Two-Sided Generalized Confidence Intervals for C pk

Transcription

1 CHAPTER 4 Two-Sided Generalized Confidence Intervals for C pk 4.1 Introduction 9 4. Existing Methods Two-Sided Generalized Confidence Intervals for C pk Simulation Results An Illustration Conclusions Introduction As mentioned earlier, the main objective of this venture is to examine the performance of GCIs in the context of constructing confidence intervals for PCIs, with special reference to the holding of repeated sampling properties. This investigation has become necessary, because of the fact that the idea of GCIs is not based on the conventional repeated sampling considerations, but rather on exact probability statements described in section 1.6. As desired, it has been verified through adequate simulation experiments in chapter, that the method of GCIs works well in the case of PCIs in general and very well in the case of C pk in particular, in the context of constructing lower confidence limits. The performance with respect to coverage probability was most promising in the case of C pk than in the case of the other two indices C pk and C pmk included in the study. 9

2 We have also seen in chapter 3 that the idea of GCIs, was equally competent when the data used for computing the lower confidence limits were following a one-way random model set up; that is, when the data were suffering from batch-wise variation besides the inherent within variation. It is quite natural, in this context to examine the performance of GCIs in the case of constructing two-sided confidence intervals for PCIs. To examine the performance of GCIs in the context of two-sided confidence intervals, we have decided to do it in the case of C pk alone. This was both due to the unique role that C pk plays among the class of all PCIs defined so far, and due to the most optimistic performance of GCIs that we witnessed in the case of C pk than in the case of the other two indices in the previous chapters in connection with the construction of lower confidence limits. Though an avalanche of new-generation PCIs with added features have been proposed so far to suit the various practical needs recently, this time-tested second-generation capability index still attracts the attention of theoreticians as well as practitioners throughout the world; see Kotz and Lovelace (1998). The number of articles published so far on C pk, is a proof of its unparallel acceptability among all PCIs. After reviewing the existing methods available in the literature for the construction of two-sided confidence intervals for C pk in the next section, we derive the two-sided confidence intervals by the method of GCIs in section 4.3. The detailed results and analysis of a big simulation experiment are reported in section 4.4. Besides assessing the performance of GCIs, a comparison is also made with that of the existing methods. The methodology is applied in a practical context, and the findings are reported in section Existing Methods Exact confidence intervals for C pk are difficult to obtain mainly because of the difficulty in obtaining pivotal quantities of the conventional type with the property that the probability distribution of which are free of the presence of nuisance parameters. Following Owen (1965), Chou and Owen (1989) had shown that the probability distribution of the natural estimator of C pk defined by [ X LSL Ĉ pk = min, 3S ] USL X 3S where X and S are respectively the mean and the standard deviation of the sample used to estimate, itself is too complicated as it is the joint distribution of two dependent non-central student s t random variables. Kotz and Johnson (1993) writes, difficulties in the construction of confidence intervals for C pk arise from the rather complicated way 93

3 in which the parameters µ and σ appear in the expression of C pk. Therefore, several authors have come up with their own approximate confidence intervals based on various arguments. Incidently, it may be noticed that in the entire collection of confidence intervals proposed so far for various PCIs, a good number of them are for this time-tested capability index. Zhang et al. (1990) considered the construction of two-sided confidence intervals for C pk from a different point of view. They had studied the skewness of the distribution of the natural estimator of Ĉpk, using the coefficient of skewness given by β = E[Ĉpk E(Ĉpk)] 3 σ 3 pk where σ pk is the standard deviation of Ĉpk. It was shown by them that Cˆ pk is consistent and asymptotically unbiased. They had obtained the expressions of E( C ˆ pk ) and σpk as E( ˆ C pk ) = n 1 n n 1 C pk σpk = (n 1) 1 n 3 1 n n 1 C pk. The value of β has been calculated for various combinations of n, µ, σ, LSL and USL, and found that the absolute value of β was smaller than 0.35 for samples of sizes greater than 100. This showed that the distribution of Ĉ pk is only slightly skewed. Based on the properties of Ĉ pk such as consistency, asymptotic unbiasedness and weak skewness, they have suggested the following symmetric two-sided confidence interval for C pk ˆ C pk ± kˆσ pk (4..1) where k is a constant to be chosen by the user. After having conducted a big simulation experiment with n = 5, 50, 100, 150 and 00; and C pk = 0.75, 1.00, 1.33 and.00, it has been noticed by them that the confidence intervals obtained for k = and k = 3, namely Ĉpk ± ˆσ pk and Ĉpk ± 3ˆσ pk, respectively contained the true value of C pk, 95.8% and 99.8% of the total number of cases. They also noticed that the proportion of time the true value of C pk was above the interval was not too different from the proportion of time that it was below the interval. 94

4 Prompted by these studies, Kushler and Hurley (199) later modified the confidence intervals given by (4..1) as Ĉ pk ± Z 1 α/ σˆ pk (4..) to get the two-sided 100(1 α)% confidence interval, where Z 1 α/ is the 100(1 α/)th percentile point of standard normal distribution. Bootstrap confidence intervals for PCIs were first proposed by Franklin and Wasserman (1991), when they applied it to construct two-sided confidence intervals for the index C pk. Bootstrap confidence intervals have the additional advantage that they can be calculated under whatever the underlying process distribution be. The method of Bootstrap was introduced in 1979 by Bradly Efron (1979), and later Efron and Tibshirani (1986) further developed three types of bootstrap confidence intervals: the standard bootstrap (SB) confidence interval, the percentile bootstrap (PB) confidence interval, and the biased- corrected percentile bootstrap (BCPB) confidence interval. Franklin and Wasserman (199) had constructed all these three types of bootstrap confidence intervals for C pk, corresponding to 90% confidence level. These confidence intervals have been based on 1000 bootstrap samples corresponding to two samples of sizes n = 0 and n = 40. To compare the distributional differences on the bootstrap method, two samples each of the above sizes have been selected from a normal distribution and also from a highly skewed distribution. As an initial study, their results were only satisfactory with respect to coverage probability and average width of the interval. Since the coverage probabilities were a little less than the nominal values, sample sizes greater than 40 were recommended by them. Recently, Chen and Chen (004) also used the method of bootstrap to calculate the SB, PB and BCPB confidence intervals for comparing the capability of two processes and showed that SB method was performing well especially for sample sizes greater than 50. The confidence intervals obtained by the above two methods by Zhang et al. (1990) and Franklin and Wasserman (199) were assuming little distributional aspects of Ĉpk, and hence they were that much non-parametric. But Kotz and Johnson (1993) and later Kotz and Lovelace (1998) made detailed studies of the distributional aspects of most of the PCIs in general, and that of Ĉpk in particular. See also Chan et al. (1990), Li et al. (1990), Gruner (1991), Johnson et al. (1994), Pearn and Chen (1996), Chen and Pearn (1997), Pearn et al. (1999). Using the distributional aspects of Ĉpk, many approximate confidence intervals are available in the literature. The following are three important ones among them. Heavlin (1988) suggested the following as a two-sided 100(1 α)% confidence 95

5 interval for C pk : [ { Cˆ n 1 pk Z 1 α/ 9n(n 3) + C ˆ 1 pk (n 3) { Cˆ n 1 pk + Z 1 α/ 9n(n 3) + C ˆ 1 pk (n 3) ( n 1 ( n 1 )} 1, )} 1 ], (4..3) where Z 1 α/ is the 100(1 α/)th percentile point of a standard normal distribution. By making use of a simple asymptotic approximation to the non-central t distribution by Johnson and Kotz (1970), a two-sided confidence interval for C pk was given by Bissel (1990) as Cˆ 1 pk Z 1 α/ 9n + Ĉ pk (n 1), Cˆ pk + Z 1 α/ 1 9n + Ĉ pk (n 1). After studying the entire class of available two-sided confidence intervals for C pk till date, Nagata and Nagahata (1994) suggested the following modification to the above confidence interval of Bissel: 1 1 Z 1 α/ 5(n 1)Ĉpk 5(n 1)Ĉpk + Z 1 α/ 1 9n + Ĉ pk (n 1), 1 9n + Ĉ pk (n 1) (4..4) 4.3 Two-Sided Generalized Confidence Intervals for C pk The method of GCIs described in Section 1.6 has been used to obtain lower confidence limits for PCIs in two different contexts in the previous chapters and now it can be extended to construct two-sided intervals also. The method of construction may be explained in the context of C pk as follows. Assume that the process characteristic X follows a normal distribution with mean µ and variance σ. Let (X 1, X,..., X n ) be an observable random vector from the process and (X1, X,..., Xn) be an independent copy of it. Consider also, the observable statistic S = ( X, S) and its independent copy S = ( X, S ). The parameter of interest whose GCI is to be computed is θ = d M µ 3σ = C pk = π(µ, σ) 96

6 The generalized pivotal quantity for constructing GCIs may be obtained using the recipe given Section 1.7. To this end, define R µ (S, S, µ, σ) = X n 1 n = X n 1 n n ( X µ) σ Z U S, and R σ (S, S σ, µ, σ) = (n 1)S (n 1)S = (n 1) S, U σ S, (n 1)S where Z = X µ σ/ n N(0, 1), U = (n 1)S σ χ n 1, and Z and U are statistically independent. It may be noticed that both R µ and R σ satisfy (GPQ1) and (GPQ) of section 1.6 and hence, are GPQs of µ and σ respectively. Similarly, it may be verified that R Cpk (S, S, µ, σ) defined by R Cpk (S, S, µ, σ) = d M R µ 3R σ is a GPQ of C pk by fulfilling (GPQ1) and (GPQ). So, a two-sided 100(1 α)% GCI of C pk, with confidence coefficient 1 α is given by the interval [R Cpk,α/, R Cpk,1 α/] where P [R Cpk,α/ < R Cpk < R Cpk,1 α/] = 1 α. In other words, R Cpk,α/ and R Cpk,1 α/ are the 100(α/)th and 100(1 α/)th percentiles of R Cpk so that P [R Cpk R Cpk,α/] = P [R Cpk R Cpk,1 α/] = α/. Closed form expressions for R Cpk,α/ and R Cpk,1 α/ cannot be obtained. But, using numerical integration one can compute P [R Cpk a] for known values of a, as follows: We may also write R Cpk = min{r 1, R } where R 1 = Rµ LSL 3R σ Therefore, P [R Cpk a] = P [min(r 1, R ) a] = 0 and R = USL Rµ 3R σ. = P [R 1 a, R a] [ Rµ LSL = P a, USL R ] µ a 3R σ 3R σ = P [LSL + 3aR σ R µ USL 3aR σ ] n 1 n 1 Z = P [LSL + 3a s x s USL 3a U n U [ USL 3a ] n 1 s v LSL+ 3a n 1 s v n(y : x, n 1 s n v )dy f(v; n 1)dv, n 1 U s] 97

7 where s = ( x, s) is an observed value of S = ( X, S), n(y; b, c) denote a normal distribution with mean b and variance c, and f(v; n 1) is the density of a central chi-square variable with n 1 degrees of freedom. Then, we have [ (USL x) nv P [R Cpk a] = E v {Φ n 1 s Φ [( LSL x) nv n 1 s 3a ] n ]} + 3a n where Φ denote the standard normal distribution function and E v denote the expectation under V χ n 1. Thus, R C pk,α/ and R Cpk,1 α/ can be computed by solving P [R Cpk R Cpk,α/] = 1 α/ and P [R Cpk R Cpk,1 α/] = α/. This can be done by trial and error method and repeated use of Monte-Carlo simulation. 4.4 Simulation Results The performance of the proposed GCIs has been examined through a simulation study. We considered the same hypothetical situation that we had considered in the previous chapters, where we assumed LSL = 7, USL = 14, µ = 10. We had investigated the coverage probability and the length of two-sided confidence intervals of the method of GCIs for various values of σ that will provide C pk = 1, 1.33, 1.5,,.5 and 3. The estimates of the coverage probabilities, the two-sided GCIs and their length, have been obtained as follows. A random sample of size n from N(µ, σ ), having a fixed value of C pk has been generated first. The sample mean x and sample standard deviation s are calculated. For the fixed n, C pk, x and s; 10,000 pairs of (Z, U ) values have been generated to calculate the two-sided GCI and its length of confidence level 1 α as per the method mentioned in the previous section 4.3. This process is repeated for 10,000 different samples of the said size to calculate 10,000 two-sided 100(1 α)% GCIs. The coverage probability (CG) of the method of GCIs is the proportion of time the calculated GCI contains the true value of the parameter C pk. The confidence intervals provided by the method of GCIs will be exact or the method of GCIs will have exact freqentist coverage, if this coverage probability is exactly equal to the nominal value 1 α. The length of each of these 10,000 two-sided confidence intervals is also calculated, and lastly its average length (LG). Together with the calculation of the the two-sided GCI, for each fixed n, C pk, x and s, the 100(1 α)% two-sided confidence intervals of the methods of Heavlin, and Nagata and Nagahata have also been calculated by using equations (4..3) and (4..4). Their coverage probabilities and average length covering the entire 10,000 two-sided 98

8 Table 4.1: Coverage probabilities of the two-sided confidence intervals of C pk for the three methods corresponding to 90% confidence level 1 α C pk n Coverage probability CG CH CN * indicates a value different from 0.90 at a significance level of confidence intervals are, calculated as (CH) and (CN); and (LH) and (LN) respectively. All the Computations have been repeated for 90% and 95% confidence levels and for n = 30, 50 and 100. The results of the 90% and 95% coverage probabilities, (CG), (CH) and (CN) of the three methods are given in Table 4.1 and 4. respectively; and the corresponding average length (LG), (LH),(LN) in Table 4.3 and Table 4.4. A detailed study of the coverage probabilities and average length of the intervals is, now undertaken. Consider the 90% coverage values given in Table 4.1, first: In connection with 90% coverage probabilities, the frequency of coverage for an accurate confidence interval is binomially distributed with N = 10, 000 and p = Therefore, a 99% confidence band for the coverage percentage is 0.90±.575 = 0.90± Hence, one ,000 could be 99% confident that a true 90% confidence interval would have a proportion of coverage between and We see in Table 4.1 that no coverage value based on the generalized method is lying outside this interval. Though the method due to Nagata and Nagahata is also having this property, but it is with more dispersion than that of the generalized. At the same time, it may also be noted that not even a single coverage probability based on Heavlin s method, is observed in the interval covering the entire 10,000 simulations! Next, we are considering the impact of the size of the sample used, on the coverage probabilities. For this, the graph of the coverage proba- 99

9 Table 4.: Coverage probabilities of the two-sided confidence intervals of C pk for the three methods corresponding to 95% confidence level 1 α C pk n Coverage probability CG CH CN * indicates a value different from 0.95 at a significance level of bilities of the three methods are considered together for each of the sample sizes 30, 50 and 100, and given in figure 4.1. To make the comparisons more easier, the three graphs are drawn on the same format. It may be seen in figure 4.1 that the coverage probabilities of the method of GCIs are fully within the confidence band (0.893, ) and are more close to the nominal value than that of the other methods for each of the three sample sizes. A comparison of the three situations in the figure shows that an improvement in coverage probability is possible definitely with an increase in the sample size for all the three methods. Even at the smallest sample size of 30 used in the experiment, the coverage of GCIs is found performing well. On comparing the method of GCIs, with that of the other two existing methods, it may be seen that the performance of the method of Nagata and Nagahata is comparable to that of GCIs though good only at larger values of n. Thus the repeated sampling property of the method of GCIs is more encouraging than that of the existing approximate methods. Let us consider, now, the case of coverage probabilities at 95% confidence level given in Table 4.. Since the frequency of coverage is binomially distributed, the 99% confidence interval for the coverage percentage is ±.575 = (0.9444, ). All the conclusions that we drew in the case 10,000 of 90% confidence intervals are true in this case also. No coverage probability due to 100

10 Figure 4.1: Plot of coverage probabilities of C pk for the three methods corresponding to 90% confidence level 101

11 the method of GCIs is outside the 99% confidence band. In fact, the coverages come closer to the nominal value as n becomes large. On comparing with the performance of the existing methods, the method of Nagata and Nagathata has the second best coverages next to that of GCIs though with an aberrant coverage of corresponding to n = 30 and C pk = 1.5. As in the case of 90% coverages, the method of Heavlin is not at all comparable with the other two methods as no coverage due to this method is seen in the 99% confidence interval. To get a more clear picture of these ideas, let us concentrate on figure 4.. Here also, to make the comparison more clear, the plot of the 95% coverages of the three methods for each of the sample sizes n = 30, 50, and 100, are drawn on the same format. As seen in the other case of 90%, the coverage probabilities of GCIs are well within the confidence bands for each of the three sample sizes and it come closer to the nominal value as n increases. The graph of method of GCIs is consistently close to the nominal value for all values of C pk and n. This once again establishes that the method of GCIs possesses the repeated sampling property of confidence intervals in a more realistic way than that of the other existing methods. Next, we consider the way the three types of confidence intervals behave with respect to the other characteristic of a good confidence interval, viz length of the interval. Table 4.3 gives the expected length of the 10,000 confidence intervals corresponding to each of the methods with confidence coefficient 0.90 obtained in the simulation experiment involving 10,000 samples. It can be seen from Table 4.3 that the confidence intervals provided by the method of GCIs have the shortest average length consistently for all values of n and C pk, and hence, the best. It may also be noted that the length considerably improves with increase in the sample size. Though the confidence intervals provided by the method of Nagata and Nagahata have the next shortest average length, the same given by the method of Heavlin comes nowhere near that of GCIs. Since the coverage probabilities of the method of GCIs were also more close to the nominal value corresponding 90% confidence level, we can say that the method of GCIs can be recommended for computing two-sided confidence intervals for C pk. Similarly, it may be seen from Table 4.4 that the average length of the GCIs is the shortest one, followed by that of Nagata and Nagahata, in the case of 95% confidence level also. Here also the method of GCIs has a stunning performance, as not even a single average length of GCIs is worse than that of the other two existing methods, corresponding to even the smallest sample attempted in this study, namely 30. So a sample of size 30 may be proposed for two-sided confidence interval estimation of C pk by the method of GCIs. 10

12 Figure 4.: Plot of coverage probabilities of C pk for the three methods corresponding to 95% confidence level 103

13 Table 4.3: Average length of the two-sided confidence intervals of C pk for the three methods corresponding to 90% confidence level 1 α C pk n Average Length of the Confidence Intervals LG LH LN Table 4.4: Average length of the confidence intervals of C pk for the three methods corresponding to 95% confidence level 1 α C pk n Average Length of the Confidence Intervals LG LH LN

14 Table 4.5: Two-sided confidence intervals of C pk provided by the three methods for the piston rings data corresponding to 90% and 95% confidence levels Confidence Method of GCI Method of Nagata and Method of Heavlin Coefficient Nagahata C.I. Length C.I. Length C.I. Length 95% (1.4130,1.8306) (1.4111,1.8319) (1.4073,1.8410) % (1.4447,1.7963) (1.4449,1.7981) (1.44,1.8061) An Illustration The results of the simulation study, have been verified through an example. Montgomery (005, Example 5.3) gives a data pertaining to the diameter of piston rings produced by a forging process for an automotive engine. It consists of 5 batches of 5 observations each. Since the process was in control when the data were taken, we treat the collection as a set of n = 15 observations. The data are given in Table.7. Here LSL = 73.95, USL=74.05, x = , and s = The 90% and 95% confidence intervals as per the three methods of GCIs, Nagata and Nagahata, and Heavlin are given in Table 4.5. It is clear that the method of GCIs, gives the shortest confidence interval for both 90% and 95% confidence coefficients. Thus, this example reaffirms the fact that GCIs do possess the repeated sampling property, and the results of even this isolated example are in perfect agrement with the findings of the simulation study. 4.6 Conclusions Difficulties in the construction of conventional two-sided confidence intervals with exact frequentist coverage, have been felt for a long time in the case of C pk. So, only approximate methods were in use with that much sacrifice to various repeated sampling properties. Motivated by the proposal of an exact method for the construction of confidence intervals based on the new idea of GCIs by Weerahandi (1993, 1995, 004), and its successful applications in a wide variety of real life contexts, we have decided to apply this idea to construct two-sided confidence intervals for C pk. We have succeeded in deriving the generalized pivotal quantity required for constructing the confidence intervals. Since the performance of a method for constructing confidence intervals depends on the extent to which it holds various repeated sampling properties, we undertook an extensive simulation study and the findings are given below. The simulation experiment undertaken in this study involved 10,000 10,000 repetitions,and it revealed that the performance of the method of GCIs, in the context of providing two-sided confidence intervals for C pk, is exceptionally good. The coverage 105

15 probabilities of the confidence intervals by the method of GCIs were very close to the nominal values and not even a single coverage was outside the 99% confidence band for each of the GCIs of confidence level 0.90 and On comparing the coverage probabilities of the existing methods with that of GCIs, it is further established that the method of GCIs has at least a slight edge over the best of the existing methods. These observations were true for even the smallest sample of size 30 used in the experiment and for all values of C pk. This once again confirms that the method of GCIs possesses the repeated sampling property possessed by the conventional confidence intervals, and hence it can be used to construct two-sided confidence intervals for PCIs. The most remarkable feature of the two-sided confidence intervals by the method of GCIs we noticed, was with respect to the length of the confidence intervals and that is that not even the length of a single confidence interval of the generalized method, was greater than that of the existing methods. The fact that these findings are true for all values of C pk, all levels of confidence coefficients, and for even the smallest sample of size 30 used in the experiment, establish that the method of GCIs may be proposed as the best alternative method for constructing two-sided confidence intervals for the PCI C pk. More details about these procedures and computations of two-sided confidence intervals for C pk are available in Sebastian and Kurian (006). Bibliography Bissell, A.F. (1990). How reliable is your capability index? Applied Statistics, Chan, L.K., Xiong, Z. and Zhang, D. (1990). On the asymptotic distributions of some process capability indices. Communications in Statistics Theory and Methods, 19(1) Chen, J.P. and Chen, K.S. (004). Comparing the capability of two processes using C pm. Journal of Quality Technology, 36(3) Chen, S.M. and Pearn, W.L. (1997). The asymptotic distribution of the estimated process capability index C pk. Communications in Statistics Theory and Methods, Chou, Y.M. and Owen, D.B. (1989). On the distributions of the estimated process capability indices. Commun. Statist. Thoery Methods, 18(1) Efron B. (1979). Bootstrap methods. Another look at the Jackknife. Annals of Statistics,

16 Efron, B. and Tibshirani, R. (1986). Bootstrap methods for standard errors, confidence intervals, and other Measures of statistical accuracy. Statistical Science, 1(1) Franklin, L.A. and Wasserman, G.S. (1991). Bootstrap confidence interval estimates of C pk : An introduction. Communications in Statistics-Simulation and Computation, Franklin, L.A. and Wasserman, G.S. (199). Bootstrap lower confidence limits for capability indices. Journal of Quality Technology, Gruner, G. (1991). Calculation of C pk under conditions of variable tolerances. Quality Engineering, Heavlin, W.D. (1988). Statistical properties of capability indices. Technical Report 30, Technical Library, Advanced Micro Devices, Inc., Sunnyvale, CA. Johnson, N.L. and Kotz, S. (1970). Distributions in Statistics. Continues Univariate Distributions, John Wiley, New York. Johnson, N.L., Kotz, S. and Pearn, W.L. (1994). Flexible process capability indices. Pakistan Journal of Statistics, Kotz, S. and Lovelace, C.R. (1998). Process Capability Indices in Theory and Practice. London Arnold. Kushler, R.H. and Hurley, P. (199). Confidence bounds for capability indices. Journal of Quality Technology, Li, N., Owen, D.B. and Borrego, S.A. (1990). Lower confidence limits on process capability indices based on the range. Communications in Statistics - Simulation and Computation, Nagata, Y. and Nagahata, H. (1994). Approximation formulas for the confidence intervals of process capability indices. Okayama Economic Review, Owen, D.B. (1965). A special case of a bivariate noncentral t-distribution. Biometrika, Pearn, W.L. and Chen, K.S. (1996). A Bayesian-like estimator of C pk. Communications in Statistics Simulations and Computation,

17 Pearn, W.L., Chen, K.S. and Lin, P.C. (1999). The probability density function of the estimated process capability index C pk. Far East Journal of Theoretical Statistics, Sebastian, G. and Kurian, K.M. (006). Two-sided confidence intervals for the process capability index C pk. Statistical Methods, special issue Weerahandi, S. (1993). Generalized confidence intervals. Journal of the American Statistical Association, Weerahandi, S. (1995). Exact Statistical Methods for Data Analysis. New York Springer Verlag. Weerahandi, S. (004). Generalized Inference in Repeated Measures. New York Wiley. Zhang, N.F., Stenback, G.A. and Wardrop, D.M. (1990). Interval estimation of process capability index C pk. Communications in Statistics - Theory and Methods, 19(1)