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International Journal of Production
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Lower confidence bounds for C
PU
and C
PL
based on multiple samples
with application to production yield
assurance
W. L. Pearn a , M. H. Shu b & B. M. Hsu c a
Department of Industrial Engineering & Management , National Chiao Tung University , Taiwan, ROC
b
Department of Commerce Automation & Management , National Pingtung Institute of Commerce , Taiwan, ROC c
Department of Industrial Engineering and Management , Cheng Shiu University , Taiwan, ROC
Published online: 21 Feb 2007.
To cite this article: W. L. Pearn , M. H. Shu & B. M. Hsu (2004) Lower confidence
bounds for C PU and C PL based on multiple samples with application to production yield assurance, International Journal of Production Research, 42:12, 2339-2356, DOI: 10.1080/00207540310001652888
To link to this article: http://dx.doi.org/10.1080/00207540310001652888
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int. j. prod. r es.,15 June 2004, vol. 42, no. 12, 2339–2356
Lower confidence bounds for C
PUand C
PLbased on multiple samples
with application to production yield assurance
W. L. PEARNy*, M. H. SHUz and B. M. HSU}
For stably normal processes with one-sided specification limits, capability indices
CPU and CPL have been used to provide numerical measures on production
yield assurance. Statistical properties of the estimators of CPU and CPL have
been investigated extensively for cases with a single sample. It is shown that for
multiple samples, the uniformly minimum-variance unbiased estimators of CPU
and CPL are consistent and asymptotically efficient. Based on the uniformly
minimum-variance unbiased estimators, an algorithm is developed with an efficient program using a direct search method to compute the lower confidence
bounds for CPUand CPL. The lower confidence bounds convey critical information
to the minimum capability of a process, providing a necessary yield assurance of production. The lower confidence bounds are tabulated for some commonly used capability requirement so that engineers/practitioners can use them for their in-plant applications. An example of a high-speed buffer amplifier is presented to illustrate the practicality of the approach to data collected from the factories for production yield assurance.
1. Introduction
Process capability indices have been used in the manufacturing industry to mea-sure the capability of a process to reproduce items satisfying the requirement preset by the product designers or customer’s specifications. Several capability indices, including Cp, CPU, CPL, Cpk, Cpmand Cpmk, are developed for this purpose (Kane
1986, Chan et al. 1988, Cheng 1992, 1994–95, Pearn et al. 1992). Those indices essentially compare the predefined product specifications with the actual process distribution characteristics, which have been defined as follows:
Cp¼ USL LSL 6 , CPU¼ USL 3 , CPL¼ LSL 3 , Cpk¼min USL 3 , LSL 3 , Cpm¼ USL LSL 6 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þð TÞ2 q , Cpmk¼min USL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þð TÞ2 q , LSL 3 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2þð TÞ2 q ( ) ;
Revision received October 2003.
yDepartment of Industrial Engineering & Management, National Chiao Tung University, Taiwan, ROC.
zDepartment of Commerce Automation & Management, National Pingtung Institute of Commerce, Taiwan, ROC.
}Department of Industrial Engineering and Management, Cheng Shiu University, Taiwan, ROC.
*To whom correspondence should be addressed. e-mail: wlpearn@mail.nctu.edu.tw
International Journal of Production ResearchISSN 0020–7543 print/ISSN 1366–588X online # 2004 Taylor & Francis Ltd
http://www.tandf.co.uk/journals DOI: 10.1080/00207540310001652888
where USL is the upper specification limit, LSL is the lower specification limit, is the process mean, is the process standard deviation (overall process variation) and T is the target value.
The indices Cp, Cpk, Cpmand Cpmkare appropriate for a product with two-sided
specification limits, where both USL and LSL are required. However, the indices CPU and CPL are designed specifically for a product with a one-sided specification
limit, and only one USL and LSL is required in this case. Many quality/reliability and statistics literatures have addressed the statistical properties of the estimators of CPU and CPL, and studied their industrial applications based on a single sample.
Examples include Chou and Owen (1989) for obtaining the sampling distributions and other statistical properties, Chou (1994) for developing a procedure for selecting better suppliers, Pearn and Chen (2002) for obtaining the uniformly minimum-var-iance unbiased estimator (UMVUE) and developing a test based on the UMVUE, Lin and Pearn (2002) for implementing the statistical testing with application to capability determination of the voltage level translator, and Pearn and Shu (2002) for proposing an algorithm to calculate the lower confidence bounds (LCBs) with application to low drop-out regulators. However, their investigations on CPU and
CPL are restricted to a single sample.
In practice, manufacturing information about product quality characteristics is often derived from multiple samples rather than from a single sample, particularly when a daily-based production control plan is implemented for monitoring process stability. The purpose of the present paper is to consider the capability estimation and testing of the one-sided capability indices CPU and CPL for multiple samples
with variable sample sizes, and to apply the proposed LCB approach to real-world manufacturing applications for production yield assurance.
2. Capability requirements for production processes
In current practice (Kotz and Lovelace 1998), a process is called inadequate if CI<1.00, where CI¼CPUor CPL; it indicates that the process is not adequate with
respect to the production tolerances; either the process variation, 2, needs to be reduced or the process mean, , needs to be shifted closer to the target value, T. A process is called marginally capable if 1.00 CI<1.33; it indicates that caution
needs to be taken about the process distribution and some process control is required. A process is called satisfactory if 1.33 CI<1.67; it indicates that process
quality is satisfactory, material substitution may be allowed, and no stringent quality control is required. A process is called excellent if 1.67 CI<2.00; it indicates that
process quality exceeds satisfactory. Finally, a process is called super if CI2.00.
Table 1 summarizes the above five conditions and the corresponding CI values.
However, Montgomery (2001) recommended some minimum quality requirements on CPU and CPL (table 2) for specific process types that must run under some
designated capability conditions. Therefore, it would be desirable to determine a bound that practitioners would be expected to find the true value of the process capability no less than the bound value with certain level of confidence.
For normally distributed processes with one-sided specification limit USL, the process yield P(X<USL) is:
P X 3 < USL 3 ¼P 1 3Z < CPU ¼P Z <ð 3CPUÞ ¼F 3Cð PUÞ; 2340 W. L. Pearnet al.
where Z is the standard normal distribution N(0, 1). Similarly, for normally distrib-uted processes with one-sided specification limit LSL, the process yield P(X>LSL) can be obtained as follows:
P X 3 < LSL 3 ¼P 1 3Z < CPL ¼P Z > 3Cð PLÞ ¼F 3Cð PLÞ:
Therefore, the corresponding non-conforming units in parts per million (NCPPM) for a well-controlled normally distributed process can be calculated, exactly, as NCPPM ¼106½1 Fð3CIÞ. Table 3 shows various CPU and CPL
values and the corresponding NCPPM. Consequently, the production yield for usual existing processes should target no more than 88 NCPPM, noting that NCPPM 200 is the common standard used in most electronic industries for pro-ducts with two-sided specifications. The production yield for newly set-up processes on safety, strength or with critical parameters, however, should target no more than 0.8 NCPPM, a more stringent requirement set for possible mean shift or variation change.
Quality condition CIvalues
Inadequate CI<1.00
Marginally capable 1.00 CI<1.33
Satisfactory 1.33 CI<1.67
Excellent 1.67 CI<2.00
Super 2.00 CI
Table 1. Some commonly used capability requirement and
quality conditions.
Index value Production process types
1.25 Existing processes
1.45 New processes, orexisting processes on safety,
strength, or critical parameters
1.6 New processes on safety, strength, or critical
parameters
Table 2. Some minimum capability requirements of CPUand
CPLfor new and special processes.
CPUor CPL NCPPM CPUor CPL NCPPM
1.00 1349.90 1.45 6.81
1.15 280.29 1.60 0.7933
1.25 88.42 1.67 0.2722
1.33 33.04 2.00 0.0010
Table 3. Various CPUor CPLvalues and the corresponding
NCPPM.
2341 Lower confidence bounds for CPUand CPL
3. Estimating CPUand CPLbased on multiple samples
To estimate the indices CPUand CPLin the presence of single samples, Chou and
Owen (1989) considered the following natural estimators of CPUand CPL:
^ C CPU¼ USL X 3S , ^CCPL¼ X LSL 3S ; where n is the sample size, X ¼Pni¼1xi=n, and S ¼ ðn 1Þ1
Pn
i¼1ðxiXÞ
2
h i1=2
are conventional estimators of and , which may be obtained from a process that is demonstrably stable (in control). Chou and Owen (1989) showed that under normality assumption, the estimators 3pffiffiffinCC^PUand 3
ffiffiffi n p ^
C
CPL are distributed as
tðn 1, UÞand tðn 1, LÞ respectively, a non-central t distribution with n 1
degrees of freedom and non-centrality parameters U¼3
ffiffiffi n p ^ C CPUand L¼
3pffiffiffinCC^PL. However, both estimators are biased. Pearn and Chen (2002) considered
the indices CPU and CPL and obtained their UMVUEs. Lin and Pearn (2002)
developed efficient SAS/Maple computer programs for calculating the critical values and the p values using those UMVUEs for capability testing. Pearn and Shu (2002) proposed an algorithm for calculating the exact LCBs for CPUand CPL.
Kirmani et al. (1991) indicated that a common practice of the process capability estimation in most manufacturing industries is first to implement a daily-based data collection program for monitoring/controlling the process stability, then to analyse the past ‘in control’ data. Following the designated sampling plan, multiple samples of ms groups each of with sizes ni, ðxi1, xi2, . . . , xiniÞ, are chosen randomly from a stable process which follows a normal distribution N(, 2) for i ¼ 1, 2, . . . , ms.
We consider the following natural estimators of CPU and CPL. Let Xi¼
Pni
j¼1xij=ni and [Si¼ ðni1Þ
1Pni
j¼1ðxijXiÞ21=2 be the ith sample mean and
the sample standard deviation, respectively. Then, X ¼Pms
i¼1Xi=ms and
S2p¼
Pms
i¼1ðni1Þ Si2=
Pms
i¼1ðni1Þ are the unbiased estimators of and 2
, respec-tively, and the estimators of CPUand CPLcan be written as:
~ C CM PU¼ bNm sðUSL XÞ 3Sp , ~CC M PL¼ bNm sðX LSLÞ 3Sp : where N ¼Pms
i¼1niand bNms is the correction factor defined as:
bNms ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 N ms s G½ðN msÞ=2 G½ðN ms1Þ=2 :
Pearn et al. (2002) showed that if the process follows the normal distribution N(, 2), then the estimators ð3pffiffiffiffiN=bNmsÞ ~CC
M PUand ð3 ffiffiffiffi N p =bNmsÞ ~CC M PLare distributed
as the non-central t distribution with N ms degrees of freedom and non-central
parameters U¼3 ffiffiffiffi N p CPUand L¼3 ffiffiffiffi N p
CPL, respectively. The rth moment (about
zero) can be obtained as (1). Pearn et al. (2003) showed that for a fixed total number of observations, with ðN ms1Þ 3, if the number of samples ms1 > ms2, then Varð ~CCMPUÞm
s1>Varð ~CC
M
PUÞms2. Further, since both estimators depend only on the
suffi-cient and complete statistics (X, S2P) of (, 2), then ~CCMPUand ~CCMPL are UMVUEs of
CPUand CPL, respectively, where ZU¼
ffiffiffiffi N p ðUSL XÞ=: E ~hCCMPUir¼ðG ðN m½ sÞ=2Þ r1 G N m½ð srÞ=2 3pffiffiffiffiN r G ðN m½ s1Þ=2 ð Þr E Zð UÞ r : ð1Þ 2342 W. L. Pearnet al.
The probability density function (PDF) and cumulative density function (CDF) of UMVUEs of CPUand CPLcan be easily attained as (2) and (3), respectively, where
v ¼ N ms: f ðxÞ ¼ 3 ffiffiffiffi N p vðv=2Þe2=2 bvpffiffiffiðv=2Þðv þ9b2v x2NÞðvþ1Þ=2 X1 j¼0 2Gðv þ j þ 1Þ=2Þ j! ! 2x2 b2vð9NÞ1v þ x2 !j=2 : ð2Þ Fðt0Þ ¼ Zt0 1 3pffiffiffiffiNvðv=2Þe2=2 bv ffiffiffi p v=2 ð Þ v þ9b2v x2N ðvþ1Þ=2 X 1 j¼0 2Gðv þ j þ 1Þ=2Þ j! ! 2x2 b2vð9NÞ1v þ x2 !j=2 dx: ð3Þ
In the following, it is also shown that both UMVUEs are consistent and asymptotically efficient. Let !L and !P denote convergence in distribution and convergence in probability, respectively. The proofs of Lemmas 1 and 2 and Theorem are shown in appendix 1.
Lemma 1: Define Mk¼E(x )kas the kth central moment. If M4exists, then as
N ! 1,pffiffiffiffiN X , S2P2 !L Nð0, Þ, where ¼ 2 M 3 M3 M44 :
Lemma 2: Let =ðnÞ be a sequence of random vectors and b a fixed vector such that pffiffiffiffiffinð=ðn
Þ bÞ has a limiting distribution N(0, T) as n ! 1. Let f ð=Þ be a vector-valued function of = such that each component fjð=Þ has a non-zero differ-ential at = ¼ b, and let @f jð=Þ=@==¼b
be the (i, j )th component of Fb. Then,
ffiffiffiffiffi n
p
f ð=ðnÞÞ f ðbÞ
½ has a limiting distribution N(0, FbTF0bÞ.
Theorem: If the process characteristic follows a normal distribution, then (a) ~CCMPUis consistent.
(b) pffiffiffiffiN CC~MPUCPU
converges to N(0,19þC2PU=2) in distribution.
(c) ~CCMPU is asymptotically efficient.
4. Lower confidence bounds for CPUand CPL
For cases with a single sample, Pearn and Shu (2002) established the LCBs on CPU and CPL based on the UMVUEs of CPU and CPL. For cases with multiple
samples, ~CCM
PU and ~CCMPL are used, the UMVUEs of CPU and CPL, which is
consis-tent and asymptotically efficient, to obtain the LCB on CPU and CPL. Let USL ¼
X þ kUSP and LSL ¼ X kLSP so that kU¼3 ~CCMPU=bNms and kL¼3 ~CC
M
PU=bNms.
A 100 % LCB CMU for CPUsatisfies Pr CPUCMU
¼. It can be written as: Pr USL 3 C M U ¼Pr X þ kUSP 3 C M U ! Pr ¼ Z 3 ffiffiffiffi N p CMU Sp= kUpffiffiffiffiN ! ¼Pr Z 3 ffiffiffiffi N p CMU Sp= 3 ~CC M PU bNms ffiffiffiffi N p ! ¼PrðtðN ms, UÞ tUÞ ¼; 2343 Lower confidence bounds for CPUand CPL
and Pr(t(N ms, U) tU) ¼ 1 . Similarly, a 100 % LCB CML for CPL satisfies
Pr CPLCML
¼. It can be shown that Pr(t(N ms, L) tL) ¼ , where
Z N(0, 1), tL¼kL ffiffiffiffi N p , U¼ 3 ffiffiffiffi N p CMU and L¼3 ffiffiffiffi N p
CML: Thus, to obtain the
LCB, one can proceed as follows: 4.1. Algorithm for the LCB
To compute the LCBs, CMU, an algorithm called the LCB is developed. An
auxiliary function for evaluating CMU, the cumulative distribution function of the
non-central chi-square distribution (Lenth 1989) is required. The step sizes for numerical computation are t1and t2, where 0<t2<t10.1.
Step1. Read the sample data (x1, x2, . . . , xN), USL (or LSL), ms, and .
Step2. Calculate Xi¼ Xni j¼1xij=niSi¼ ½ðni1Þ 1Xni j¼1ðxijXiÞ 2 1=2 Xi¼Xms i¼1Xi=ms, S 2 p¼ Xms i¼1ðni1ÞS 2 i . Xms i¼1ðni1Þ: bNmsffi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN ms1Þ=ðN msÞ p ð1 1=ð4ðN ms1ÞÞ þ1=ð32ðN ms1Þ2Þ þ5=ð128ðN ms1Þ3Þ, and ~CCMPU¼bNmsðUSL XÞ=3SP: Step3. Compute an initial guess for CMU.
For i ¼ 1, 2, . . . , evaluate CMUðiÞ ¼ it1, tU¼3
ffiffiffiffi N p ~ C CM PU=bNms, and U¼3 ffiffiffiffi N p CMUðiÞ, until (t(N ms, U) tU) 0.
Step4. Find the LCB CMU on CPUthrough numerical iterations.
For j ¼ 0, 1, . . . , evaluate CMUðjÞ ¼ CMUðiÞ jt2 and U¼3
ffiffiffiffi N p
CMUðjÞ
until (t(N ms, U) tU) 0. Set CMU ¼CMUðjÞ.
Step 5. Output the conclusive message, ‘The true value of the process capability CPU
is no less than the CMU with 100g% level of confidence.’
We implement the algorithm and develop a Fortran program to compute the LCBs (see appendix 2). Tables 4–6 summarize the LCBs CMU on CPUand CPL for
~ C CM
PUðor ~CCMPLÞ ¼0.8(0.1)3.0, for the total number of observations N ¼ 100, 150 and
200 with various msand ¼ 0.95. The results indicate that the LCB CMU decreases as
msincreases and increases as N increases in all cases. Figures 1 and 2 plot the curves
of the LCB CMU on CPU and CPL with the sample sizes N ¼ 100 and 200 versus
various subgroups ms, respectively, for ~CCMPUðor ~CCMPLÞ ¼0.8, 1.2, 1.5, 2.0, 2.5, 3.0
with confidence level ¼ 0.95. For bottom curve 1, ~CCM
PU ðor ~CCMPLÞ ¼0.8; for bottom
curve 2, ~CCM
PUðor ~CCMPLÞ ¼1.2; for bottom curve 3, ~CCMPUðor ~CCMPLÞ ¼1.5; for top curve 3,
~ C CM
PUðor ~CCMPLÞ ¼2.0; for top curve 2, CC~MPUðor ~CCMPLÞ ¼2.5; and for top curve 1,
~ C CM
PUðor ~CCMPLÞ ¼3.0. For example, if ~CCPUM ðor ~CCMPLÞ ¼1.5 for N ¼ 100 with ms¼25,
then from table 4 and figure 1, the LCB CMU¼1.302, and so one can conclude
that CPU(or CPL)>1.302, with 95% confidence.
4.2. Sample size determination
The sample size determination is essential to most factory applications, particu-larly for those implementing a routine-basis data collection plan for monitoring and controlling process quality. It directly relates to the sampling cost of a data collection plan. Extensive calculations are performed for ~CCMPUðor ~CCMPLÞ ¼0.8(0.1)3.0 with the
2344 W. L. Pearnet al.
ms/ ~CC M PU 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 1 0.696 0.786 0.877 0.966 1.056 1.146 1.235 1.325 1.414 1.504 1.593 1.682 1.771 1.861 1.950 2.039 2.128 2.217 2.306 2.395 2.484 2.573 2.662 2 0.696 0.786 0.876 0.966 1.056 1.145 1.235 1.324 1.413 1.503 1.592 1.681 1.770 1.860 1.949 2.038 2.127 2.216 2.305 2.394 2.483 2.572 2.661 3 0.695 0.786 0.876 0.965 1.055 1.145 1.234 1.323 1.413 1.502 1.591 1.680 1.769 1.858 1.947 2.036 2.125 2.214 2.303 2.392 2.481 2.570 2.659 4 0.695 0.785 0.875 0.965 1.054 1.144 1.233 1.323 1.412 1.501 1.590 1.679 1.768 1.857 1.946 2.035 2.124 2.213 2.302 2.391 2.480 2.569 2.658 5 0.695 0.785 0.875 0.964 1.054 1.143 1.233 1.322 1.411 1.500 1.589 1.678 1.767 1.856 1.945 2.034 2.123 2.212 2.301 2.389 2.478 2.567 2.656 10 0.693 0.782 0.872 0.961 1.050 1.140 1.229 1.318 1.406 1.495 1.584 1.673 1.761 1.850 1.939 2.027 2.116 2.204 2.293 2.382 2.470 2.559 2.647 15 0.690 0.780 0.869 0.958 1.047 1.136 1.224 1.313 1.401 1.490 1.578 1.667 1.755 1.844 1.932 2.020 2.108 2.197 2.285 2.373 2.461 2.550 2.638 20 0.688 0.777 0.866 0.954 1.043 1.131 1.220 1.308 1.396 1.484 1.572 1.660 1.748 1.836 1.924 2.012 2.100 2.188 2.276 2.364 2.452 2.540 2.627 25 0.685 0.774 0.862 0.951 1.039 1.127 1.215 1.302 1.390 1.478 1.566 1.653 1.741 1.829 1.916 2.004 2.091 2.179 2.266 2.354 2.441 2.529 2.616 30 0.682 0.770 0.858 0.946 1.034 1.122 1.209 1.296 1.384 1.471 1.558 1.646 1.733 1.820 1.907 1.994 2.081 2.168 2.255 2.342 2.429 2.517 2.604 35 0.679 0.767 0.854 0.941 1.029 1.116 1.203 1.290 1.377 1.463 1.550 1.637 1.724 1.810 1.897 1.984 2.070 2.157 2.243 2.330 2.417 2.503 2.590 40 0.675 0.762 0.849 0.936 1.023 1.109 1.196 1.282 1.368 1.455 1.541 1.627 1.713 1.799 1.886 1.972 2.058 2.144 2.230 2.316 2.402 2.488 2.574 45 0.671 0.758 0.844 0.930 1.016 1.102 1.188 1.274 1.359 1.445 1.531 1.616 1.702 1.787 1.873 1.958 2.044 2.129 2.215 2.300 2.386 2.471 2.557 50 0.666 0.752 0.838 0.923 1.008 1.094 1.179 1.264 1.349 1.434 1.519 1.604 1.689 1.773 1.858 1.943 2.028 2.113 2.198 2.282 2.367 2.452 2.537 55 0.660 0.745 0.830 0.915 0.999 1.084 1.168 1.252 1.337 1.421 1.505 1.589 1.673 1.757 1.841 1.926 2.010 2.094 2.178 2.262 2.346 2.430 2.514 60 0.654 0.738 0.822 0.905 0.989 1.072 1.156 1.239 1.322 1.406 1.489 1.572 1.655 1.739 1.822 1.905 1.988 2.071 2.154 2.237 2.320 2.403 2.486 65 0.646 0.728 0.811 0.894 0.976 1.059 1.141 1.223 1.305 1.387 1.470 1.552 1.634 1.716 1.798 1.880 1.962 2.044 2.126 2.208 2.290 2.372 2.454 70 0.636 0.717 0.798 0.880 0.961 1.042 1.123 1.203 1.284 1.365 1.446 1.527 1.607 1.688 1.769 1.849 1.930 2.011 2.091 2.172 2.253 2.333 2.414 75 0.623 0.702 0.782 0.861 0.941 1.020 1.099 1.178 1.257 1.336 1.416 1.495 1.574 1.653 1.732 1.811 1.889 1.968 2.047 2.126 2.205 2.284 2.363 80 0.605 0.683 0.760 0.837 0.914 0.991 1.068 1.145 1.221 1.298 1.375 1.452 1.528 1.605 1.682 1.759 1.835 1.912 1.989 2.065 2.142 2.219 2.295 85 0.580 0.654 0.728 0.802 0.876 0.949 1.023 1.096 1.170 1.243 1.317 1.390 1.464 1.537 1.611 1.684 1.757 1.831 1.904 1.978 2.051 2.124 2.198 90 0.539 0.608 0.676 0.745 0.813 0.881 0.949 1.018 1.086 1.154 1.222 1.290 1.358 1.427 1.495 1.563 1.631 1.699 1.767 1.835 1.903 1.971 2.039 95 0.452 0.509 0.566 0.624 0.681 0.738 0.795 0.852 0.909 0.966 1.023 1.080 1.137 1.194 1.251 1.308 1.365 1.422 1.479 1.536 1.593 1.650 1.707
Table 4. Lower confidence bounds for N ¼ 100, with ms¼1(1)5, 10(5)95, ¼ 0.95, and ~CC
M PU¼0.8(0.1)3.0. 2345 Lower confidence bounds for C PU an d C PL
ms/ ~CC M PU 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 1 0.715 0.807 0.899 0.990 1.082 1.173 1.265 1.356 1.447 1.539 1.630 1.721 1.812 1.903 1.994 2.085 2.176 2.267 2.358 2.482 2.540 2.631 2.722 2 0.714 0.806 0.898 0.990 1.081 1.173 1.264 1.356 1.447 1.538 1.629 1.720 1.811 1.903 1.994 2.085 2.176 2.267 2.358 2.482 2.540 2.631 2.722 3 0.714 0.806 0.898 0.990 1.081 1.173 1.264 1.355 1.446 1.538 1.629 1.720 1.811 1.902 1.993 2.084 2.175 2.266 2.357 2.481 2.539 2.630 2.721 4 0.714 0.806 0.898 0.989 1.081 1.172 1.264 1.355 1.446 1.537 1.628 1.719 1.810 1.901 1.992 2.083 2.174 2.265 2.356 2.481 2.538 2.629 2.720 5 0.714 0.806 0.897 0.989 1.080 1.172 1.263 1.354 1.445 1.537 1.628 1.719 1.810 1.901 1.992 2.083 2.174 2.264 2.355 2.480 2.537 2.628 2.719 10 0.713 0.804 0.896 0.987 1.079 1.170 1.261 1.352 1.443 1.534 1.625 1.716 1.807 1.897 1.988 2.079 2.170 2.261 2.351 2.477 2.533 2.624 2.714 15 0.712 0.803 0.895 0.986 1.077 1.168 1.259 1.350 1.441 1.531 1.622 1.713 1.803 1.894 1.985 2.075 2.166 2.257 2.347 2.474 2.528 2.619 2.709 20 0.710 0.802 0.893 0.984 1.075 1.166 1.257 1.347 1.438 1.528 1.619 1.710 1.800 1.891 1.981 2.071 2.162 2.252 2.343 2.472 2.523 2.614 2.704 25 0.709 0.800 0.891 0.982 1.073 1.164 1.254 1.345 1.435 1.525 1.616 1.706 1.796 1.887 1.977 2.067 2.157 2.248 2.338 2.469 2.518 2.608 2.699 30 0.708 0.799 0.889 0.980 1.071 1.161 1.251 1.342 1.432 1.522 1.612 1.703 1.793 1.883 1.973 2.063 2.153 2.243 2.333 2.465 2.513 2.603 2.693 35 0.706 0.797 0.888 0.978 1.068 1.159 1.249 1.339 1.429 1.519 1.609 1.699 1.789 1.878 1.968 2.058 2.148 2.238 2.328 2.462 2.507 2.597 2.687 40 0.705 0.795 0.886 0.976 1.066 1.156 1.246 1.336 1.425 1.515 1.605 1.695 1.784 1.874 1.963 2.053 2.143 2.232 2.322 2.459 2.501 2.591 2.680 45 0.703 0.793 0.883 0.973 1.063 1.153 1.243 1.332 1.422 1.511 1.601 1.690 1.780 1.869 1.958 2.048 2.137 2.226 2.316 2.455 2.494 2.584 2.673 50 0.701 0.791 0.881 0.971 1.060 1.150 1.239 1.328 1.418 1.507 1.596 1.685 1.775 1.864 1.953 2.042 2.131 2.220 2.309 2.451 2.487 2.576 2.665 60 0.697 0.787 0.876 0.965 1.054 1.143 1.231 1.320 1.409 1.498 1.586 1.675 1.763 1.852 1.941 2.029 2.118 2.206 2.295 2.443 2.472 2.560 2.649 70 0.692 0.781 0.869 0.958 1.046 1.134 1.222 1.310 1.399 1.486 1.574 1.662 1.750 1.838 1.926 2.014 2.102 2.190 2.277 2.434 2.453 2.541 2.629 80 0.686 0.774 0.862 0.949 1.037 1.124 1.212 1.299 1.386 1.473 1.560 1.647 1.735 1.822 1.909 1.996 2.083 2.170 2.257 2.424 2.431 2.518 2.605 90 0.679 0.766 0.853 0.939 1.025 1.112 1.198 1.284 1.370 1.457 1.543 1.629 1.715 1.801 1.887 1.973 2.059 2.145 2.231 2.412 2.403 2.489 2.575 100 0.670 0.755 0.841 0.926 1.011 1.096 1.181 1.266 1.351 1.436 1.520 1.605 1.690 1.775 1.860 1.945 2.029 2.114 2.199 2.399 2.368 2.453 2.538 110 0.657 0.741 0.824 0.908 0.991 1.074 1.158 1.241 1.324 1.407 1.490 1.574 1.657 1.740 1.823 1.906 1.989 2.072 2.155 2.384 2.321 2.404 2.487 120 0.638 0.720 0.801 0.882 0.963 1.043 1.124 1.205 1.286 1.366 1.447 1.528 1.609 1.689 1.770 1.850 1.931 2.012 2.092 2.366 2.254 2.334 2.415 130 0.608 0.685 0.762 0.839 0.916 0.992 1.069 1.146 1.223 1.299 1.376 1.453 1.529 1.606 1.683 1.759 1.836 1.913 1.989 2.066 2.143 2.219 2.296 140 0.541 0.609 0.678 0.746 0.814 0.882 0.950 1.019 1.087 1.155 1.223 1.291 1.359 1.427 1.495 1.563 1.632 1.700 1.768 1.836 1.904 1.972 2.040
Table 5. Lower confidence bounds for N ¼ 150, with ms¼1(1)5, 10(5)50, 60(10)140, ¼ 0.95, and ~CC
M PU¼0.8(0.1)3.0. 2346 W. L. Pearn et al.
ms/ ~CC M PU 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 1 0.726 0.819 0.912 1.005 1.097 1.190 1.282 1.375 1.467 1.560 1.652 1.744 1.837 1.929 2.021 2.113 2.205 2.298 2.390 2.482 2.574 2.666 2.759 2 0.726 0.819 0.912 1.004 1.097 1.190 1.282 1.375 1.467 1.559 1.652 1.744 1.836 1.928 2.021 2.113 2.205 2.297 2.389 2.482 2.574 2.666 2.758 3 0.725 0.819 0.911 1.004 1.097 1.189 1.282 1.374 1.467 1.559 1.651 1.744 1.836 1.928 2.020 2.112 2.205 2.297 2.389 2.481 2.573 2.665 2.757 4 0.725 0.818 0.911 1.004 1.097 1.189 1.282 1.374 1.466 1.559 1.651 1.743 1.835 1.928 2.020 2.112 2.204 2.296 2.388 2.481 2.573 2.665 2.757 5 0.725 0.818 0.911 1.004 1.096 1.189 1.281 1.374 1.466 1.558 1.651 1.743 1.835 1.927 2.019 2.111 2.204 2.296 2.388 2.480 2.572 2.664 2.756 10 0.725 0.817 0.910 1.003 1.095 1.188 1.280 1.372 1.464 1.557 1.649 1.741 1.833 1.925 2.017 2.109 2.201 2.293 2.385 2.477 2.569 2.661 2.753 15 0.724 0.817 0.909 1.002 1.094 1.186 1.279 1.371 1.463 1.555 1.647 1.739 1.831 1.923 2.015 2.107 2.199 2.291 2.383 2.474 2.566 2.658 2.750 20 0.723 0.816 0.908 1.001 1.093 1.185 1.277 1.369 1.461 1.553 1.645 1.737 1.829 1.921 2.013 2.104 2.196 2.288 2.380 2.472 2.563 2.655 2.747 25 0.722 0.815 0.907 0.999 1.092 1.184 1.276 1.368 1.459 1.551 1.643 1.735 1.827 1.918 2.010 2.102 2.194 2.285 2.377 2.469 2.560 2.652 2.744 30 0.721 0.814 0.906 0.998 1.090 1.182 1.274 1.366 1.458 1.549 1.641 1.733 1.824 1.916 2.008 2.099 2.191 2.282 2.374 2.465 2.557 2.648 2.740 35 0.721 0.813 0.905 0.997 1.089 1.181 1.272 1.364 1.456 1.547 1.639 1.730 1.822 1.913 2.005 2.096 2.188 2.279 2.371 2.462 2.554 2.645 2.736 40 0.720 0.812 0.904 0.996 1.087 1.179 1.271 1.362 1.454 1.545 1.637 1.728 1.819 1.911 2.002 2.093 2.185 2.276 2.367 2.459 2.550 2.641 2.732 45 0.719 0.811 0.903 0.994 1.086 1.177 1.269 1.360 1.452 1.543 1.634 1.725 1.817 1.908 1.999 2.090 2.182 2.273 2.364 2.455 2.546 2.637 2.728 50 0.718 0.810 0.901 0.993 1.084 1.176 1.267 1.358 1.449 1.541 1.632 1.723 1.814 1.905 1.996 2.087 2.178 2.269 2.360 2.451 2.542 2.633 2.724 60 0.716 0.807 0.898 0.990 1.081 1.172 1.263 1.354 1.445 1.536 1.626 1.717 1.808 1.899 1.989 2.080 2.171 2.262 2.352 2.443 2.534 2.624 2.715 70 0.713 0.804 0.895 0.986 1.077 1.168 1.258 1.349 1.439 1.530 1.620 1.711 1.801 1.892 1.982 2.072 2.163 2.253 2.344 2.434 2.524 2.615 2.705 80 0.710 0.801 0.892 0.982 1.073 1.163 1.253 1.343 1.433 1.524 1.614 1.704 1.794 1.884 1.974 2.064 2.154 2.244 2.334 2.424 2.514 2.604 2.694 90 0.707 0.798 0.888 0.978 1.068 1.157 1.247 1.337 1.427 1.516 1.606 1.696 1.785 1.875 1.965 2.054 2.144 2.233 2.323 2.412 2.502 2.591 2.681 100 0.704 0.793 0.883 0.973 1.062 1.151 1.241 1.330 1.419 1.508 1.597 1.687 1.776 1.865 1.954 2.043 2.132 2.221 2.310 2.399 2.488 2.577 2.666 110 0.700 0.789 0.878 0.967 1.055 1.144 1.233 1.322 1.410 1.499 1.587 1.676 1.764 1.853 1.941 2.030 2.118 2.207 2.295 2.384 2.472 2.561 2.649 120 0.695 0.783 0.871 0.960 1.048 1.136 1.224 1.312 1.400 1.488 1.576 1.663 1.751 1.839 1.927 2.015 2.103 2.190 2.278 2.366 2.454 2.542 2.629 150 0.672 0.757 0.842 0.927 1.012 1.097 1.182 1.267 1.352 1.437 1.521 1.606 1.691 1.776 1.860 1.945 2.030 2.115 2.199 2.284 2.369 2.454 2.538 180 0.609 0.686 0.763 0.840 0.916 0.993 1.070 1.147 1.223 1.300 1.377 1.453 1.530 1.607 1.683 1.760 1.836 1.913 1.990 2.066 2.143 2.220 2.296
Table 6. Lower confidence bounds for N ¼ 200, with ms¼1(1)5, 10(5)50, 60(10)120, 150, 180, ¼ 0.95, and ~CC
M PU¼0.8(0.1)3.0. 2347 Lower confidence bounds for C PU an d C PL
sample sizes N ¼ 20(10)220, ms¼10(10)120 with confidence level ¼ 0.90, 0.95,
0.975 and 0.99, and analyse the estimation precision RPU¼CMU= ~CCMPU. The parameter
values investigated cover a wide range of applications. The results indicate that in all cases investigated, the estimating precision RPU decreases as ms increases, and
increases as N and ~CCM
PUðor ~CCMPLÞincreases. Hence, for practical purpose, one can take
the minimum among those to obtain a quick reference on the minimum CPUand CPL
without further calculations.
Table 7 shows the sample size, N, and number of samples, ms, required and the
corresponding minimal (conservative) precision of the estimation RPU. For example,
¼0.95, N ¼ 150, ms¼30 gives RPU¼0.885. Thus, the true value of CPU is no
less than ~CCMPUðor ~CCMPLÞ 0.885. However, if RPU¼0.885 is chosen, then one can
determine N ¼ 140 with ms¼10, or N ¼ 150 with ms¼30, or N ¼ 160 with ms¼40.
Similarly, if RPU¼0.9 is chosen, then one can determine N ¼ 180 with ms¼10, or
N ¼190 with ms¼20, or N ¼ 200 with ms¼40, or N ¼ 210 with ms¼50, or N ¼ 220
with ms¼60, depending on which sampling plan is more appropriate to the
application. 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 0 10 20 30 40 50 60 70 80 90 ms LCB
Figure 1. Lower confidence bound CMU with N ¼ 100 versus subgroup msfor
~ C
CMPUðor ~CCMPLÞ ¼0.8, 1.2, 1.5, 2.0, 2.5, 3.0 (from bottom to top).
0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 ms LCB
Figure 2. Lower confidence bound CMU with N ¼ 200 versus subgroup msfor
~ C
CMPUðor ~CCMPLÞ ¼0.8, 1.2, 1.5, 2.0, 2.5, 3.0 (from bottom to top).
2348 W. L. Pearnet al.
5. HSBA production yield assurance
The product investigated is a monolithic open-loop unity-gain buffer amplifier with a high symmetrical slew rate of up to 3600 V/ms and a very wide bandwidth of 320 MHz at 5 Vp–p output swing called the high-speed buffer amplifier (HSBA). A complementary bipolar IC process is used that incorporates pn-junction-isolated high-frequency NPN (a layer of P-doped semiconductor between two N-doped layers) and PNP (a layer of N-doped semiconductor between two P-doped layers) transistors to achieve high-frequency performance previously unattainable with con-ventional integrated circuit technology. The unique design offers a high-performance alternative to expensive discrete or hybrid solutions. The HSBA features low quies-cent currents, low input bias current, small signal delay time and phase shift, and low differential gain and phase errors.
The two types of HSBA with a 3 or 6 mA quiescent current are well suited for the operation between high-frequency processing stages. The HSBA demonstrates out-standing performance even in feedback loops of wide-band amplifiers or phase-locked loop systems. The type II HSBA, with 6 mA quiescent current and, therefore, a lower output impedance, can easily drive 50 inputs or 75 systems and cables. The broad range of analogue and digital applications extends from the decoupling of signal processing stages, impedance transformation and input amplifiers for radio frequency (RF) equipment and automatic test equipment (ATE) systems to video systems, distribution fields, communications systems and output drivers for graphic cards. The HSBA is available in an industry standard pin-out SO–8 package (figure 3). The simplified circuit diagram and quiescent current versus temperature for HSBA are shown in figure 4.
The quiescent current is an essential product characteristic, which has a signifi-cant impact on product quality. For the particular type II HSBA, the USL placed on quiescent current is set to 6 mA. A previous study verified that the measurement
N/ms 10 20 30 40 50 60 70 80 90 100 110 120 20 0.649 — — — — — — — — — — — 30 0.737 0.659 — — — — — — — — — — 40 0.779 0.744 0.664 — — — — — — — — — 50 0.805 0.784 0.748 0.668 — — — — — — — — 60 0.824 0.809 0.787 0.751 0.670 — — — — — — — 70 0.838 0.827 0.812 0.790 0.753 0.671 — — — — — — 80 0.849 0.840 0.829 0.814 0.792 0.754 0.672 — — — — — 90 0.858 0.851 0.842 0.831 0.816 0.793 0.756 0.673 — — — — 100 0.866 0.860 0.853 0.844 0.833 0.817 0.794 0.757 0.674 — — — 110 0.872 0.867 0.861 0.854 0.845 0.834 0.818 0.795 0.757 0.674 — — 120 0.878 0.873 0.868 0.863 0.855 0.846 0.835 0.819 0.796 0.758 0.675 — 130 0.883 0.879 0.875 0.870 0.864 0.856 0.847 0.836 0.820 0.797 0.759 0.675 140 0.887 0.884 0.880 0.876 0.871 0.864 0.857 0.848 0.837 0.821 0.798 0.759 150 0.891 0.888 0.885 0.881 0.877 0.871 0.865 0.858 0.849 0.837 0.821 0.798 160 0.894 0.892 0.889 0.885 0.882 0.877 0.872 0.866 0.859 0.850 0.838 0.822 170 0.898 0.895 0.893 0.890 0.886 0.882 0.878 0.873 0.867 0.859 0.850 0.838 180 0.901 0.898 0.896 0.893 0.890 0.887 0.883 0.879 0.873 0.867 0.860 0.851 190 0.903 0.901 0.899 0.897 0.894 0.891 0.887 0.884 0.879 0.874 0.868 0.860 200 0.906 0.904 0.902 0.900 0.897 0.894 0.891 0.888 0.884 0.880 0.874 0.868 210 0.908 0.906 0.904 0.902 0.900 0.898 0.895 0.892 0.888 0.885 0.880 0.875 220 0.910 0.909 0.907 0.905 0.903 0.901 0.898 0.895 0.892 0.889 0.885 0.880
Table 7. Total number of sample observations, N (left), number of samples, ms(top), and
precision of estimation with ¼ 0.95.
2349 Lower confidence bounds for CPUand CPL
system used introduces an ignorable measurement error. Sample data are collected from 20 subgroups of five observations each by measuring quiescent currents of HSBAs (table 8). Figures 5 and 6 show the histogram and normal probability plot of the 100 HSBA data with no observations outside the USL, and both figures show that the sample data appear to be approximately normal. A Shapiro–Wilk test is also applied to verify the normality assumption. Figure 7 shows the corresponding X and S control charts and indicates the process under ‘in control’. Thus, it is concluded that the sample data can be regarded as taken from a stably normal process. To obtain the LCB on CPU, the Fortran program shown in appendix
1 is executed. The program reads the sample data file and the input of the sample size N ¼ 100, USL ¼ 6 mA, ms¼20 and confidence level ¼ 0.95, then outputs the
overall sample mean X ¼ 5.610, and the pooled sample standard deviation, SP¼0.082, the estimator CC~MPU¼1.5712 and the LCB C
M
U¼1.3707. The actual
program execution output is shown in appendix 3. It is therefore concluded that the true process capability CPUis no less than 1.3707 with a 95% level of confidence.
Hence, the number of product items conforming to the manufacturing specifications
Figure 3. High-speed buffer amplifier (HSBA).
Figure 4. Simplified circuit diagram and quiescent current versus temperature for the
HSBA.
2350 W. L. Pearnet al.
are assured to be no greater than 20 parts per million. Equivalently, the production yield is assured to be no less than 99.9980%. These product items conformed to the manufacturing specifications (with the quiescent current, USL, not exceeding 6 mA, which obviously satisfies the preset quality requirements) and are considered as reliable products.
6. Conclusions
Process capability indices CPU and CPL have been widely used in the
manu-facturing industry to provide quantitative measures on process performance,
Figure 5. Histogram of 100 HSBA data.
Subgroups Observations (mA) X S
1 5.90 5.73 5.72 5.54 5.64 5.706 0.13259 2 5.68 5.51 5.62 5.66 5.49 5.592 0.087006 3 5.52 5.67 5.60 5.64 5.72 5.63 0.075498 4 5.64 5.53 5.66 5.59 5.53 5.59 0.060415 5 5.78 5.61 5.52 5.65 5.74 5.66 0.103682 6 5.65 5.67 5.61 5.59 5.67 5.638 0.036332 7 5.58 5.61 5.56 5.54 5.58 5.574 0.026077 8 5.50 5.62 5.61 5.60 5.41 5.548 0.09094 9 5.56 5.62 5.63 5.62 5.60 5.606 0.027928 10 5.46 5.63 5.59 5.64 5.65 5.594 0.078294 11 5.59 5.63 5.57 5.57 5.57 5.586 0.026077 12 5.52 5.62 5.71 5.55 5.56 5.592 0.075299 13 5.66 5.60 5.73 5.46 5.51 5.592 0.109407 14 5.54 5.79 5.64 5.56 5.60 5.626 0.099398 15 5.57 5.68 5.60 5.67 5.66 5.636 0.04827 16 5.58 5.59 5.61 5.53 5.58 5.578 0.029496 17 5.59 5.51 5.68 5.50 5.54 5.564 0.073689 18 5.50 5.69 5.54 5.45 5.67 5.57 0.105594 19 5.72 5.66 5.61 5.53 5.54 5.612 0.080436 20 5.70 5.70 5.55 5.67 5.92 5.708 0.133679
Table 8. Twenty groups of five observations (100 sample data).
2351 Lower confidence bounds for CPUand CPL
particularly for processes with one-sided specification limits. Statistical properties of the estimators of CPUand CPL have been investigated extensively but are restricted
to cases with single samples. The indices CPU and CPL provide yield assurance of
production. The present paper considered the UMVUE of CPUand CPLfor cases of
multiple samples and showed that this UMVUE is consistent and asymptotically efficient. An efficient algorithm/program is presented to compute the LCBs on CPU
and CPL, which presents a measure on the minimum capability of the process. The
paper also provided tables for engineers/practitioners to use for in-plant factory
Figure 6. Normal probability plot for 100 HSBA data.
5. 50 5. 55 5. 60 5. 65 5. 70 5. 75 X=5.610 UCL=5.717 LCL=5.503 0 2 4 6 8 10 12 14 16 18 20 0 0. 05 0. 10 0. 15 0. 20 X -B a r S t d D e v Subgroup S=.075 UCL=.157 LCL=0 3 Limits For n=5: Subgroup Sizes: * n=5
Figure 7. Xand S control charts for HSBA data.
2352 W. L. Pearnet al.
applications. An example of HSBA is presented to illustrate the practicality of the LCB approach to actual data collected from the real-world applications. The imple-mentation of the existing statistical theory for the process yield assessment makes it possible for the production industry to apply the complicated theoretical results to the factory actual productions.
Acknowledgements
The authors thank the anonymous referees for careful reading of the manuscript and for constructive comments that significantly improved the paper. Research was supported in part by the National Science Council, Taiwan, ROC, Grant No. NSC 91-2218-E-230-005.
Appendix 1
A.1. Proof of Lemma 1
See Serfling (1980, p. 72). For the process following the normal distribution, M3¼0 and M4¼34, hence: ¼ 2 0 0 24 :
A.2. Proof of Lemma 2
See Anderson (1984, Theorem 4.2.3). An estimator ~Nof is said to be consistent
if for all e>0. pðj ~Nj >eÞ ! 0 as N ! 1 for all . A sufficient condition for the
consistency is that Eð ~NÞ ! and Varð ~NÞ !0. Under the regular conditions, the
estimator ~ffiffiffiffi N is said to be asymptotically efficient if ~N is asymptotically normal,
N p
ð ~NEð ~NÞÞ !0, and N Varð ~NEð ~NÞÞ converges to the Cramer–Rao lower
bound (CRLB).
A.3. Proof of Theorem
(a) Using the Chebyshev inequality: p CC~M PUCPU > e <Eð ~CC M PUCPUÞ2 e2 : Since Eð ~CCM
PUCPUÞ2¼Varð ~CCMPUÞ ¼Eð ~CCMPUÞ2C2PU, then by Stirling’s
for-mula, one obtains Eð ~CCM
PUÞ2, which converges to C2PU. Hence, Eð ~CCMPUCPUÞ2
converges to zero. Therefore, ~CCM
PU!
P
CPU and so ~CCMPU is consistent.
(b) Note that the following function is a real-valued function, which is differ-entiable for and 2, thus:
CPUð, 2Þ ¼ USL 3ð2Þ1=2 with @CPU @ ¼ 1 3ð2Þ1=2, @CPU @2 ¼ USL 6ð2Þ3=2 , 2353 Lower confidence bounds for CPUand CPL
If F ¼ ð@CPU=@ @CPU=@2Þ is defined, then by Lemmas 1 and 2, one has
the following. From (a) CC~MPU!P CPU and Slutsky’s theorem (Bain and
Engehardt 1992), it can be shown thatpffiffiffiffiN CC~M
PUCPU !L Nð0, C2PUÞ: ffiffiffiffi N p ~ C CMPUCPU ¼pffiffiffiffiN CC~MPUðX, S2PÞ CPUðu, 2Þ h i !L Nð0, C2PUÞ 2CPU ¼FF0¼1 9 1 þ ðUSL Þ2 22 ! ¼1 9þ C2PU 2 (c) For a normal distribution, the log-maximum likelihood function is:
L ¼lnf f ðx; u, 2Þg ¼ 1 2lnð2
2Þ ðx Þ2
22 :
The information matrix is: I ð, 2Þ ¼ E @ 2 L=@2 @2L=@@2 @2L=@@2 @2L=@ð2Þ2 ¼ 1= 2 0 0 1=22 : Hence, CRLB ¼ 1 NFI 1ðu, 2ÞF0 ¼ 1 9N 1 þ ððUSL Þ2Þ 22 ! ¼ 2 CPU N : From (b), pffiffiffiffiN CC~MPUCPU !L Nð0, C2PUÞ, and therefore ~CC M PU is asymptoti-cally efficient. Appendix 2
! Fortran 90 Program for lower confidence bounds based on multiple samples ! Read the sample data (x1, x2, . . . , xN), USL (or LSL), ms, and g.
REAL, DIMENSION(1:5, 1:20) :: A INTEGER IDF, NOUT
REAL DELTA, P, T, TNIN, b, bf, y, y1, Icpu, Tcpu, mean(20), var(20), std(20) EXTERNAL TNIN, UMACH
A ¼ reshape((/5.8971, 5.7319, 5.7176, 5.5356, 5.6376, & &5.6774, 5.5087, 5.6171, 5.6648, 5.4905, 5.519, 5.6661, 5.5995, 5.6446, 5.716, & &5.641, 5.5331, 5.6639, 5.5858, 5.5306, 5.778, 5.6068, 5.5203, 5.654, 5.743, & &5.6505, 5.6672, 5.6089, 5.5865, 5.674, 5.5834, 5.6064, 5.5552, 5.5417, 5.5772, & &5.499, 5.6211, 5.6087, 5.5965, 5.409, 5.5566, 5.6212, 5.6314, 5.6152, 5.5968, & &5.4637, 5.6288, 5.5889, 5.6412, 5.6509, 5.5927, 5.6276, 5.5715, 5.5726, 5.5675, & &5.519, 5.6199, 5.705, 5.5508, 5.5574, 5.6614, 5.6034, 5.731, 5.4554, 5.5121, & &5.5351, 5.785, 5.6367, 5.5623, 5.6025, 5.5732, 5.6831, 5.6016, 5.6695, 5.6597, & &5.5803, 5.5903, 5.6109, 5.5321, 5.5797, 5.5924, 5.513, 5.6797, 5.5038, 5.5396, & &5.5006, 5.6906, 5.5422, 5.4533, 5.6691, 5.7226, 5.6571, 5.6118, 5.5269, 5.5361, & &5.7045, 5.6978, 5.5483, 5.6685, 5.9185/), (/5, 20/))
PRINT*, ‘Please Enter: Overall Sample Observations, of Subsamples, & &USL, alpha risk (1 r).’
READ*, N, m, USL, alpha
2354 W. L. Pearnet al.
! Calculate Xi Si, XXXX, S2p, bnms, and ~CC M PU. b ¼ 0; bf ¼ 0 b ¼ sqrt((N m 1)/2.0)*(1 1.0/(4*(N m 1)) þ & &1.0/(32*(N m 1)**2) þ 5.0/(128*(N m 1)**3)) bf ¼ sqrt(2.0/(N m))*b DO e ¼ 1, 20
mean(e) ¼ sum(A(:, e))/5 DO ee ¼ 1, 5
var(e) ¼ var(e) þ ((A(ee, e) mean(e))**2) ENDDO
std(e) ¼ sqrt(var(e)/(5.0 1.0)) ENDDO
amean ¼ sum(mean)/20.0
pstd ¼ sqrt(sum(var)/(100.0 20.0))
PRINT*, ‘The Overall Sample Mean ¼ ’, amean
PRINT*, ‘The Pooled Sample Standard Deviation ¼ ’, pstd ecpu ¼ bf*(USL amean)/(3*pstd)
print*, ‘The Estimated Cpu Based on Multiple Samples ¼ ’, ecpu ! Compute an initial guess for CMU.
Do i ¼ 1, 100 Cpu ¼ i*0.1
IDF ¼ 0; DELTA ¼ 0; P ¼ 0; T ¼ 0; y ¼ 0; y1 ¼ 0; Icpu ¼ 0; Tcpu ¼ 0 CALL UMACH (2, NOUT)
IDF ¼ N m
DELTA ¼ 3*sqrt(N/1.0)*Cpu P ¼ 1 alpha
T ¼ 3*sqrt(N/1.0)*eCpu/bf y ¼ TNIN(P, IDF, DELTA) IF ((yc).GE. 0) Then
Icpu ¼ DELTA/(3*sqrt(N/1.0)) EXIT
ENDIF ENDDO
! Find the lower confidence bound CMU on CPUthrough numerical iterations.
Do J ¼ 1, 10000 Tcpu ¼ Icpu 0.0001*J
DELTA1 ¼ 3*sqrt(N/1.0)*TCpu y1 ¼ TNIN(P, IDF, DELTA1) IF ((y1 T).LE. 0) Then EXIT
ENDIF ENDDO
! Output the Lower Confidence Bounds
PRINT*, ‘The Lower Confidence Bounds on Cpu Based on Multiple & & Samples ¼ ’, Tcpu
END ! End
2355 Lower confidence bounds for CPUand CPL
Appendix 3 Input:
Please enter: Overall sample observations, ] of subsamples, USL, alpha risk (1 r). 100, 20, 6.0, 0.05.
Output:
Overall sample mean ¼ 5.609857.
Pooled sample standard deviation ¼ 8.198889E 02. Estimated Cpu-based on multiple samples ¼ 1.571239.
Lower confidence bounds on Cpu-based on multiple samples ¼ 1.370700.
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