A note on Bayesian estimation of process capability indices

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A note on Bayesian estimation of process capability indices

Jyh-Jen Horng Shiau

a;∗

_{, Hui-Nien Hung}

a

_{, Chun-Ta Chiang}

b

a_{Institute of Statistics, National Chiao Tung University, Hsinchu, Taiwan}

b_{Department of Quality Management Technology, Center of Aviation and Space Technology, Industrial Technology Research}

Institute, Hsinchu, Taiwan

Received September 1998; received in revised form January 1999

Abstract

Process capability indices are useful for assessing the capability of manufacturing processes. Most traditional methods are obtained from the frequentist point of view. We view the problem from the Bayes and empirical Bayes approaches by using non-informative and conjugate priors, respectively. c 1999 Elsevier Science B.V. All rights reserved

MSC: 62N10; 62A15

Keywords: Process capability indices; Bayesian approach; Bayes estimators; Non-informative prior; Conjugate prior

1. Introduction

Process capability indices (PCIs), as a process performance measure, have become very popular in assessing the capability of manufacturing processes in practice during the past decade. More and more eorts have been devoted to studies and applications of PCIs. For example, Rado (1989) demonstrated how Imprimis Technology, Inc. used the PCIs for program planning and growth to enhance product development. The Cp

and Cpk indices have been used in Japan and in the US automotive industry such as Ford Motor Company

(see Kane, 1986a, b). For more information on PCIs, see Kotz and Johnson (1993), Kotz et al. (1993), and the references cited therein.

The usual practice is to estimate these PCIs from data and then judge the capability of the process by these estimates. Commonly used estimators are reviewed in Section 2. Most studies on PCIs are based on the traditional frequentist point of view. The main objective of this note is to provide both point and interval estimators of some popular PCIs from the Bayesian point of view. We believe this eort is well justied since Bayesian estimation has become one of popular approaches in estimation nowadays and resulting Bayes estimators in general have good theoretical properties, such as admissibility (Bernardo and Smith, 1993). In addition, the Bayesian approach has one great advantage over the traditional frequentist approach—the posterior distribution is very easy to derive and then credible intervals, which is the Bayesian analogue of

∗_{Corresponding author. Present address: Department of Statistics, Sequoia Hall, Stanford University, Stanford, CA 94305-4065, USA.}

Tel.: +1-650-725-5976; fax: +1-650-725-8977.

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the classical condence interval, can be easily obtained either by theoretical derivation or by Monte Carlo methods (Tanner, 1993). A simple estimate of the index is not very useful in making reasonable decision on the capability of a process. An interval estimate approach is more appropriate.

This paper is organized as follows. We give a brief review on the most popular PCIs, Cp, Cpk, and Cpm in

Section 2. In Section 3, we derive Bayes estimators for C2

p, Cpm2 (with the process mean being the target

value T) and C2

pk (with the process mean being the middle point m of the two specication limits), with

respect to some chosen priors. In Sections 3.1 and 3.2, we consider the non-informative and the Gamma prior, respectively, and derive Bayes estimators and credible intervals for each prior. For the Gamma prior, the maximum-likelihood method in empirical Bayes approach is adopted for choosing the parameters in the prior. The derivation is given in the appendix. In Section 4, we propose a Bayesian procedure based on the credible intervals derived in Section 3. An example is given to demonstrate the application of the proposed Bayesian procedure. Finally, we conclude this note by a brief summary in Section 5.

Throughout this paper, it is assumed that the process measurements are independently and identically dis-tributed from a normal distribution. In other words, the process is under statistical control. We remark that estimation of PCIs is meaningful only when the process is under statistical control.

2. A review of some process capability indices

The most popular PCIs are Cp, Cpk, Cpm, and Cpmk. The Cp index is dened as

Cp=USL − LSL₆ ;

where LSL and USL are the lower and upper specication limits, respectively, and is the process standard deviation. Note that Cp does not depend on the process mean. The Cpk is then introduced to re ect the impact

of on the process capability indices. The Cpk index is dened as

Cpk= min USL −  3 ;  − LSL 3 :

The Cpm index was introduced by Chan et al. (1988). This index takes into account the in uence of the

departure of the process mean from the process target T. The Cpm is dened as

Cpm= USL − LSL

6p2_{+ ( − T)}2:

Combining the three indices, Cp, Cpk, and Cpm, Pearn et al. (1992) proposed the Cpmk index. This index is

dened as Cpmk= min ( USL −  3p2_{+ ( − T)}2;  − LSL 3p2_{+ ( − T)}2 ) :

The natural and most commonly used estimators of Cp, Cpk, Cpm, and Cpmk are

ˆCp=USL − LSL_6s ; ˆCpk= min USL − x 3s ; x − LSL 3s ; ˆCpm= USL − LSL 6ps2_{+ ( x − T)}2

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and ˆCpmk= min ( USL − x 3ps2_{+ ( x − T)}2; x − LSL 3ps2_{+ ( x − T)}2 ) ;

respectively, where x is the sample mean and s is the sample standard deviation. 3. Bayesian estimation for some PCIs

In this section, we derive the Bayes estimators for C2

p, Cpm2 with = T, and Cpk2 with = m with respect

to some priors. Two prior distributions are considered. The rst prior is the non-informative prior, and the second prior is Gamma(a; b). Reasons for choosing these priors are given at the beginning of the following two subsections, respectively. For the Gamma prior, the maximum-likelihood method in the empirical Bayes approach is adopted for choosing the parameters a and b in the prior.

Recall that the natural (most common) estimator of Cp is ˆCp=(USL−LSL)=(6s). Assuming that the process

measurements follow a N(; 2_{), Cheng and Spiring (1989) derived that the probability density function (p.d.f.)}

for ˆCp is f(y|Cp) = 2 n − 1 2 −1 _{(n − 1)C}2 p 2 !(n−1)=2 y−n_exp " −(n − 1)C_2y₂ p2 # for 0 ¡ y ¡ ∞: Let W = ˆC2 p, then the p.d.f. of W is f(w|Cp) = n − 1 2 −1 _{(n − 1)C}2 p 2 !_(n−1)=2 w−((n+1)=2)_exp " −(n − 1)C_2w p2 # for 0 ¡ w ¡ ∞: That is, ˆC2

p follows an Inverse Gamma(; ) with parameters = (n − 1)=2 and = ((n − 1)Cp2=2)−1.

Set the parameter = C2

p. Them the likelihood function L(|w) of ,

L(|w) = n − 1 2 −1_{(n − 1)} 2 (n−1)=2 (n−1)=2_w−((n+1)=2)_exp₋(n − 1) 2w : We now derive the posterior distributions for under two dierent prior distributions. 3.1. Non-informative prior

For the choice of the prior, in this subsection, we consider the prior () = 1=, for 0 ¡ ¡ ∞. There are two reasons for choosing this prior. The rst one is that the above prior can maximize the dierence between the information (entropy) of the parameter provided by the prior and posterior distributions. In other words, the above prior allows the prior to provide information about the parameter as little as possible (see Bernardo and Smith, 1993). This prior is usually referred as a reference prior. The second reason is that with the above prior, the p × 100% credible interval has coverage probability p up to the second order (in contrast to the rst order for any other priors) in the frequentist sense (Welch and Peers, 1963). In other words, the credible interval obtained from the above prior has a more precise coverage probability than that obtained from any other priors.

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With this non-informative prior, we have the joint p.d.f. of (; w) f(; w) = f(w|) × () = ( ())−1n − 1 2 (n−1)=2 (−1)_w−((n+1)=2)_exp₋ ; (1) where = (n − 1)=2 and = 2w=(n − 1) = 2 ˆC2 p=(n − 1).

Hence the posterior distribution of given w is f(|w) = (−1)exp(−=)₍₎ :

That is, the posterior distribution of given w is a Gamma(; ). So, the posterior mean for = C2

p is E(| ˆC2 p) = =n − 1₂ · 2 ˆC 2 p n − 1 = ˆC2p: Therefore, ˆC2

p is a Bayes estimator of Cp2, and we have a nice Bayesian interpretation for the estimator ˆC2p

of C2 p.

In addition, it can be shown that the mode of f(|w) is ( − 1) = ((n − 3)=(n − 1)) ˆC2

p, which is the Bayes

estimator of C2

p is the sense of extended zero-one loss.

Next, consider Cpm under = T and Cpk under = m. Recall that these two indices (with the

spe-cial restrictions on ) are both reduced to Cp. Chan et al. (1988) considered ˆCpm= (USL − LSL)=(6 ˆ0),

where ˆ0= ((1=(n − 1))Pn_i=1(xi− T)2)1=2. The p.d.f. of ˆCpm when = T, given by Theorem 8 in Chan et al.

(1988), is f(y|Cpm) = 2 n₂ −1 _{(n − 1)C}_pm2 2 !n=2 y−(n+1)_exp ₋(n − 1)Cpm2 2y2 ! for 0 ¡ y ¡ ∞:

By the same technique as that for Cp, we can obtain that the posterior p.d.f. of = Cpm2 given ˆCpm is

f(| ˆC2 pm) =

˜−1_{exp(−= ˜)}

( ˜) ˜˜ ;

which is a Gamma( ˜; ˜) distribution with ˜ = n=2 and ˜ = 2 ˆC2pm=(n − 1). So, the posterior mean for Cpm2 is

E(C2 pm| ˆC2pm) = ˜ ˜ =n₂ ·2 ˆC 2 pm n − 1= n n − 1 ˆC2pm; and thus a Bayes estimator for C2

pm is (n=(n − 1)) ˆC2pm.

However, it seems more natural to consider the estimator ˆ00= ((1=n)Pn_i=1(xi− T)2)1=2, since, under the

assumption =T; ˆ00 2 is both an unbiased estimator and the maximum-likelihood estimator (MLE) of 2_{. Then,}

in this case, it is easily seen that the posterior mean of C2

pm is exactly ˆC2pm. Also, by the same technique

as that for Cp, the posterior mode for Cpm2 is (n=2 − 1) · (2 ˆC2pm=n) = ((n − 2)=n) ˆC2pm. Likewise, if we let

ˆCpk= (USL − LSL)=(6 ˆ∗) with ˆ∗= ((1=n)P_i=1n (xi− m)2)1=2, we can also obtain that ˆCpk is the posterior

mean of C2

pk and that the posterior mode is ((n − 2)=n) ˆC2pk.

Next, we consider the interval estimation of these PCIs. Recall that the posterior distribution of =C2 p given

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Table 1

Summary of the point and interval estimators for PCIs under non-informative prior

Index Posterior mean Postetrior mode Credible interval C2 p Cˆ2p n − 3_{n − 1}Cˆ2p _ˆ C2_p n − 12n−1;1−p; ∞ C2 pm Cˆ2pm n − 2_n Cˆ2pm _ˆ C2_pm n n;1−p2 ; ∞ C2 pk Cˆ2pk n − 2_n Cˆ2pk _C_ˆ₂ pk n 2n;1−p; ∞

n − 1. Denote the (1 − p) × 100th percentile of a 2 _{distribution with degrees of freedom n − 1 by}2 n−1;1−p.

Then a useful p × 100% credible interval of C2

p is [p; ∞), where p= (=2)2n−1;1−p= ( ˆC2p=(n − 1))2n−1;1−p.

Similarly, [( ˆC2

pm=n)2n;1−p; ∞) is the corresponding p × 100% credible interval of Cpm2 , when = T; and

[( ˆC2

pk=n)2n;1−p; ∞) is the interval for Cpk2 , when = m.

We summarize the results derived above in Table 1 for quick reference. Note that in this table ˆCp=(USL−

LSL)=(6s); ˆCpm= (USL − LSL)=(6 ˆ00); and ˆCpk= (USL − LSL)=(6 ˆ∗).

3.2. Gamma prior

In the Bayesian literature, in addition to the non-informative prior, the conjugate prior is another important prior (Bernardo and Smith, 1993). The most important reason for using the conjugate prior is that, with the conjugate prior, the prior and posterior are in the same distribution family. That is, the prior and posterior distribution functions have the same mathematical form. Since ˆC2

p follows an Inverse Gamma distribution, we

know that the conjugate prior must be a Gamma prior. Assume that is distributed as Gamma(a; b) with p.d.f. () = ( (a)ba₎−1a−1_exp₋

b

for 0 ¡ ¡ ∞; 0 ¡ a ¡ ∞; 0 ¡ b ¡ ∞: Then, the joint p.d.f. for = C2

p and W is

f(; w) =((n − 1)=2)_{() (a)b}_a(n−1)=2_w₊₁+a−1_exp

− 1 + 1 b : Thus, the posterior distribution becomes

f( | w) =a

0₋₁

exp(−=b0₎

(a0_)b0a0 ;

where a0_{= + a = (n − 1)=2 + a and b}0_{= (1= + 1=b)}−1_{= ((n − 1)=2w + 1=b)}−1_.

Note that f(|w) is a Gamma(a0_{; b}0_{) density. Therefore, the posterior mean for C}2 p is E(C2 p| ˆC2p) = a0b0= n − 1 2 + a n − 1 2 ˆC2 p +1_b !₋₁ :

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And the posterior mode for C2 p is (a0_{− 1)b}0₌n − 3 2 + a n − 1 2 ˆC2 p +1_b !₋₁ :

The parameters a and b in the prior distribution can be given either subjectively or objectively. To obtain the hyperparameters a and b objectively, we may adopt the maximum-likelihood method in the empirical Bayes approach (Bernardo and Smith, 1993). Consider for any xed a and b,

f(w | a; b) = Z _∞ 0 f(; w|a; b) d = Z _∞ 0 ((n − 1)=2)(n−1)=2

() (a)ba_w+1+a−1exp

− 1 + 1 b d =((n − 1)=2)_{() (a)b}_a(n−1)=2_w₊₁ (a0_)b0a0 : (2)

If a is given, then, by maximizing (2) when w is xed, we obtain that the maximum-likelihood estimator of b is ˆb = ˆC2

p=a. The derivation is given in the appendix.

So, when b = ˆb, the posterior mean for C2 p is n − 1 2 + a n − 1 2 ˆC2 p +1 ˆb !₋₁ = ˆC2 p:

This shows that ˆC2

p is the Bayes estimator of Cp2 in the sense of the empirical Bayes.

Also, the posterior mode for C2 p is n − 3 2 + a n − 1 2 ˆC2 p +1 ˆb !−1 = n − 3 2 + a n − 1 2 + a −1 ˆC2 p:

For Gamma(a; b) prior, again consider the estimator ˆ00_{= ((1=n)}Pn

i=1(xi− T)2)1=2 for ˆCpm under = T,

and the estimator ˆ∗_{= ((1=n)}Pn

i=1(xi− m)2)1=2 for ˆCpk under = m. Then, in these cases, it can be easily

seen that the posterior distribution of = C2

pm is a gamma distribution with p.d.f.

f(|w) = a

00₋₁

exp(−=b00₎

(a00_)b00a00 ;

where a00_{= ˜ + a = n=2 + a and b}00_{= (n=(2 ˆC}2

pm) + 1=b)−1. Then we obtain that the posterior mean for Cpm2 is

a00_b00_{=(n=2+a) (n=(2 ˆC}2

pm)+1=b)−1 and the posterior mode for Cpm2 is (a00−1)b00=((n−2)=2+a) (n=(2 ˆC2pm)+

1=b)−1_{. Similarly, the posterior mean for C}2

pk is (n=2 + a)(n=(2 ˆC2pk) + 1=b)−1 and the posterior mode for Cpk2

is ((n − 2)=2 + a) (n=(2 ˆC2

pk) + 1=b)−1.

Assume a is given. To estimate b empirically for Cpm under = T and Cpk under = m, we adopt the

maximum-likelihood method. Similar results as that for Cp hold for these two cases. That is, for Cpm2 ; ˆb= ˆC2pm=a;

the posterior mean is ˆC2

pm, and the posterior mode is ((n−2)=2+a) (n=2+a)−1ˆC2pm(=((n−2+2a)=(n+2a)) ˆC2pm)

in the sense of the empirical Bayes. Similarly, for C2

pk; ˆb = ˆC2pk=a, the posterior mean is ˆC2pk, and the posterior

mode is ((n − 2)=2 + a) (n=2 + a)−1_ˆC2

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Table 2

Summary of the point and interval estimators for PCIs under gamma prior when a is given and b is estimated by the maximum-likelihood method

Index Posterior mean Postetrior mode Credible interval C2 p Cˆ2p n − 3 + 2a_{n − 1 + 2a}Cˆ2p _ˆ C2_p n − 1 + 2a2n−1;1−p; ∞ C2 pm Cˆ2pm n − 2 + 2a_{n + 2a} Cˆ2pm _ˆ C2_pm n + 2a2n;1−p; ∞ C2 pk Cˆ2pk n − 2 + 2a_{n + 2a} Cˆ2pk _C_ˆ₂ pk n + 2a2n;1−p; ∞

Next, we consider the interval estimation. Again assume that a is given. If 2a is an integer, then a p×100% credible interval for C2

p is [((n − 1)= ˆC2p+ 2= ˆb)−12n−1+2a;1−p; ∞); which can be simplied to [( ˆC2p=(n − 1 +

2a))2

n−1+2a;1−p; ∞) when ˆb = ˆC2p=a. For Cpm2 and Cpk2 under the special cases, the p × 100% credible interval

are [(n= ˆC2

pm+ 2= ˆb))−12n+2a;1−p; ∞); and [(n= ˆC2pk+ 2= ˆb)−12n+2a;1−p; ∞); respectively. These two intervals can

also be simplied to [( ˆC2pm=(n + 2a))2n+2a;1−p; ∞); when ˆb = ˆC2p=a and [( ˆC2pk=(n + 2a))2n+2a;1−p; ∞); when

ˆb = ˆC2

pk=a; respectively. If 2a is not an integer, then we may approximate n+2a;1−p2 by interpolating values of

2

n+d2ae;1−p and 2n+d2ae+1;1−p, where dxe denotes the largest integer less than or equal to x.

If both a and b need to be estimated, there is no explicit form for ˆa, the MLE of a. ˆa can only be obtained numerically.

We now summarize the above results in Table 2. As in Table 1, here we use ˆCp=(USL−LSL)=(6s); ˆCpm=

(USL − LSL)=(6 ˆ00), and ˆCpk= (USL − LSL)=(6 ˆ∗).

Note that Table 2 is reduced to Table 1 when a = 0. This is not surprising since the non-informative prior considered in Section 3.1 is the limiting case of Gamma(a; ˆb) when a goes to 0.

When using these point and interval estimates in practice, all the quantities in Tables 1 and 2 should be square-rooted for better interpretation.

4. A Bayesian procedure and an example

In this section, we describe how to use the estimators described in the previous section in real-life appli-cations. Point estimates can give some assessment on the process capability. However, as mentioned before, it is more appropriate to use interval estimates when it comes to determine whether the process is capable or not. With these interval estimates at hand, we now describe a Bayesian procedure in the following.

A p × 100% credible interval means the posterior probability that the true PCI lies in this interval is p. Let p be a high probability, say, 0.95. Suppose for this particular process under consideration to be capable, the process index must reach at least a certain level C∗_{, say, 1.33. Now, from the process data, we compute}

the lower bound of the credible interval for the index (not for the squared index) and denote it by C. The Bayesian procedure is very simple — if C ¿ C∗_{, then we say that the process is capable in a Bayesian sense.}

We use the data given in Table 6–1 of Montgomery (1990, p. 207) to demonstrate this Bayesian procedure. This example is about a manufacturing process which produced piston rings for an automotive engine. The

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Table 3

Point estimates of the three indices obtained by the posterior mean and posterior mode under the non-informative prior

Index Estimate by posterior mean Estimate by posterior mode

Cp 1.6551 1.6417

Cpm 1.6439 1.6307

Cpk 1.6162 1.6032

Table 4

The lower bound of the interval estimates of the three indices under the non-informative prior obtained with the posterior probability being 0.9, 0.95, 0.99, and 0.999 Posterior probability p Index 0.9 0.95 0.99 0.999 Cp 1.5179 1.4810 1.4126 1.3374 Cpm 1.5082 1.4717 1.4040 1.3296 Cpk 1.4827 1.4468 1.3803 1.3071 Table 5

The lower bound of the credible intervals C of the three indices for various a and p a Index p 0.01 0.1 1 10 50 100 Cp 0.9 1.5179 1.5180 1.5190 1.5279 1.5535 1.5708 0.95 1.4810 1.4811 1.4824 1.4936 1.5257 1.5476 0.99 1.4126 1.4128 1.4145 1.4299 1.4742 1.5045 0.999 1.3374 1.3376 1.3340 1.3597 1.4172 1.4567 Cpm 0.9 1.5082 1.5083 1.5094 1.5195 1.5478 1.5668 0.95 1.4717 1.4718 1.4732 1.4854 1.5203 1.5438 0.99 1.4040 1.4042 1.4060 1.4223 1.4690 1.5008 0.999 1.3296 1.3299 1.3322 1.3528 1.4124 1.4532 Cpk 0.9 1.4828 1.4829 1.4844 1.4972 1.5331 1.5564 0.95 1.4468 1.4470 1.4487 1.4637 1.5058 1.5335 0.99 1.3803 1.3805 1.3827 1.4015 1.4550 1.4909 0.999 1.3072 1.3074 1.3100 1.3330 1.3989 1.4436

measurements are the inside diameter of the rings manufactured in this process. 125 measurements were taken from the process when the process was in control. The upper specication limit USL = 74:05 and the lower specication limit LSL = 73:95: The target value T = 74: From the process data, we obtain that sample mean x = 74:00118 and the sample standard deviation s = 0:01006997. ˆCp= (USL − LSL)=(6s) = 1:655086; ˆCpm=

(USL − LSL)=(6 ˆ00) = 1:643914; and ˆCpk= (USL − LSL)=(6 ˆ∗) = 1:616159.

Table 3 reports the point estimates and Table 4 reports the interval estimates under the non-informative prior with p = 0:9; 0:95; 0:99; 0:999 for the piston ring example. Numbers given in Table 4 are the lower

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bound C of the credible interval for the indices (i.e., not squared). For C∗_{= 1:33, these C values indicate}

that the process is capable in the Bayesian sense, except for the two cases when the “condence level” is very high (0.999). We also notice that these estimates are not much dierent for the three indices. This can be explained by the fact that the three values x; T, and m are very close in this example.

To demonstrate the importance of the interval estimate, we now turn to a hypothetical example. Suppose that the index ˆCp= 1:4, which is greater than the presumed level of capability 1.33. The C value obtained in

this case is 1.2840, which is below 1.33. So we cannot conclude that the process is capable. Point estimate does not give us clue on how big the estimation error is, while a credible interval estimate can provide us a statement about the true index based on the posterior probability.

Now, under the Gamma prior, Table 5 gives the C values for various a and p. It is noticed from this table that the prior parameter a seems not aecting the C values much. Again, for p = 0:9; 0:95; 0:99, the conclusions are all the same—the process is capable in the Bayesian sense.

5. Summary

PCIs are getting more and more popular in the eorts of quality and productivity improvement. In this paper, we provide both point and interval estimators by the Bayesian approach. We derive the posterior distributions for C2

p; Cpm2 with = T, and Cpk2 with = m with respect to the two priors. We then derive

the Bayes estimators, posterior mean and mode, for each of these PCIs. We remark that in the Gamma(a; b) prior with a xed, the parameter b is estimated by the maximum-likelihood method in the empirical Bayes approach. It is found that these Bayes estimators either are the traditional estimators themselves or just dier from the traditional estimators by a constant multiplier that converges to 1 as the sample size goes to innity. In addition, Bayesian credible interval estimate are obtained analytically. Based on these interval estimates, a simple Bayesian procedure for determining if the process is capable is proposed for practitioners to use.

For the special cases that we consider in this paper, interval estimators can also be easily derived from the frequentist approach. We do not intend to replace the frequentist approach on PCIs. We simply provide a Bayesian alternative. However, when the distribution of the PCI estimators are very complicated, as they often are for some favorable estimators, such as Cpm with no restriction on the process mean, then our approach

becomes very valuable in obtaining an interval estimate of the index. Results for this particular problem is given in a subsequent paper.

Acknowledgements

The authors would like to express their gratitude to a referee for the valuable comments, which greatly improved the quality of the paper. This research work was partially supported by the National Science Council of the Republic of China. Grant No. NSC87-2118-M-009-004 and NSC87-2118-M-009-003.

Appendix

In this appendix, we derive the MLE of b by maximizing (2) given in Section 3.2. Let f(b) = b−a_{(1= +}

1=b)−(+a)_{= e}l(b)_{, where l(b) = −a log b − ( + a)log(1= + 1=b). Then we have}

dl(b) db = − a b + ( + a) b2_{+ b}= −a(b − =a) b2_{+ b} :

Since a; b, and are positive numbers, we have dl(b)=db ¿ 0 if b ¡ =a; dl(b)=db = 0 if b = =a; and dl(b)=db ¡ 0 if b ¿ =a. Hence ˆb = =a = ˆC2

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