The idea of measuring process capability is not new, but the use of indices to communicate information about processes and the specified requirements has become widespread. In this paper, to measure the degree of process centering (the ability to cluster around the center), the process accuracy index C is investigated. An explicit a form for the cumulative distribution function of the estimator Cˆa is derived.
Subsequently, the distributional and inferential properties of the estimated process accuracy index C are provided. The calculations of critical value, p-value and lower a confidence bound are developed for testing process quality. Furthermore, a new generalization of C for cases with asymmetric tolerances is proposed to measure the a process accuracy, and the distributional properties of the new estimator are investigated.
The results obtained are useful for the practitioners in performing the process accuracy testing, and decisions making rule therefore can be obtained on whether actions should be taken to improve the process quality. For illustrative purpose, three application examples are given to show how to test process accuracy using the actual data collected from the factory.
20
References
[1]. Boyles, R. A. (1991). The Taguchi capability index. Journal of Quality Technology, 23, 17-26.
[2]. Boyles, R. A. (1994). Process capability with asymmetric tolerances.
Communications in Statistics: Simulation and Computation, 23(3), 615-643.
[3]. Chan, L. K., Cheng, S. W. and Spiring, F. A. (1988). A new measure of process capability Cpm. Journal of Quality Technology, 20(3), 162-175.
[4]. Chen, K. S. (1998). Incapability index with asymmetric tolerances. Statistica Sinica, 8(1), 253-262.
[5]. Chen, K. S., Pearn, W. L. and Lin, P. C. (1999). A new generalization of the capability index Cpm for asymmetric tolerances. International Journal of Reliability, Quality & Safety Engineering, 6(4), 383-398.
[6]. Choi, B. C. and Owen, D. B. (1990). A study of a new capability index.
Communications in Statistics: Theory & Methods, 19, 1231-1245.
[7]. Chou, Y. M., Owen, D. B. and Borrego, A. S. (1990). Lower confidence limits on process capability indices. Journal of Quality Technology, 22, 223-229.
[8]. Chou, Y. M., Polansky, A. M. and Mason, R. L. (1998). Transforming non-normal data to normality in statistical process control. Journal of Quality Technology, 30(2), 133-141.
[9]. Hoffman, L. L. (2001). Obtaining confidence intervals for Cpk using percentiles of the distribution of Cp. Quality & Reliability Engineering International, 17(2), 113-118.
[10]. Jessenberger, J. and Weihs, C. (2000). A note on the behavior of Cpmk with asymmetric specification limits. Journal of Quality Technology, 32(4), 440-443.
[11]. Juran, J. M. (1974). Quality Control Handbook (3rd Edition), McGraw-Hill, New York.
[12]. Kane, V. E. (1986). Process capability indices. Journal of Quality Technology, 18(1), 41-52.
[13]. Kotz, S. and Johnson, N. L. (2002). Process capability indices – a review, 1992-2000. Journal of Quality Technology, 34(1), 1-19.
[14]. Kotz, S. and Lovelace, C. (1998). Process Capability Indices in Theory and Practice. Arnold, London, U.K.
[15]. Kotz, S., Pearn, W. L. and Johnson, N. L. (1993). Some process capability indices are more reliable than one might think. Journal of the Royal Statistical Society C, 42(1), 55-62.
[16]. Kushler, R. and Hurley, P. (1992). Confidence bounds for capability indices.
Journal of Quality Technology, 24, 188-195.
21
[17]. Leone, F. C., Nelson, L. S. and Nottingham, R. B. (1961). The folded normal distribution. Technometrics, 3, 543-550.
[18]. Nagata, Y. and Nagahata, H. (1994). Approximation formulas for the lower confidence limits of process capability indices. Okayama Economic Review, 25, 301-314.
[19]. Palmer, K. and Tsui, K. L. (1999). A review and interpretations of process capability indices. Annals of Operations Research, 87, 31-47.
[20]. Pearn, W. L. and Chen, K. S. (1998). New generalization of process capability index Cpk. Journal of Applied Statistics, 25(6), 801-810.
[21]. Pearn, W. L., Chen, K. S. and Lin, P. C. (1999). The probability density function of the estimated process capability index Cpk. Far East Journal of Theoretical Statistics, 3(1), 67-80.
[22]. Pearn, W. L., Kotz, S. and Johnson, N. L. (1992). Distributional and inferential properties of process capability indices. Journal of Quality Technology, 24(4), 216-231.
[23]. Pearn, W. L., Lin, G. H. and Chen, K. S. (1998). Distributional and inferential properties of process accuracy and process precision indices. Communications in Statistics: Theory & Methods, 27(4), 985-1000.
[24]. Pearn, W. L. and Lin, P. C. (2002). Computer program for calculating the p-values in testing process capability Cpmk. Quality & Reliability Engineering International, 18(4), 333-342.
[25]. Spiring, F. A. (1997). A unifying approach to process capability indices. Journal of Quality Technology, 29(1), 49-58.
[26]. Spiring, F., Leung, B., Cheng, S. and Yeung, A. (2003). A bibliography of process capability papers. Quality & Reliability Engineering International, 19(5), 445-460.
[27]. Sullivan, L. P. (1984). Targeting variability - a new approach to quality. Quality Progress, 17(7), 15–21.
[28]. Sullivan, L. P. (1985). Letters. Quality Progress, 18(4), 7–8.
[29]. Tang, L. C., Than, S. E. and Ang, B. W. (1997). A graphical approach to obtaining confidence limits of Cpk. Quality & Reliability Engineering International, 13, 337-346.
[30]. Vännman, K. (1995). A unified approach to capability indices. Statistica Sinica, 5, 805-820.
[31]. Vännman, K. (1997). Distribution and moments in simplified form for a general class of capability indices. Communications in Statistics: Theory & Methods, 26, 159-179.
[32]. Vännman, K. (1997). A General Class of Capability Indices in the Case of Asymmetric Tolerances. Communications in Statistics: Theory & Method, 26,
22
2049–2072.
[33]. Vännman, K. and Kotz, S. (1995). A superstructure of capability indices
distributional properties and implications. Scandinavian Journal of Statistics, 22, 477-491.
[34]. Vännman, K. and Hubele, N. F. (2003). Distributional properties of estimated capability indices based on subsamples. Quality & Reliability Engineering International, 19(5), 445-460.
[35]. Wu, C. C. and Tang, G. R. (1998). Tolerance design for products with asymmetric quality losses. International Journal of Production Research, 36(9), 2529-2541.
[36]. Zimmer, L. S., Hubele, N. F. and Zimmer, W. J. (2001). Confidence intervals and sample size determination for Cpm. Quality & Reliability Engineering
International, 17, 51-68.
23
Appendix
Table 3 The values of C′′ , a B C′′ and
( )
ˆa MSE C′′ for b = 2,( )
ˆa ξ= -1.0(0.5)1.0, : 6 : 4l u
D D = , and n = 10(10)50
Table 4. The values of C′′ , a B C′′ and
( )
ˆa MSE C′′ for b = 3,( )
ˆa ξ= -1.0(0.5)1.0, : 6 : 4l u
D D = , and n = 10(10)50
Table 5. The values of C′′ , a B C′′ and
( )
ˆa MSE C′′ for b = 4,( )
ˆa ξ= -1.0(0.5)1.0, : 6 : 4l u
D D = , and n = 10(10)50
ξ = −1.0 0.5ξ = − 0.0ξ = 0.5ξ = 1.0ξ = n
Bias MSE Bias MSE Bias MSE Bias MSE Bias MSE 10 0.0000 0.0109 -0.0043 0.0061 -0.0701 0.0128 -0.0043 0.0029 0.0000 0.0048
20 0.0000 0.0056 -0.0005 0.0046 -0.0496 0.0064 -0.0005 0.0020 0.0000 0.0025
30 0.0000 0.0037 -0.0001 0.0035 -0.0405 0.0043 -0.0001 0.0015 0.0000 0.0016
40 0.0000 0.0028 0.0000 0.0027 -0.0350 0.0032 0.0000 0.0012 0.0000 0.0012
50 0.0000 0.0022 0.0000 0.0022 -0.0313 0.0026 0.0000 0.0010 0.0000 0.0010
C ′′ a 0.6667 0.8333 1.0000 0.8889 0.7778
ξ = −1.0 ξ = −0.5 ξ =0.0 ξ =0.5 ξ =1.0
n
Bias MSE Bias MSE Bias MSE Bias MSE Bias MSE 10 -0.0001 0.0245 -0.0064 0.0128 -0.1051 0.0250 -0.0064 0.0055 -0.0001 0.0109 20 -0.0000 0.0125 -0.0008 0.0102 -0.0743 0.0125 -0.0008 0.0044 -0.0000 0.0056 30 -0.0000 0.0083 -0.0001 0.0078 -0.0607 0.0083 -0.0001 0.0034 -0.0000 0.0037 40 -0.0000 0.0062 -0.0000 0.0061 -0.0526 0.0065 0.0000 0.0027 -0.0000 0.0028 50 -0.0000 0.0050 -0.0000 0.0050 -0.0470 0.0050 0.0000 0.0022 -0.0000 0.0022
C ′′a 0.5000 0.7500 1.0000 0.8333 0.6667
ξ = −1.0 0.5ξ = − 0.0ξ = 0.5ξ = 1.0ξ = n
Bias MSE Bias MSE Bias MSE Bias MSE Bias MSE 10 0.0000 0.0061 -0.0032 0.0036 -0.0526 0.0076 -0.0032 0.0018 0.0000 0.0027
20 0.0000 0.0031 -0.0004 0.0026 -0.0372 0.0038 -0.0004 0.0012 0.0000 0.0014
30 0.0000 0.0021 -0.0001 0.0020 -0.0303 0.0025 -0.0001 0.0009 0.0000 0.0009
40 0.0000 0.0016 0.0000 0.0015 -0.0263 0.0019 0.0000 0.0007 0.0000 0.0007
50 0.0000 0.0012 0.0000 0.0012 -0.0235 0.0015 0.0000 0.0006 0.0000 0.0006
C ′′ a 0.7500 0.8750 1.0000 0.9167 0.8333
24
Table 6. The values of C′′ , a B C′′ and
( )
ˆa MSE C′′ for b = 5,( )
ˆa ξ= -1.0(0.5)1.0, : 6 : 4l u
D D = , and n = 10(10)50
Figure 15. MSE plot of ˆC ′′ (versus n) for a d* σ = 2, D Dl: u =6 : 4, with ξ=1.0, 0, and 0.5 (from bottom to top).
Figure 16. MSE plot of ˆC ′′ (versus n) for a ξ=0.5, :D Dl u =6 : 4, withd* σ = 5, 4, 3 and 2 (from bottom to top).
ξ = −1.0 0.5ξ = − 0.0ξ = 0.5ξ = 1.0ξ = n
Bias MSE Bias MSE Bias MSE Bias MSE Bias MSE 10 0.0000 0.0039 -0.0026 0.0023 -0.0421 0.0051 -0.0026 0.0012 0.0000 0.0017
20 0.0000 0.0020 -0.0003 0.0017 -0.0297 0.0025 -0.0003 0.0008 0.0000 0.0009
30 0.0000 0.0013 -0.0001 0.0013 -0.0243 0.0017 -0.0001 0.0006 0.0000 0.0006
40 0.0000 0.0010 0.0000 0.0010 -0.0210 0.0013 0.0000 0.0004 0.0000 0.0004
50 0.0000 0.0008 0.0000 0.0008 -0.0188 0.0010 0.0000 0.0004 0.0000 0.0004
C ′′ a 0.8000 0.9000 1.0000 0.9333 0.8667
25
Table 7. Lower confidence bounds C of aL Cˆa= 0.75 for various parameter values, with α = 0.05, ξ= 1.0:0.1:2.0, and n = 10:10:100.
Table 8. Lower confidence bounds C of aL Cˆa= 0.75 for various parameter values, with α = 0.01, ξ= 1.0:0.1:2.0, and n = 10:10:100.
ξ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
n=10 0.479 0.526 0.559 0.583 0.602 0.617 0.630 0.640 0.649 0.656 0.662 n=20 0.605 0.625 0.640 0.651 0.661 0.669 0.675 0.681 0.686 0.690 0.694 n=30 0643 0.656 0.667 0.675 0.682 0.688 0.692 0.696 0.700 0.703 0.706 n=40 0.662 0.673 0.681 0.688 0.693 0.698 0.702 0.705 0.708 0.710 0.713 n=50 0.674 0.683 0.690 0.696 0.700 0.704 0.708 0.710 0.713 0.715 0.717 n=60 0.683 0.690 0.696 0.701 0.705 0.709 0.712 0.714 0.717 0.719 0.720 n=70 0.689 0.696 0.701 0.706 0.709 0.712 0.715 0.717 0.719 0.721 0.723 n=80 0.694 0.700 0.705 0.709 0.712 0.715 0.718 0.720 0.722 0.723 0.725 n=90 0.698 0.703 0.708 0.712 0.715 0.717 0.720 0.722 0.723 0.725 0.726 n=100 0.701 0.706 0.710 0.714 0.717 0.719 0.721 0.723 0.725 0.726 0.728
ξ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0
n=10 0.100 0.246 0.354 0.424 0.473 0.510 0.537 0.559 0.577 0.592 0.605 n=20 0.479 0.526 0.559 0.583 0.602 0.617 0.630 0.640 0.649 0.656 0.662 n=30 0.566 0.593 0.613 0.629 0.641 0.651 0.660 0.667 0.673 0.678 0.683 n=40 0.605 0.625 0.640 0.651 0.661 0.669 0.675 0.681 0.686 0.690 0.694 n=50 0.628 0.643 0.656 0.665 0.673 0.680 0.685 0.690 0.694 0.698 0.701 n=60 0.643 0.656 0.667 0.675 0.682 0.688 0.692 0.696 0.700 0.703 0.706 n=70 0.654 0.666 0.675 0.682 0.688 0.693 0.698 0.701 0.704 0.707 0.710 n=80 0.662 0.673 0.681 0.688 0.693 0.698 0.702 0.705 0.708 0.710 0.713 n=90 0.669 0.678 0.686 0.692 0.697 0.701 0.705 0.708 0.711 0.713 0.715 n=100 0.674 0.683 0.690 0.696 0.700 0.704 0.708 0.710 0.713 0.715 0.717
26
Table 9. Lower confidence bounds C of aL Cˆa= 0.50 for various parameter values, with α = 0.05, ξ= 2.0:0.1:3.0, and n = 10:10:100.
Table 10. Lower confidence bounds C of aL Cˆa= 0.50 for various parameter values, with α = 0.01, ξ= 2.0:0.1:3.0, and n = 10:10:100.
ξ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
n=10 0.324 0.335 0.345 0.354 0.362 0.369 0.375 0.381 0.386 0.391 0.395 n=20 0.387 0.394 0.400 0.405 0.410 0.414 0.418 0.421 0.424 0.427 0.430 n=30 0.412 0.417 0.421 0.425 0.429 0.432 0.435 0.438 0.440 0.442 0.444 n=40 0.425 0.429 0.433 0.436 0.439 0.442 0.445 0.447 0.449 0.451 0.453 n=50 0.434 0.438 0.441 0.444 0.446 0.449 0.451 0.453 0.455 0.456 0.458 n=60 0.441 0.444 0.447 0.449 0.452 0.454 0.456 0.457 0.459 0.461 0.462 n=70 0.446 0.448 0.451 0.453 0.455 0.457 0.459 0.461 0.462 0.464 0.465 n=80 0.449 0.452 0.454 0.457 0.459 0.460 0.462 0.464 0.465 0.466 0.467 n=90 0.453 0.455 0.457 0.459 0.461 0.463 0.464 0.466 0.467 0.468 0.469 n=100 0.455 0.458 0.460 0.462 0.463 0.465 0.466 0.468 0.469 0.470 0.471
ξ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0
n=10 0.209 0.231 0.249 0.265 0.279 0.292 0.303 0.313 0.322 0.330 0.338 n=20 0.324 0.335 0.345 0.354 0.362 0.369 0.375 0.381 0.386 0.391 0.395 n=30 0.365 0.373 0.380 0.387 0.393 0.398 0.402 0.407 0.411 0.414 0.418 n=40 0.387 0.394 0.400 0.405 0.410 0.414 0.418 0.421 0.424 0.427 0.430 n=50 0.402 0.407 0.412 0.417 0.421 0.424 0.428 0.431 0.434 0.436 0.439 n=60 0.412 0.417 0.421 0.425 0.429 0.432 0.435 0.438 0.440 0.442 0.444 n=70 0.419 0.424 0.428 0.431 0.435 0.438 0.440 0.443 0.445 0.447 0.449 n=80 0.425 0.429 0.433 0.436 0.439 0.442 0.445 0.447 0.449 0.451 0.453 n=90 0.430 0.434 0.437 0.440 0.443 0.446 0.448 0.450 0.452 0.454 0.456 n=100 0.434 0.438 0.441 0.444 0.446 0.449 0.451 0.453 0.455 0.456 0.458
27
Table 11. Critical value c of 0 C ′′ for various parameter values, with a Dl =Du = , d α= 0.01, 0.05 and n = 10, 25(25)150.
|µ−T | /σ 0.5 1.0 1.5
c 0 c 0 c 0
n d
σ C ′′ a α=0.01 α =0.05 C ′′a
α =0.01 α=0.05 C ′′ a
α=0.01 α=0.05 2 0.750 0.993 0.965 0.500 0.868 0.760 0.250 0.618 0.510 3 0.833 0.995 0.977 0.667 0.912 0.840 0.500 0.745 0.673 4 0.875 0.997 0.983 0.750 0.934 0.880 0.625 0.809 0.755 10
5 0.900 0.997 0.986 0.800 0.947 0.904 0.700 0.847 0.804 2 0.750 0.973 0.914 0.500 0.733 0.664 0.250 0.483 0.414 3 0.833 0.982 0.943 0.667 0.822 0.777 0.500 0.655 0.610 4 0.875 0.987 0.957 0.750 0.866 0.832 0.625 0.741 0.707 25
5 0.900 0.989 0.966 0.800 0.893 0.866 0.700 0.793 0.766 2 0.750 0.914 0.866 0.500 0.664 0.616 0.250 0.414 0.366 3 0.833 0.943 0.911 0.667 0.777 0.744 0.500 0.610 0.578 4 0.875 0.957 0.933 0.750 0.832 0.808 0.625 0.707 0.683 50
5 0.900 0.966 0.947 0.800 0.866 0.847 0.700 0.766 0.747 2 0.750 0.884 0.845 0.500 0.634 0.595 0.250 0.384 0.345 3 0.833 0.923 0.896 0.667 0.756 0.730 0.500 0.590 0.563 4 0.875 0.942 0.922 0.750 0.817 0.797 0.625 0.692 0.672 75
5 0.900 0.954 0.938 0.800 0.854 0.838 0.700 0.754 0.738 2 0.750 0.866 0.832 0.500 0.616 0.582 0.250 0.366 0.332 3 0.833 0.911 0.888 0.667 0.744 0.722 0.500 0.578 0.555 4 0.875 0.933 0.916 0.750 0.808 0.791 0.625 0.683 0.666 100
5 0.900 0.947 0.933 0.800 0.847 0.833 0.700 0.747 0.733 2 0.750 0.854 0.824 0.500 0.604 0.574 0.250 0.354 0.324 3 0.833 0.902 0.882 0.667 0.736 0.716 0.500 0.569 0.549 4 0.875 0.927 0.912 0.750 0.802 0.787 0.625 0.677 0.662 125
5 0.900 0.942 0.929 0.800 0.842 0.829 0.700 0.742 0.729 2 0.750 0.845 0.817 0.500 0.595 0.567 0.250 0.345 0.317 3 0.833 0.896 0.878 0.667 0.730 0.712 0.500 0.563 0.545 4 0.875 0.922 0.909 0.750 0.797 0.784 0.625 0.672 0.659 150
5 0.900 0.938 0.927 0.800 0.838 0.827 0.700 0.738 0.727
28
Table 12. Critical value c of 0 C ′′ for various parameter values, with a D Dl: u =6 : 4, α = 0.01, 0.05 and n = 10, 25, 50 and 75.
/
µ−T σ 0.5 1.0 1.5
c 0 c 0 c 0
n d*
σ C ′′a α =0.01 α =0.05 C ′′a α=0.01 α=0.05 C ′′a α=0.01 α=0.05 2 0.750 0.989 0.951 0.500 0.863 0.759 0.250 0.617 0.510 3 0.833 0.993 0.967 0.667 0.909 0.839 0.500 0.745 0.673 4 0.875 0.994 0.975 0.750 0.931 0.879 0.625 0.808 0.755 10
5 0.900 0.995 0.980 0.800 0.945 0.903 0.700 0.847 0.804 2 0.750 0.965 0.909 0.500 0.732 0.664 0.250 0.482 0.414 3 0.833 0.977 0.939 0.667 0.821 0.776 0.500 0.655 0.609 4 0.875 0.982 0.954 0.750 0.866 0.832 0.625 0.741 0.707 25
5 0.900 0.986 0.963 0.800 0.893 0.865 0.700 0.793 0.765 2 0.750 0.914 0.866 0.500 0.665 0.616 0.250 0.414 0.366 3 0.833 0.942 0.910 0.667 0.776 0.744 0.500 0.609 0.577 4 0.875 0.957 0.933 0.750 0.832 0.808 0.625 0.707 0.683 50
5 0.900 0.965 0.946 0.800 0.866 0.846 0.700 0.765 0.746 2 0.750 0.884 0.844 0.500 0.634 0.595 0.250 0.387 0.345 3 0.833 0.922 0.896 0.667 0.756 0.730 0.500 0.591 0.563 4 0.875 0.942 0.922 0.750 0.817 0.797 0.625 0.693 0.672 75
5 0.900 0.953 0.937 0.800 0.853 0.838 0.700 0.754 0.738
29
0.1 0.993 0.968 0.995 0.976 0.996 0.979 0.996 0.981 0.996 0.982 0.2 0.996 0.981 0.996 0.983 0.996 0.983 0.996 0.982 0.996 0.979 0.3 0.996 0.983 0.996 0.981 0.995 0.973 0.993 0.960 0.990 0.943 0.4 0.996 0.982 0.994 0.967 0.989 0.937 0.976 0.912 0.955 0.895 0.5 0.995 0.975 0.988 0.933 0.962 0.900 0.933 0.880 0.914 0.866 0.6 0.993 0.960 0.966 0.903 0.926 0.875 0.903 0.858 0.887 0.846 0.7 0.988 0.935 0.935 0.881 0.901 0.857 0.881 0.842 0.867 0.833 0.8 0.976 0.912 0.912 0.864 0.882 0.843 0.864 0.831 0.852 0.822 09 0.954 0.894 0.894 0.852 0.867 0.833 0.852 0.822 0.841 0.814 1.0 0.933 0.880 0.880 0.841 0.856 0.825 0.841 0.815 0.832 0.808
Table 14. Critical Values c for various parameter values, 0 C ′′ =0.67, :a D Dl u =7 : 3,
0.1 0.991 0.958 0.993 0.968 0.994 0.973 0.995 0.975 0.995 0.976 0.2 0.995 0.975 0.995 0.978 0.995 0.978 0.995 0.976 0.995 0.973 0.3 0.995 0.978 0.995 0.975 0.993 0.965 0.991 0.948 0.987 0.925 0.4 0.995 0.976 0.992 0.957 0.985 0.917 0.968 0.884 0.941 0.861 0.5 0.994 0.968 0.984 0.912 0.949 0.868 0.912 0.841 0.887 0.823 0.6 0.991 0.948 0.955 0.872 0.903 0.835 0.872 0.813 0.850 0.797 0.7 0.985 0.914 0.915 0.843 0.870 0.811 0.843 0.792 0.825 0.779 0.8 0.968 0.884 0.884 0.821 0.845 0.793 0.821 0.777 0.805 0.765 09 0.939 0.860 0.860 0.804 0.825 0.780 0.804 0.765 0.790 0.755 1.0 0.912 0.841 0.841 0.791 0.810 0.769 0.791 0.755 0.778 0.746
30
0.1 0.987 0.936 0.990 0.952 0.992 0.959 0.992 0.962 0.993 0.965 0.2 0.992 0.962 0.993 0.967 0.993 0.967 0.993 0.964 0.992 0.959 0.3 0.993 0.967 0.993 0.962 0.990 0.947 0.987 0.921 0.981 0.886 0.4 0.993 0.964 0.989 0.934 0.978 0.874 0.952 0.825 0.911 0.790 0.5 0.991 0.951 0.976 0.867 0.924 0.800 0.867 0.760 0.829 0.732 0.6 0.987 0.921 0.932 0.806 0.853 0.750 0.806 0.716 0.774 0.693 0.7 0.977 0.871 0.871 0.762 0.803 0.714 0.762 0.685 0.734 0.666 0.8 0.952 0.825 0.825 0.729 0.765 0.687 0.729 0.662 0.705 0.645 09 0.908 0.788 0.789 0.704 0.735 0.666 0.704 0.644 0.682 0.629 1.0 0.867 0.760 0.760 0.683 0.712 0.650 0.683 0.630 0.664 0.616
Table 16. Critical Values c for various parameter values, 0 C ′′ =0.25, :a D Dl u =7 : 3,
0.1 0.981 0.905 0.986 0.929 0.988 0.939 0.989 0.944 0.989 0.947 0.2 0.989 0.944 0.990 0.951 0.990 0.950 0.989 0.946 0.988 0.939 0.3 0.990 0.951 0.989 0.943 0.986 0.921 0.980 0.882 0.972 0.829 0.4 0.989 0.946 0.983 0.902 0.967 0.812 0.928 0.737 0.866 0.686 0.5 0.987 0.927 0.964 0.801 0.886 0.700 0.801 0.640 0.743 0.598 0.6 0.980 0.882 0.898 0.709 0.780 0.625 0.709 0.575 0.661 0.540 0.7 0.966 0.806 0.807 0.644 0.705 0.571 0.644 0.528 0.602 0.499 0.8 0.928 0.737 0.737 0.594 0.648 0.531 0.594 0.493 0.558 0.468 09 0.862 0.683 0.683 0.556 0.603 0.500 0.556 0.466 0.524 0.444 1.0 0.801 0.640 0.640 0.525 0.568 0.475 0.525 0.445 0.497 0.424
31
0.1 0.991 0.958 0.993 0.968 0.994 0.973 0.995 0.975 0.995 0.976 0.2 0.995 0.975 0.995 0.978 0.995 0.978 0.995 0.977 0.995 0.974 0.3 0.995 0.978 0.995 0.976 0.994 0.968 0.991 0.956 0.988 0.941 0.4 0.995 0.977 0.992 0.962 0.986 0.936 0.973 0.912 0.955 0.895 0.5 0.994 0.970 0.985 0.932 0.961 0.900 0.933 0.880 0.914 0.866 0.6 0.991 0.956 0.964 0.903 0.926 0.875 0.903 0.858 0.887 0.846 0.7 0.986 0.934 0.935 0.881 0.901 0.857 0.881 0.842 0.867 0.833 0.8 0.973 0.912 0.912 0.864 0.882 0.843 0.864 0.831 0.852 0.822 09 0.953 0.894 0.894 0.852 0.867 0.833 0.852 0.822 0.841 0.814 1.0 0.933 0.880 0.880 0.841 0.856 0.825 0.841 0.815 0.832 0.808
Table 18. Critical Values c for various parameter values, 0 C ′′ =0.67, :a D Dl u =6 : 4,
0.1 0.989 0.944 0.991 0.958 0.992 0.964 0.993 0.967 0.993 0.969 0.2 0.993 0.967 0.994 0.971 0.994 0.971 0.994 0.969 0.993 0.966 0.3 0.994 0.972 0.993 0.968 0.992 0.958 0.989 0.942 0.984 0.922 0.4 0.994 0.969 0.990 0.950 0.982 0.915 0.965 0.884 0.940 0.861 0.5 0.992 0.961 0.980 0.911 0.948 0.868 0.912 0.841 0.887 0.823 0.6 0.989 0.942 0.953 0.872 0.903 0.835 0.872 0.813 0.850 0.797 0.7 0.981 0.913 0.915 0.843 0.870 0.811 0.843 0.792 0.825 0.779 0.8 0.965 0.884 0.884 0.821 0.845 0.793 0.821 0.777 0.805 0.765 09 0.939 0.860 0.860 0.804 0.825 0.780 0.804 0.765 0.790 0.755 1.0 0.912 0.841 0.841 0.791 0.810 0.769 0.791 0.755 0.778 0.746
32
0.1 0.983 0.916 0.987 0.937 0.989 0.946 0.990 0.951 0.990 0.953 0.2 0.990 0.951 0.991 0.957 0.991 0.957 0.991 0.954 0.990 0.949 0.3 0.991 0.957 0.990 0.952 0.988 0.937 0.983 0.913 0.976 0.883 0.4 0.991 0.954 0.985 0.924 0.973 0.872 0.947 0.824 0.910 0.790 0.5 0.988 0.941 0.971 0.865 0.922 0.800 0.867 0.760 0.829 0.732 0.6 0.983 0.913 0.929 0.806 0.853 0.750 0.806 0.716 0.774 0.693 0.7 0.972 0.869 0.871 0.762 0.803 0.714 0.762 0.685 0.735 0.666 0.8 0.947 0.824 0.825 0.729 0.765 0.687 0.729 0.662 0.705 0.645 09 0.907 0.788 0.789 0.704 0.735 0.666 0.704 0.644 0.682 0.629 1.0 0.867 0.760 0.760 0.683 0.712 0.650 0.683 0.630 0.664 0.616
Table 20. Critical Values c for various parameter values, 0 C ′′ =0.25, :a D Dl u =6 : 4,
0.1 0.975 0.874 0.981 0.906 0.984 0.919 0.985 0.926 0.986 0.930 0.2 0.985 0.926 0.987 0.936 0.987 0.935 0.986 0.931 0.985 0.924 0.3 0.987 0.936 0.986 0.928 0.982 0.905 0.975 0.869 0.964 0.824 0.4 0.986 0.931 0.978 0.887 0.960 0.809 0.921 0.737 0.865 0.686 0.5 0.983 0.911 0.956 0.798 0.883 0.700 0.801 0.640 0.743 0.598 0.6 0.975 0.869 0.894 0.709 0.780 0.625 0.709 0.575 0.661 0.540 0.7 0.958 0.803 0.807 0.644 0.705 0.571 0.644 0.528 0.602 0.499 0.8 0.921 0.737 0.737 0.594 0.648 0.531 0.594 0.493 0.558 0.468 09 0.861 0.683 0.683 0.556 0.603 0.500 0.556 0.466 0.524 0.443 1.0 0.801 0.640 0.640 0.525 0.568 0.475 0.525 0.445 0.496 0.424
33
0.1 0.987 0.937 0.990 0.953 0.992 0.960 0.992 0.964 0.993 0.966 0.2 0.992 0.964 0.993 0.969 0.993 0.969 0.993 0.968 0.992 0.965 0.3 0.993 0.969 0.993 0.966 0.991 0.959 0.988 0.948 0.984 0.936 0.4 0.993 0.968 0.989 0.953 0.982 0.932 0.969 0.911 0.953 0.895 0.5 0.991 0.961 0.981 0.929 0.958 0.899 0.933 0.880 0.914 0.866 0.6 0.988 0.948 0.961 0.902 0.926 0.875 0.903 0.858 0.887 0.846 0.7 0.981 0.931 0.935 0.881 0.901 0.857 0.881 0.842 0.867 0.833 0.8 0.969 0.911 0.912 0.864 0.882 0.843 0.864 0.831 0.852 0.822 09 0.952 0.894 0.894 0.852 0.868 0.833 0.852 0.822 0.841 0.814 1.0 0.933 0.880 0.880 0.841 0.856 0.825 0.841 0.815 0.832 0.808
Table 22. Critical Values c for various parameter values, 0 C ′′ =0.67, :a D Dl u =4 : 6,
0.1 0.983 0.917 0.987 0.939 0.989 0.947 0.990 0.952 0.991 0.955 0.2 0.990 0.952 0.991 0.959 0.991 0.959 0.991 0.957 0.990 0.954 0.3 0.991 0.959 0.990 0.956 0.988 0.946 0.984 0.932 0.979 0.916 0.4 0.991 0.957 0.986 0.939 0.976 0.910 0.959 0.883 0.938 0.861 0.5 0.989 0.949 0.974 0.907 0.945 0.867 0.912 0.841 0.887 0.823 0.6 0.984 0.932 0.949 0.871 0.903 0.835 0.872 0.813 0.850 0.797 0.7 0.975 0.909 0.914 0.843 0.870 0.811 0.843 0.792 0.825 0.779 0.8 0.959 0.883 0.884 0.821 0.845 0.793 0.821 0.777 0.805 0.765 09 0.936 0.860 0.860 0.804 0.825 0.780 0.804 0.765 0.790 0.755 1.0 0.912 0.841 0.841 0.791 0.810 0.769 0.791 0.755 0.778 0.746
34
0.1 0.975 0.875 0.981 0.907 0.984 0.920 0.985 0.928 0.986 0.932 0.2 0.985 0.928 0.987 0.938 0.987 0.938 0.986 0.936 0.985 0.931 0.3 0.987 0.938 0.986 0.933 0.982 0.918 0.976 0.897 0.968 0.873 0.4 0.986 0.936 0.979 0.907 0.964 0.865 0.939 0.823 0.906 0.790 0.5 0.983 0.922 0.962 0.859 0.917 0.799 0.867 0.760 0.828 0.732 0.6 0.976 0.897 0.923 0.805 0.853 0.750 0.806 0.716 0.774 0.693 0.7 0.963 0.862 0.870 0.762 0.803 0.714 0.762 0.685 0.735 0.666 0.8 0.939 0.823 0.825 0.729 0.765 0.687 0.729 0.662 0.705 0.645 09 0.904 0.788 0.789 0.704 0.736 0.666 0.704 0.644 0.682 0.629 1.0 0.867 0.760 0.760 0.683 0.712 0.650 0.683 0.630 0.664 0.616
Table 24. Critical Values c for various parameter values, 0 C ′′ =0.25, :a D Dl u =4 : 6,
0.1 0.962 0.813 0.972 0.861 0.976 0.881 0.978 0.892 0.979 0.898 0.2 0.978 0.892 0.981 0.907 0.981 0.908 0.980 0.904 0.978 0.896 0.3 0.981 0.908 0.979 0.900 0.973 0.878 0.965 0.846 0.952 0.810 0.4 0.980 0.904 0.969 0.861 0.946 0.797 0.908 0.735 0.859 0.685 0.5 0.975 0.884 0.943 0.789 0.875 0.699 0.800 0.640 0.743 0.598 0.6 0.965 0.846 0.885 0.708 0.780 0.625 0.709 0.575 0.661 0.540 0.7 0.945 0.793 0.806 0.644 0.705 0.571 0.644 0.528 0.602 0.499 0.8 0.908 0.735 0.737 0.594 0.648 0.531 0.594 0.493 0.558 0.468 09 0.856 0.683 0.683 0.556 0.604 0.500 0.556 0.466 0.524 0.443 1.0 0.800 0.640 0.640 0.525 0.568 0.475 0.525 0.445 0.496 0.424
35
0.1 0.985 0.927 0.989 0.946 0.990 0.954 0.991 0.958 0.992 0.961 0.2 0.991 0.958 0.992 0.964 0.992 0.965 0.992 0.963 0.991 0.961 0.3 0.992 0.964 0.992 0.962 0.990 0.955 0.986 0.945 0.982 0.934 0.4 0.992 0.963 0.988 0.950 0.980 0.930 0.967 0.910 0.952 0.895 0.5 0.990 0.957 0.979 0.927 0.957 0.899 0.933 0.879 0.914 0.866 0.6 0.986 0.945 0.960 0.902 0.926 0.875 0.903 0.858 0.887 0.846 0.7 0.979 0.928 0.935 0.881 0.901 0.857 0.881 0.842 0.867 0.833 0.8 0.967 0.910 0.912 0.864 0.882 0.843 0.864 0.831 0.852 0.822 09 0.951 0.894 0.894 0.852 0.868 0.833 0.852 0.822 0.841 0.814 1.0 0.933 0.879 0.880 0.841 0.856 0.825 0.841 0.815 0.832 0.808
Table 26. Critical Values c for various parameter values, 0 C ′′ =0.67, :a D Dl u =3 : 7,
0.1 0.980 0.904 0.985 0.929 0.987 0.939 0.988 0.945 0.989 0.948 0.2 0.988 0.945 0.990 0.953 0.990 0.954 0.989 0.952 0.989 0.949 0.3 0.990 0.953 0.989 0.950 0.986 0.941 0.982 0.927 0.976 0.913 0.4 0.989 0.952 0.984 0.934 0.974 0.907 0.957 0.882 0.936 0.861 0.5 0.987 0.943 0.972 0.904 0.943 0.867 0.911 0.841 0.887 0.823 0.6 0.982 0.927 0.947 0.871 0.903 0.835 0.872 0.813 0.850 0.797 0.7 0.973 0.906 0.914 0.843 0.870 0.811 0.843 0.792 0.825 0.779 0.8 0.957 0.882 0.884 0.821 0.845 0.793 0.821 0.777 0.805 0.765 09 0.935 0.860 0.860 0.804 0.825 0.780 0.804 0.765 0.790 0.755 1.0 0.911 0.841 0.841 0.791 0.810 0.769 0.791 0.755 0.778 0.746
36
0.1 0.970 0.855 0.978 0.892 0.981 0.908 0.983 0.916 0.984 0.922 0.2 0.983 0.916 0.985 0.929 0.985 0.930 0.984 0.927 0.983 0.922 0.3 0.985 0.929 0.984 0.925 0.980 0.910 0.973 0.890 0.964 0.868 0.4 0.984 0.927 0.976 0.900 0.960 0.860 0.935 0.821 0.904 0.790 0.5 0.981 0.914 0.958 0.855 0.914 0.799 0.866 0.759 0.828 0.732 0.6 0.973 0.890 0.920 0.804 0.853 0.750 0.806 0.716 0.774 0.693 0.7 0.959 0.857 0.870 0.762 0.803 0.714 0.762 0.685 0.735 0.666 0.8 0.935 0.821 0.825 0.729 0.765 0.687 0.729 0.662 0.705 0.645 09 0.902 0.788 0.789 0.704 0.736 0.666 0.704 0.644 0.682 0.629 1.0 0.866 0.759 0.760 0.683 0.712 0.650 0.683 0.630 0.664 0.616
Table 28. Critical Values c for various parameter values, 0 C ′′ =0.25, :a D Dl u =3 : 7,
0.1 0.956 0.782 0.967 0.839 0.972 0.862 0.974 0.875 0.976 0.883 0.2 0.974 0.875 0.978 0.893 0.978 0.895 0.977 0.891 0.975 0.884 0.3 0.978 0.894 0.976 0.888 0.970 0.866 0.960 0.836 0.946 0.802 0.4 0.977 0.891 0.965 0.850 0.941 0.790 0.903 0.732 0.856 0.685 0.5 0.971 0.872 0.937 0.783 0.871 0.698 0.799 0.639 0.743 0.598 0.6 0.960 0.836 0.880 0.707 0.780 0.625 0.709 0.575 0.661 0.540 0.7 0.939 0.786 0.805 0.643 0.705 0.571 0.644 0.528 0.602 0.499 0.8 0.903 0.732 0.737 0.594 0.648 0.531 0.594 0.493 0.558 0.468 09 0.853 0.682 0.683 0.556 0.604 0.500 0.556 0.466 0.524 0.443 1.0 0.799 0.639 0.640 0.525 0.568 0.475 0.525 0.445 0.496 0.424
37
20 0.025 0.354 0.027 0.474 0.286 0.538 0.408 0.578 0.479 0.605 30 0.145 0.500 0.365 0.563 0.468 0.600 0.527 0.625 0.566 0.643 40 0.354 0.559 0.474 0.603 0.538 0.630 0.578 0.648 0.605 0.663 50 0.447 0.592 0.529 0.626 0.576 0.648 0.606 0.663 0.628 0.675
Table 30. Lower Confidence Bounds C ′′ for various parameter values, ˆaL C ′′ =0.75, a
20 0.025 0.355 0.027 0.474 0.286 0.538 0.408 0.578 0.479 0.605 30 0.144 0.500 0.365 0.563 0.467 0.600 0.527 0.625 0.566 0.643 40 0.354 0.559 0.474 0.603 0.538 0.630 0.578 0.649 0.605 0.663 50 0.447 0.592 0.529 0.626 0.576 0.648 0.606 0.663 0.628 0.675
Table 31. Lower Confidence Bounds C ′′ for various parameter values, ˆaL C ′′ =0.75, a
20 0.027 0.357 0.033 0.474 0.286 0.538 0.408 0.578 0.479 0.605 30 0.146 0.500 0.365 0.563 0.467 0.600 0.527 0.625 0.566 0.643 40 0.354 0.559 0.474 0.603 0.538 0.630 0.578 0.649 0.605 0.663 50 0.447 0.592 0.529 0.626 0.576 0.648 0.606 0.663 0.628 0.675
38
Table 32. Lower Confidence Bounds C ′′ for various parameter values, ˆaL C ′′ =0.75, a : 3 : 7
l u
D D = , α =0.01, 0.05, n=20(10)50, and ξ =0.6(0.1)1.0.
ξ =0.6 0.7ξ = 0.8ξ = 0.9ξ = 1.0ξ = n
0.01 α =
0.05 α =
0.01 α =
0.05 α =
0.01 α =
0.05 α =
0.01 α =
0.05 α =
0.01 α =
0.05 α =
20 0.027 0.360 0.038 0.474 0.286 0.538 0.408 0.578 0.479 0.605 30 0.148 0.500 0.365 0.563 0.467 0.600 0.527 0.625 0.566 0.643 40 0.354 0.559 0.474 0.603 0.538 0.630 0.578 0.649 0.605 0.663 50 0.447 0.592 0.529 0.626 0.576 0.648 0.606 0.663 0.628 0.675