A ootstrap Confdence Lmt for Process Capablty Indces YANG Janfeng School of usness, Zhengzhou Unversty, P.R.Chna, 450001 Abstract The process capablty ndces are wdely used by qualty professonals as an estmate of process capablty. The lower confdence lmts of PCIs are dffcult to be estmated by parametrc methods for some non-normal dstrbuted processes. The non-parametrc but computer ntensve ootstrap technques are utlzed for these cases. The Percentle-t ootstrap (PT) method s used to estmate the confdence nterval for PCIs of these non-normal dstrbuted processes n the paper. A seres of smulatons usng dfferent dstrbutons, parameters and sample sze for the ndex C pk was undertaken to compare the performance and relablty of PT to S, P and CP. The results show that the PT method s better to estmate lower confdence lmts of PCIs for most non-normal dstrbuted processes. Keywords Process capablty ndex, ootstrap, Confdence lmt, non-normal dstrbuton 1 Introducton A seres of process capablty ndces (PCI) have receved substantal attenton snce Juran ntroduced the frst ndex C p. PCIs was wdely used n manufacture ndustry all over the world. PCIs are used to determne the capablty of a process. The most basc modfcaton s C p, C pk (Kane, 1986). The most basc PCIs can be defned as the followng: USL LSL C p = (1) 6σ USL µ µ LSL C p = mn, (2) 6σ 6σ where USL s the upper specfcaton lmt, LSL s the lower specfcaton lmt, µ s the process mean, σ s the process standard devaton, and T s target value. The mportant part of PCI s the confdence nterval because PCI s an estmated value. Chou et al. (1990) provded tables for constructng the 95% lower confdence lmts for C p, C pk. oyles (1991) provded an approxmate method for fndng lower confdence nterval. ut the calculaton of all these confdence nterval assumes a normal dstrbuted process. There s not a good parametrc method to confrm the confdence nterval of process capablty ndex for the non-normal dstrbuted processes. ootstrappng s a non-parametrc, but computer-ntensve estmaton method, ntroduced by Efron. ootstrap confdence ntervals may be obtaned wthout any assumpton of underlyng dstrbuton. Frankln and Wasserman (1992) ntroduced the ootstrap method to calculate confdence nterval of PCI and constructed the confdence nterval for some basc PCIs. Cho, et al. (1996), Han and Chou (2000), Tong (1998), Nam et al. (2001) and alamural (2002, 2003) dscussed ootstrap confdence nterval for non-normal dstrbutons and non-normal PCIs. ut the researchs show that some standard ootstrap methods are not precse for non-normal dstrbuted processes. The artcle reported the performance and relablty of Percentle-t ootstrap (PT) method (Lm, Shn and Km, 2004) for estmatng confdence nterval of PCIs n non-normal dstrbuted processes. 2 ootstrap Confdence Intervals of PCI ootstrap method s a computer-based method for estmatng the standard error of a summary statstc. The ootstrap estmate of standard error requres no theoretcal calculatons, and s avalable no matter how mathematcally complcated the summary statstc may be. Several types of ootstrap method for constructng confdence ntervals have been developed: the standard method (S), the percentle method (P), the bas-corrected percentle method (CP), and percentle-t method. 1201
Let x 1, x 2,, x n be a sample X taken from a process,.e., a sequence of n..d. random varables. A ootstrap sample X, denoted by x 1, x 2,, x n, s a sample of sze n drawn wth replacement from the orgnal sample X. There are a total of n n such resamples possble. In our case, these resamples would be used to calculate n n values of Ĉ. (C denotes PCIs.) Each of these would be an estmate of C and the entre collecton would consttute the complete ootstrap dstrbuton for Ĉ. ootstrap samplng s equvalent to samplng wth replacement form the emprcal probablty dstrbuton functon. Thus, the ootstrap dstrbutons of Ĉ s estmate of the dstrbuton of C. In practce, a rough mnmum of 1000 ootstrap resamples are usually suffcent to compute reasonably accurate confdence nterval estmates (Efron & Tbshran, 1986). Therefore, t s assumed that =1000 tmes bootstrap samples are taken and =1000 tmes ootstrap Ĉ are calculated and ordered from smallest to largest. The followeng s four possble constructons for the confdence nterval of PCIs usng ootstrap technques. 2.1 Standard ootstrap (S) Confdence Interval C ˆ From the =1000 ootstrap estmates, for =1,2,,, calculate the sample average 1 C = C ˆ and the sample standard devaton 1 2 s = ( Cˆ C ).The standard ootstrap c 1 = 1 confdence nterval for C s ˆ [, ˆ C Zα sc C+ Zαs c ] (3) 2.2 Percentle ootstrap (P) Confdence Interval From the ordered collecton of C, for =1, 2,, n, the percentle ootstrap confdence nterval for ˆ C s ˆ ˆ [ C ( α), C ((1 α ) )] (4) 2.3 ased-corrected Percentle ootstrap (CP) Confdence Interval It s possble that ootstrap dstrbutons obtaned usng only a sample of the complete ootstrap dstrbuton may be shfted hgher or lower than would be expected. Thus, a thrd method has been developed to correct for ths potental bas. Frst, usng the ordered dstrbuton of Ĉ, calculate the probablty ˆ ( ˆ 1 P0 = Pr C C). Then, Z 0 = Φ ( P 0 ). Thus, we can get the percentle of ordered dstrbuton F ( C ˆ ), P L = Φ( 2Z 0 Zα ) and P U = Φ( 2Z 0 + Zα ), where Φ ( ) s the standard normal cumulatve functon. Fnally, the CP confdence nterval s gven by ˆ ( ), ˆ [ C PL C ( PU )] (5) 2.4 Percentle-t ootstrap (PT) confdence nterval Cˆ ( Cˆ) ased on the ootstrap resample X, we compute Z =, = 1,2,...,, where Cˆ s the se estmate of C based on the ootstrap resample {X 1, X 2,,X }, and se s the estmated standard error of C. Thus, we can get a collecton Z, = 1,2,...,. Let Z ( ) denote the th smallest value among Z, = 1,2,...,. Then, the αth quantle of these Z values s estmated by the value t ˆα ( ) such that =1 I[ Z ( ) tˆ( α ) =α. Note that f α s not an nteger, the αth quantle s equal to the [(+1)α]th smallest value and the (1-α)th quantle s equal to the [(+1)-(+1)α]th smallest value among Z, = 1,2,...,. The 100(1-2α)% ootstrap confdence nterval for C s constructed as Cˆ + tˆ( α) s, Cˆ + tˆ(1 α) s ] (6) [ d d 1202
where s d denotes the standard error of Ĉ. However, there are no smple formulas to compute the standard error of Ĉ because of ts complcated structure. That s, t s not qute possble to compute ether estmate s d and c se for gven ootstrap resamples {X 1,X 2,,X }. Thus, we may use ootstrap s d. To estmate se, we frst compute a bootstrap estmate of standard error s nstead of for each ootstrap resample by utlzng two nested levels of ootstrap resamplng. To carry out such a resamplng, we generate 25 ootstrap samples {X 1,X 2,,X } for the purpose of estmatng the standard error. For each 25 ootstrap resample, we compute C n = 2 ˆ C j j= 1 Therefore, a 100(1-2α)% ootstrap confdence nterval for C s obtaned as ˆ ˆ [ C+ tˆ( α) s, C+ tˆ(1 α) ] c s c 3 The Smulaton for ootstrap Lower Confdence Lmts of PCI To compare the performance of ootstrap lower confdence lmts to those based on the assumpton of non-normal processes, a seres of smulatons was undertaken. In the smulaton, some dfferent dstrbutons and parameters were used to nvestgate four dfferent processes based on the assumpton that datum come from normal, skew and heavy-tal dstrbuted processes. For process capablty ndex, the lower confdence lmt s mportant not upper confdence lmt. So, we dscuss the ootstrap lower confdence lmt n the artcle. We choose normal, lognormal and t dstrbuton wth dfferent parameters to calculate the ootstrap lower confdence lmts of C pk n these smulatons. The value USL=3, LSL=-3 and T=0 was used for all smulatons. The sample sze n are 10, 30, 50 n each smulaton. We choose normal dstrbuton N(0,1) for normal processes, Lognormal(0,0.2) for slght skew processes, Lognormal(0,0.4) for severe skew processes and t 6 for heavy-taled processes. Then, there s C pk =1 for N(0,1). In order to make each dstrbuton have specfed value of µ=0 and σ=1, we transform three non-normal dstrbutons. Then, these transformed dstrbuton s gotten wth µ=0 and σ=1. Thus, these dfferent dstrbutons are comparable. Sngle smulaton was replcated N=1000 tmes. Thus, we were able to calculate the proporton of tmes the four ootstrap lower confdence lmts and the normally derved lower lmts were less than the correspondng ndex. The actual coverage proporton could be compared wth the expected value of 95%. These smulatons were undertaken by the MATLA software. The smulaton results are tabulated n Table 1-4. In these tables, Cov. denotes the coverage proporton. E ( C ) denotes the mean of 95%lower confdence lmts. E (Cˆ ) denotes the mean of estmates of C. ˆ C C ) denotes the mean of the wdth between Ĉ and C. A. Normal dstrbuted process The results show that, for a normal dstrbuted process wth n 30, all four ootstrap methods provde enough coverage proportons. ut S and PT provde wder confdence nterval than other two methods. In general, CP s better for n s larger. For small sample sze (), P and CP methods have lower coverage proportons. The two methods are not applcable to estmate ootstrap confdence nterval. For the value of ˆ C C ), PT method s better than S method for ths case. (Showed n Table1). Lognormal dstrbuted process For slght skew dstrbuted processes, S and PT methods provde stable and hgher coverage proportons, whether n s large or small. P and CP methods provde poor coverage proportons. Comparng the smulaton results of Lognormal(0,0.2) and Lognormal(0,0.4), the hger the skewness s, and se ˆ = n2 j= 1 ( Cˆ j n C 2 ) 2. (7) 1203
and the lower the coverage proporton s. For the dstrbuton Lognormal(0,0.4) wth hgh skewness, the coverage proporton of PT method s more than 0.932. Therefore, PT lower confdence lmts are relable for those underlyng skew dstrbutons. (Showed n Table 2 and 3) Table 1 Converage Proporton for 95% Lower Table 2 Converage Proporton for 95% Lower Confdence Lmts of C pk Normal Process Confdence Lmts of C pk Lognormal(0,.2) Process n Method Cov. ) ˆ C C ) n Method Cov. ) ˆ C C ) S 0.981# P 0.866 CP 0.895 PT 0.970 S 0.970 P 0.928 CP 0.937 PT 0.975# S 0.963 P 0.947 CP 0.951 PT 0.976# 0.449 0.985 0.536 0.972 0.424 1.013 0.590 S (0.273) (0.279) (0.278) (0.359) (0.306) (0.336) 0.756 0.985 0.230 0.826 0.775 1.013 0.239 P (0.242) (0.279) (0.069) (0.275) (0.306) (0.070) 0.722 0.985 0.263 0.861 0.737 1.013 0.277 CP (0.227) (0.279) (0.082) (0.259) (0.306) (0.083) 0.485 0.985 0.501 0.952 0.478 1.013 0.535 PT (0.293) (0.279) (0.264) (0.393) (0.306) (0.340) 0.748 0.976 0.229 0.967 0.736 0.987 0.251 S (0.120) (0.138) (0.056) (0.153) (0.147) (0.073) 0.801 0.976 0.176 0.932 0.800 0.987 0.188 P (0.123) (0.138) (0.034) (0.142) (0.147) (0.042) 0.794 0.976 0.183 0.942 0.790 0.987 0.197 CP (0.122) (0.138) (0.041) (0.145) (0.147) (0.048) 0.729 0.976 0.247 0.971# 0.723 0.987 0.264 PT (0.117) (0.138) (0.059) (0.146) (0.147) (0.069) 0.801 0.974 0.173 0.966 0.799 0.983 0.185 S (0.095) (0.109) (0.031) (0.119) (0.119) (0.036) 0.828 0.974 0.147 0.934 0.827 0.983 0.157 P (0.097) (0.109) (0.025) (0.117) (0.119) (0.027) 0.827 0.974 0.148 0.924 0.824 0.983 0.159 CP (0.099) (0.109) (0.028) (0.120) (0.119) (0.031) 0.786 0.974 0.189 0.971# 0.786 0.983 0.198 PT (0.094) (0.109) (0.034) (0.116) (0.119) (0.037) C. t dstrbuted process For the heavy-taled dstrbutons, P and CP methods provde poor coverage proportons, less than 0.925. PT and S methods have approxmate coverage proportons for all sample sze. ut the PT method has smaller standard devaton of confdence lmts. Therefore, PT lower confdence lmts are relable for the heavy-taled processes. (Showed n Table 4) 4 Concluson In general, S and PT methods can provde relable confdence lmts (ntervals) for underlyng dstrbuted processes whatever the sample sze. P and CP methods have poor coverage proportons for most cases. The PT method provdes hgher coverage proporton wth small confdence ntervals, whether the sample sze s large or small. In the same tme, the method has wld applcablty for most non-normal dstrbuted processes. Therefore, we suggest, for the underlyng processes wth sample sze beng more than 30, the PT confdence lmt be used frstly. ut the PT method has ts shortcomng that ts calculatng program s very complex. When the twce resample sze s more than 50, t take a lot of tme to calculate. 1204
Acknowledgment Ths artcle were supported n part by the Natonal Scence Fund Chna Grant (70572050) and Henan Soft Scence Grant (0413011500). Table 3 Converage Proporton for 95% Lower Table 4 Converage Proporton for 95% Lower Confdence Lmts of C pk Lognormal(0,.4) Process Confdence Lmts of C pk t 6 Process n Method Cov. ) ˆ C C ) n Method Cov. ) ˆ C C ) S 0.961 P 0.771 CP 0.806 PT 0.920 S 0.906 P 0.820 CP 0.838 PT 0.905 S 0.922 P 0.877 CP 0.884 PT 0.929 0.365 1.051 0.687 0.360 1.058 0.699 S 0.960 (0.485) (0.375) (0.412) (0.437) (0.363) (0.393) 0.793 1.051 0.258 0.789 1.058 0.269 P 0.757 (0.347) (0.375) (0.076) (0.326) (0.363) (0.078) 0.756 1.051 0.295 0.752 1.058 0.306 CP 0.812 (0.326) (0.375) (0.088) (0.307) (0.363) (0.090) 0.437 1.051 0.615 0.402 1.058 0.656 PT 0.921 (0.500) (0.375) (0.388) (0.468) (0.363) (0.392) 0.713 1.006 0.292 0.710 1.003 0.293 S 0.961 (0.232) (0.203) (0.088) (0.192) (0.181) (0.083) 0.800 1.006 0.206 0.794 1.003 0.209 P 0.884 (0.209) (0.203) (0.044) (0.173) (0.181) (0.042) 0.784 1.006 0.222 0.780 1.003 0.222 CP 0.893 (0.209) (0.203) (0.049) (0.175) (0.181) (0.049) 0.710 1.006 0.296 0.698 1.003 0.305 PT 0.965 (0.225) (0.203) (0.079) (0.182) (0.181) (0.077) 0.762 0.991 0.229 0.756 0.983 0.228 S 0.938 (0.181) (0.157) (0.063) (0.173) (0.152) (0.064) 0.809 0.991 0.181 0.805 0.983 0.178 P 0.902 (0.167) (0.157) (0.038) (0.154) (0.152) (0.035) 0.799 0.991 0.193 0.796 0.983 0.187 CP 0.901 (0.169) (0.157) (0.043) (0.157) (0.152) (0.042) 0.758 0.991 0.233 0.745 0.983 0.238 PT 0.952 (0.178) (0.157) (0.059) (0.165) (0.152) (0.058) References [1] V. E. Kane. Process capablty ndces. Journal of Qualty Technology, 1986, 18(1):41-45. [2] Y. Chou, D.. Owen. On the dstrbuton of the estmated process capablty ndces. Communcaton n Statstcs Theory and Methods, 1989, 18:4549-4560. [3] R. A. oyles. The Taguch capablty ndex. Journal of Qualty Technology, 1991, 23(1):107-126. [4] L. A. Frankln, G. Wasserman. ootstrap lower confdence lmts for capablty ndces. Journal of Qualty Technology, 1992, 24(4):196-240. [5] K.C. Cho, K. H. Nam, D. H. Park. Estmaton of capablty ndex based on ootstrap method. Mcroelectron Relablty, 1996, 36(9):1141-1153. [6] J. H. Han, J. J. Cho, C. S. Leem. ootstrap confdence lmts for Wrght s C s. Communcaton n Statstcs - Theory and Method, 2000, 29:485-505. [7] L. I. Tong, J. P. Chen. Lower confdence lmts of process capablty ndces for non-normal process dstrbutons. Internal Journal Qualty & Relablty Management, 1998, 15(8):907-91. [8] K. H. Nam, D. K. Km, D. H. Park. Large-sample nterval estmators for process capablty ndces. Qualty Engneerng, 2001, 14(2):213-221. [9] S. alamural, M. Kalyanasundaram. ootstrap lower confdence lmts for the process capablty ndces C p, C pk and C pm. Internatonal Journal of Qualty & Relablty Management, 2002, 19(8):1088-1097. 1205
[10] S. alamural. ootstrap confdence lmts for short-run capablty ndces. Qualty Engneerng, 2003, 15(4): 643-648. [11] J. H. Lm, S. W. Shn, D. K. Km, et al. ootstrap confdence ntervals for steady-state avalablty. Asa-Pacfc Journal of Operatonal Research, 2004, 21(3):407-419 The Author can be contacted from Emal: yangnfo@126.com 1206