝𝑦=𝑝𝑝0=𝑒𝜆𝐿𝑒𝜆𝑈𝑝0.(4.4) Then, the Bayes estimate of 𝐶𝑝𝑦 under squared error loss is given by 𝐶𝑝𝑦𝐶=𝐸𝑝𝑦=1𝑥𝑝00𝑒𝜆𝐿𝑒𝜆𝑈=1𝑔(𝜆𝑥)𝑑𝜆𝑝0𝑥𝑎+𝑖𝑥𝐿+𝑎+𝑖𝑚+𝑛𝑥𝑎+𝑖𝑥𝑈+𝑎+𝑖𝑚+𝑛.(4.5) Now, 𝐶Var𝑝𝑦𝐶𝑥=𝐸2𝑝𝑦𝑥𝐸2𝐶𝑝𝑦𝑥.(4.6) Again, 𝐸𝐶2𝑝𝑦=1𝑥𝑝020𝑒2𝜆𝐿+𝑒2𝜆𝑈2𝑒𝜆(𝐿+𝑈)=1𝑔(𝜆𝑥)𝑑𝜆𝑝02𝑥𝑎+𝑖𝑥2𝐿+𝑎+𝑖𝑚+𝑛+𝑥𝑎+𝑖𝑥2𝑈+𝑎+𝑖𝑚+𝑛𝑥2𝑎+𝑖𝑥𝐿+𝑈+𝑎+𝑖𝑚+𝑛.(4.7) Thus,𝐶Var𝑝𝑦=1𝑥𝑝02𝑥𝑎+𝑖𝑥2𝐿+𝑎+𝑖𝑚+𝑛+𝑥𝑎+𝑖𝑥2𝑈+𝑎+𝑖𝑚+𝑛𝑥2𝑎+𝑖𝑥𝐿+𝑈+𝑎+𝑖𝑚+𝑛𝑥𝑎+𝑖𝑥𝐿+𝑎+𝑖𝑚+𝑛𝑥𝑎+𝑖𝑥𝑈+𝑎+𝑖𝑚+𝑛2.(4.8) If we put 𝐿=0 that is, if only upper specification limit is given, then 𝐶𝑝𝑦=1𝑒𝜆𝑈𝑝0(4.9) with posterior distribution𝑔𝐶𝑝𝑦=𝑥𝑥𝑖+𝑎𝑚+𝑛Γ(𝑚+𝑛)𝑈1𝑝0𝐶𝑝𝑦ln1𝑝0𝐶𝑝𝑦𝑈𝑚+𝑛1𝑒(ln(1𝑝0𝐶𝑝𝑦𝑥)(𝑖+𝑎)/𝑈)𝑝0=𝑥𝑖+𝑎𝑚+𝑛Γ(𝑚+𝑛)𝐿𝐶𝑝𝑦1𝑝0𝐶𝑝𝑦𝑥𝑖+𝑎1/𝑈ln1𝑝0𝐶𝑝𝑦𝑈𝑚+𝑛1,0<𝐶𝑝𝑦<1𝑝0,(4.10) and the Bayes estimate is given by 𝐶𝑝𝑦=1𝑝0𝑥1𝑎+𝑖𝑥𝑈+𝑎+𝑖𝑚+𝑛.(4.11) Similarly, if 𝑈= that is, if only lower specification is given, then 𝐶𝑝𝑦=𝑒𝜆𝐿𝑝0(4.12) with posterior distribution 𝑔𝐶𝑝𝑦=𝑥𝑥𝑖+𝑎𝑚+𝑛Γ(𝑚+𝑛)𝐿𝑝0𝐶𝑝𝑦𝑝ln0𝐶𝑝𝑦𝐿𝑚+𝑛1𝑒(ln(𝑝0𝐶𝑝𝑦𝑥)(𝑖+𝑎)/𝐿)𝑝0=𝑥𝑖+𝑎𝑚+𝑛Γ(𝑚+𝑛)𝐿𝐶𝑝𝑦𝑝0𝐶𝑝𝑦𝑥𝑖+𝑎1/𝐿𝑝ln0C𝑝𝑦𝐿𝑚+𝑛1,0<𝐶𝑝𝑦<1𝑝0,(4.13) and the Bayes estimate is given by𝐶𝑝𝑦=1𝑝0𝑥𝑎+𝑖𝑥𝑈+𝑎+𝑖𝑚+𝑛.(4.14)

The Bayes estimate of 𝐶𝑝𝑦 and the variance depend on the parameters of the prior distribution of 𝜆. Given the random samples (𝑋1,𝑋2,,𝑋𝑛), the likelihood function of 𝜆 is gamma density with parameters 𝑋(𝑛+1,𝑖). Hence it is proposed to estimate the prior parameters 𝑚 and 𝑎 from the samples by 𝑛+1 and 𝑋𝑖 with variances 0 and 𝑛/𝜆2, respectively. Hence𝐶𝑝𝑦=1𝑝02𝑥𝑖𝑥𝐿+2𝑖2𝑛+12𝑥𝑖𝑥𝑈+2𝑖2𝑛+1,𝐶Var𝑝𝑦=1𝑥𝑝02𝑥𝑖𝑥𝐿+𝑖2𝑛+1+𝑥𝑖𝑥𝑈+𝑖2𝑛+12𝑥2𝑖𝑥𝐿+𝑈+2𝑖2𝑛+12𝑥𝑖𝑥𝐿+2𝑖2𝑛+12𝑥𝑖𝑥𝑈+2𝑖2𝑛+12.(4.15)

4.2. Noninformative Prior Distributions

In this subsection, we obtain the Bayes estimator of 𝐶𝑝𝑦 under the assumption that the parameter 𝜆 is random variable having noninformative prior 𝑔(𝜆)1/𝜆.

Hence, the Bayes estimator with respect to squared error loss function will be𝐶𝑝𝑦=1𝑝0𝑥𝑖𝑥𝐿+𝑖𝑛𝑥𝑖𝑥𝑈+𝑖𝑛,𝐶Var𝑝𝑦=1𝑥𝑝02𝑥𝑖𝑥2𝐿+𝑖𝑛+𝑥𝑖𝑥2𝑈+𝑖𝑛𝑥2𝑖𝑥𝐿+𝑈+𝑖𝑛𝑥𝑖𝑥𝐿+𝑖𝑛𝑥𝑖𝑥𝑈+𝑖𝑛2.(4.16) When only upper (lower) specification is to be given, then we will get the expressions substituting 𝐿=0(𝑈=).

5. Bayes Estimate of 𝐶𝑝𝑦 for Poisson Process

Let 𝑥1, 𝑥2, 𝑥3, …, 𝑥𝑛 be 𝑛 observations from Poisson distribution with parameter 𝜆. Then, the joint distribution of 𝑥1, 𝑥2, 𝑥3, …, 𝑥𝑛 is𝑒𝐿(𝑥𝜆)=𝜆𝜆𝑥𝑖𝑥𝑖!=𝑒𝑛𝜆𝜆𝑥𝑖𝑥𝑖!.(5.1)

5.1. Conjugate Prior Distributions

Let the prior distribution of 𝜆 is assumed to be gamma with parameter (𝑚,𝑎). Then the distribution of 𝜆 is given as𝑎𝑔(𝜆)=𝑚Γ𝑒(𝑚)𝑎𝜆𝜆𝑚1,𝜆>0.(5.2) Now, the posterior distribution of 𝜆 is given as𝑔(𝜆𝑥)=(𝑛+𝑎)(𝑥𝑖+𝑚)Γ𝑥𝑖𝑒+𝑚(𝑛+𝑎)𝜆𝜆𝑥(𝑚+𝑖1),𝜆>0.(5.3) Now, the process yield is𝑝=𝑒𝑈𝜆𝑡=𝐿𝜆𝑡𝑡!.(5.4) Then,𝐶𝑝𝑦𝐶=𝐸𝑝𝑦=1𝑥𝑝0𝑈𝑡=𝐿1𝑡!0𝑒𝜆𝜆𝑡=1𝑔(𝜆𝑥)𝑑𝜆𝑝0𝑈𝑡=𝐿1𝑡!0(𝑛+𝑎)𝑥𝑖+𝑚Γ𝑥𝑖𝑒+𝑚(𝑛+𝑎+1)𝜆𝜆𝑥𝑖+𝑚+𝑡1=1𝑑𝜆𝑝0𝑈𝑡=𝐿1𝑡!𝑛+𝑎𝑛+𝑎+1𝑥𝑖+𝑚1𝑛+𝑎+1𝑡Γ𝑥𝑖+𝑚+𝑡Γ𝑥𝑖.+𝑚(5.5) Again,𝐸𝐶2𝑝𝑦=1𝑥𝑝20𝑈𝑡=𝐿1𝑡!20𝑒2𝜆𝜆2𝑡1𝑔(𝜆𝑥)𝑑𝜆+21𝑡!𝑡!0𝑒2𝜆𝜆𝑡𝜆𝑡!=1𝑔(𝜆𝑥)𝑑𝜆𝑝20𝑈𝑡=𝐿1𝑡!20(𝑛+1)𝑥𝑖+𝑚Γ𝑥𝑖𝑒+𝑚(𝑛+𝑎+2)𝜆𝜆𝑥𝑖+𝑚+2𝑡11𝑑𝜆+21𝑡!𝑡!0(𝑛+𝑎)𝑥𝑖+𝑚Γ𝑥𝑖𝑒+𝑚(𝑛+𝑎+2)𝜆𝜆𝑥𝑖+𝑚+𝑡+𝑡1=1𝑑𝜆𝑝20𝑈𝑡=𝐿1𝑡!2𝑛+𝑎𝑛+𝑎+2𝑥𝑖+𝑚1𝑛+𝑎+22𝑡Γ𝑥𝑖+𝑚+2𝑡Γ𝑥𝑖1+𝑚+21𝑡!𝑡!𝑛+𝑎𝑛+𝑎+2𝑥𝑖+𝑚1𝑛+𝑎+2𝑡+𝑡Γ𝑥𝑖+𝑚+𝑡+𝑡Γ𝑥𝑖.+𝑚(5.6)

Here, the Bayes estimate of C𝑝𝑦 and the variance depend on the parameters of the prior distribution of 𝜆. Given the random samples (𝑋1,𝑋2,,𝑋𝑛), the likelihood function of 𝜆 is gamma density with parameters (𝑋𝑖+1,𝑛). Hence it is proposed to estimate the prior parameters 𝑚 and 𝑎 from the samples by 𝑋𝑖+1 and 𝑛 with variances 𝑛𝜆 and 0, respectively. Substituting these in the above expressions, we will have the empirical Bayes estimates.

5.2. Noninformative Prior Distributions

In this subsection, we obtain the Bayes estimator of 𝐶𝑝𝑦 under the assumption that the parameter 𝜆 is random variable having noninformative prior 𝑔(𝜆)1/𝜆.

Hence, the Bayes estimator with respect to squared error loss function will be 𝐶𝑝𝑦=1𝑝0𝑈𝑡=𝐿1𝑛𝑡!𝑛+1𝑥𝑖1𝑛+1𝑡Γ𝑥𝑖+𝑡Γ𝑥𝑖,𝐸𝐶2𝑝𝑦=1𝑥𝑝20𝑈𝑡=𝐿1𝑡!2𝑛𝑛+2𝑥𝑖1𝑛+22𝑡Γ𝑥𝑖+2𝑡Γ𝑥𝑖+2𝑡<𝑡11𝑡!𝑡!𝑛𝑛+2𝑥𝑖1𝑛+2𝑡+𝑡Γ𝑥𝑖+𝑡+𝑡Γ𝑥𝑖.(5.7) When only upper (lower) specification is to be given, then we will get the expressions substituting 𝐿=0(𝑈=).

6. Simulation and Discussion

In this section, we present some results based on the Monte Carlo Simulations to compare the performance of frequentist (maximum likelihood and minimum variance unbiased estimators) as well as the Bayesian method of estimation. All the computations were performed using R-software and Mathematica, and these are available on request from the corresponding author. The maximum likelihood estimator (MLE) and uniformly minimum variance unbiased estimator (UMVUE) and their mean square errors (MSEs) were shown in Maiti et al. [11]. We have performed the Bayes estimators and their MSEs in Tables 16. All the results are based on 25,000 replications.

tab1
Table 1: Bayes Estimates of 𝐶𝑝𝑦 and their MSEs with 𝐿=0, 𝑈=10, samples generated from normal distribution.
tab2
Table 2: Bayes Estimates of 𝐶𝑝𝑦 and their MSEs with 𝐿=0, 𝑈=8, samples generated from normal distribution.
tab3
Table 3: Bayes Estimates of𝐶𝑝𝑦 and their MSEs with 𝐿=0, 𝑈=10, samples generated from exponential distribution.
tab4
Table 4: Bayes Estimates of 𝐶𝑝𝑦 and their MSEs with 𝐿=0, 𝑈=8, samples generated from exponential distribution.
tab5
Table 5: Bayes estimates of 𝐶𝑝𝑦 and their MSEs with 𝐿=0, 𝑈=8, samples generated from exponential distribution.
tab6
Table 6: Bayes estimates of 𝐶𝑝𝑦 and their MSEs with 𝐿=0, 𝑈=8, samples generated from exponential distribution.

We represented the average 𝐶𝑝𝑦 value and the MSE for normal process in Tables 1 and 2. We take the same set up of Maiti et al. [11] to make comparable with the Bayesian approach. We take 𝑝0=0.90, for two choices of (L, U) as (0,10) and (0,8), and for sample of sizes 𝑛=25,50,100,150. We generate observations from normal distributions with choices of (𝜇,𝜎)=(5,3),(5,4),(6,3), and (6,4). First column of Tables 1 and 2 shows the values of (𝜇,𝜎) and the corresponding 𝐶𝑝𝑦. Remaining columns show average 𝐶𝑝𝑦, and its MSE, for the above-mentioned sample sizes. It is observed that in almost all the cases, MSEs of 𝐶𝑝𝑦 in the Bayesian set up using the empirical Bayes procedure of the prior parameters are larger than those obtained in the frequentist approach. 𝐶𝑝𝑦 overestimates the true 𝐶𝑝𝑦 in general. Therefore, this empirical Bayes estimate is not so encouraging compared to maximum likelihood estimator or minimum variance unbiased estimator.

We represented the average 𝐶𝑝𝑦 value and the MSE for exponential process in Tables 3 and 4. We simulate observations from the exponential distribution with rate 𝜆. We take 𝜆=0.2,0.5,0.7, and 1.0. From Tables 3 and 4, we find that for 𝜆<0.5, the empirical Bayes estimate of 𝐶𝑝𝑦 gives better result than the ML estimate of 𝐶𝑝𝑦 in MSE sence, but for 𝜆>0.5, it reverses. As soon as the mean quality characteristic gets larger (when 𝜆<0.5), the empirical Bayes estimate becomes better in MSE sense and hence, it is recommended to use it. For smaller mean quality characteristic, the use of UMVUE of 𝐶𝑝𝑦 is fair even though it is, to some extent, computation intensive.

We simulate observations from Poisson distribution with mean 𝑚. We take 𝑚=1,4,8, and 10. From Tables 5 and 6, we find that for 𝑚=1 and 4, the UMVUE of 𝐶𝑝𝑦 gives better result than the empirical Bayes estimate of 𝐶𝑝𝑦 in MSE sense, but for 𝑚=8 and 10, it is opposite. Here also if mean quality is getting larger and larger, like exponential process, the empirical Bayes estimate is estimated efficiently. So, it is advisable to use the empirical Bayes estimate of 𝐶𝑝𝑦 when mean quality characteristic is large, but for smaller mean, use of UMVUE of 𝐶𝑝𝑦 is a fair one.

It is expected that when there is prior information regarding parameters, the performance of the Bayes estimates would be better than their traditional frequentist counterpart. But here we choose empirical estimate of parameters following the approach of Lindley [13] and Awad and Gharraf [14]. Since it is an empirical approach, it may not perform uniformly better than the frequentist approach that has been reflected in simulation study. The performance is less encouraging in case of normally distributed quality characteristic whereas it performs better when the underlying distribution is exponential with larger mean and also performs better when the quality characteristic distribution is Poisson with a larger mean.

7. Data Analysis

This section is devoted for demonstrating inferential aspect of 𝐶𝑝𝑦, by analyzing some data sets. We choose two data sets fit approximately exponential and Poisson distribution, respectively.

(a) For demonstration purpose, we consider here the data that represent the number of miles to first and succeeding major motor failures of 191 buses (cf. Davis [15]) operated by a large city bus company. Failures were either abrupt, in which some part broke and the motor would not run or, by definition, when the maximum power produced, as measured by a dynamo meter, fell below a fixed percentage of the normal rated value. Failures of motor accessories which could be easily replaced were not included in these data. The bus motor failures are compared with exponential distribution, and observed chi-square index has been calculated as 3.40 with 𝑃 value 0.32.

Here, we assume that the upper specification limit (U) and lower specification limit (L) are 75 and 15, respectively. Sample size, 𝑛=85, sample mean 𝑥=35.17647058.

Then, we find out the MLE, MVUE and the Bayes estimate of the index as 𝐶𝑝𝑦=0.562372642, 𝐶𝑝𝑦=0.5623726, and 𝐶𝑏𝑝𝑦=0.5661285, respectively. And 95% lower confidence limit (lcl) of the Bayes estimate is 0.563874661.

Now, if we consider the case that only upper specification limit (UCL) has been given, then the MLE and MVUE of the index are 𝐶𝑝𝑦𝑢=0.927802988 and 𝐶𝑝𝑦𝑢=0.9280266, respectively. And 95% lower confidence limit (lcl) for the index 𝐶𝑝𝑦𝑢 is given as 0.872705746 (cf. Maiti and Saha [16]). Here, we also find out the Bayes estimate of the index as 𝐶𝑏𝑝𝑦𝑢=1.042010 and the corresponding 95% lcl, given as 1.039383292

On the other hand, if we consider the case that only lower specification limit has been given, then the MLE and MVUE of 𝐶𝑝𝑦𝑙 are 𝐶𝑝𝑦𝑙=0.687201232 and 𝐶𝑝𝑦𝑙=0.6899253, respectively. And 95% lower confidence limit (lcl) for the index is given by 0.635114903. In this case, the Bayes estimate and corresponding 95% lcl of the index are 𝐶𝑏𝑝𝑦𝑙=0.63523 and 0.629801423, respectively.

(b) Data on dates of repair calls on 15 hand electric drill motors are taken from Davis [15]. Mean number of days between failures for each drill was used as a milepost and frequency distribution compared with the theoretical Poisson distribution, and observed chi-square index has been calculated as 38 with 𝑃value 0.16. Here, we assume that the upper specification limit (U) and lower specification limit (L) are 3 and 1, respectively. Here sample size, 𝑛=164 and sample mean 𝑥=0.975609756. Then, the MLE, MVUE, and Bayes estimate of 𝐶𝑝𝑦 are 𝐶𝑝𝑦=0.637377368, 𝐶𝑝𝑦=0.6389400, and 𝐶𝑏𝑝𝑦=0.673041, respectively. And 95% lower confidence limit (lcl) of the Bayes estimate is 0.659472.

Now, if we consider the case in which only upper specification limit (UCL) has been given, then the MLE of the index is 𝐶𝑝𝑦𝑢=1.034179963 and the MVUE of the index is 𝐶𝑝𝑦𝑢=1.034559263. 95% lower confidence limit (lcl) for the index Cpyu is given as 1.024565558. Here the Bayes estimate and corresponding 95% lcl are 𝐶𝑏𝑝𝑦=1.091235 and 1.077885192, respectively.

On the other hand, if we consider the case that only lower specification limit has been given, then the MLE of the index 𝐶𝑝𝑦𝑙=0.655829023 and the MVUE is 𝐶𝑝𝑦𝑙=0.657012316. 95% lower confidence limit (lcl) for the index is given as 0.448874947. In this case, we also find out the Bayes estimate and 95% lcl of the corresponding index, which are given as 𝐶𝑏𝑝𝑦𝑢=0.6929168 and 0.626240727, respectively.

8. Concluding Remark

In this paper, the Bayesian inference aspects of generalized PCI (cf. Maiti et al. [11]) have been presented. The Bayes estimates of the generalized process capability index have been studied for normal, exponential (nonnormal), and Poisson (discrete) processes. The empirical Bayes estimation procedure has been discussed when parameters of the prior distribution are unknown. The Bayes estimates have been compared with their most frequent counterpart, and situations have been mentioned when the Bayes estimates are better through simulation study. Because of its appealing features, examining its potential use in other types of processes often arising in connection with applications would be of practical importance. Other loss functions can be used to find out the estimates in similar fashion.

Acknowledgment

The authors would like to thank the referee for a very careful reading of the paper and making a number of nice suggestions, which improved the earlier version.

References

  1. T. C. Hsiang and G. Taguchi, “Tutorial on quality control and assurance—the Taguchi methods,” in Proceedings of the Joint Meeting of the American Statistical Association, p. 188, Las Vegas, Nev, USA, 1985.
  2. B.-C. Choi and D. B. Owen, “A study of a new process capability index,” Communications in Statistics—Theory and Methods, vol. 19, no. 4, pp. 1231–1245, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  3. W. L. Pearn, S. Kotz, and N. L. Johnson, “Distributional and inferential properties of process capability indices,” Journal of Quality Technology, vol. 24, no. 4, pp. 216–231, 1992.
  4. W. L. Pearn and S. Kotz, “Application of clements' method for calculating secondand third-generation process capability indices for non-normal pearsonian populations,” Quality Engineering, vol. 7, no. 1, pp. 139–145, 1994-1995.
  5. W. L. Pearn and K. S. Chen, “Estimating process capability indices for non-normal Pearsonian populations,” Quality and Reliability Engineering International, vol. 11, no. 5, pp. 386–388, 1995.
  6. S. P. Mukherjee, “Process capability indices and associated inference problems,” in Proceedings of the International Conference on Statistical Methods and Statistical Computation, pp. 243–249, Seoul, Republic of Korea, 1995.
  7. A. B. Yeh and S. Bhattacharya, “A robust process capability index,” Communications in Statistics—Simulation and Computation, vol. 27, no. 2, pp. 565–589, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. W. Borges and L. L. Ho, “A fraction defective based capability index,” Quality and Reliability Engineering International, vol. 17, no. 6, pp. 447–458, 2001. View at Publisher · View at Google Scholar
  9. M. Perakis and E. Xekalaki, “A process capability index that is based on the proportion of conformance,” Journal of Statistical Computation and Simulation, vol. 72, no. 9, pp. 707–718, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. Perakis and E. Xekalaki, “A process capability index for discrete processes,” Journal of Statistical Computation and Simulation, vol. 75, no. 3, pp. 175–187, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. S. S. Maiti, M. Saha, and A. K. Nanda, “On generalizing process capability indices,” Journal of Quality Technology and Quantitative Management, vol. 7, no. 3, pp. 279–300, 2010.
  12. L. K. Chan, S. W. Cheng, and F. A. Spring, “A new measure of process capability: Cpm,” Journal of Quality Technology, vol. 20, no. 3, pp. 162–175, 1988.
  13. D. V. Lindely, Introduction to Probability and Statistics from a Bayesian Viewpoint, vol. 1, Cambridge University Press, Cambridge, UK, 1969.
  14. A. M. Awad and M. K. Gharraf, “Estimation of P(Y<X) in the Burr case: a comparative study,” Communications in Statistics—Simulation and Computation, vol. 15, no. 2, pp. 389–403, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. D. J. Davis, “An analysis of some failure data,” Journal of American Statistical Association, vol. 47, no. 258, pp. 113–150, 1952.
  16. S. S. Maiti and M. Saha, “Inference on generalized process capability index,” IAPQR Transactions, vol. 36, no. 1, pp. 45–67, 2011.