偏斜常態資料管制圖之研究

全文

(1)國立高雄大學統計學研究所碩士論文. 偏斜常態資料管制圖之研究 A Study of Control Charts for Skew Normal Data. 研究生：張耿睿撰指導教授：蘇志成博士. 中華民國 103 年 7 月.

(2) A Study of Control Charts for Skew Normal Data. by Keng-Jui Chang Advisor Jyh-Cherng Su. Institute of Statistics National University of Kaohsiung Kaohsiung, Taiwan 811, R.O.C. July 2014.

(3) Contents Abstract (Chinese). ii. Abstract (English). iii. 1. Introduction. 1. 2. Literature Reviews. 2. 2.1. Conventional acceptance control chart . . . . . . . . . . . . . . . . . . .. 2. 2.2. Burr acceptance control chart . . . . . . . . . . . . . . . . . . . . . . . .. 3. 2.3. Tsai’s skew normal acceptance control chart . . . . . . . . . . . . . . . .. 5. 3. Design of the Exact Skew Normal Acceptance Control Chart. 4. An Illustrative Example. 13. 5. Simulation Study and Comparisons. 17. 6. Conclusions. 19. i. 7.

(4) 偏斜常態資料管制圖之研究指導教授：蘇志成博士陸軍軍官學校管理科學系. 學生: 張耿睿國立高雄大學統計學研究所摘要管制圖在製程管制上是非常有用的工具之一，傳統管制圖的設計是建立在製程分佈為常態的假設下，推導其管制界限。但實務上，許多製程卻非常態分佈。在本論文中，我們將透過偏斜常態之樣本平均數的抽樣分布來設計偏斜常態允收管制圖，並在母體為偏斜常態或gamma分佈下，將其與現有的一些允收管制圖進行比較。模擬的結果顯示，我們所提出的精確偏斜常態允收管制圖會有較佳的表現。. 關鍵字: 允收管制圖、偏斜常態分佈、偏斜常態樣本平均數。. ii.

(5) A Study of Control Charts for Skew Normal Data Advisor: Jyh-Cherng Su Department of Management Science R.O.C. Military Academy. Student: Keng-Jui Chang Institute of Statistics National University of Kaohsiung. ABSTRACT. Control chart is one of the most powerful techniques for monitoring industrial process control. Traditionally, the control charts are designed under normal assumption. However, this assumption may not hold in some processes. In this paper, based on an exact distribution of sample mean for skew normal distribution, we will develop an exact skew normal acceptance control chart. Furthermore, the performance of the proposed exact skew normal acceptance control chart are compared with those of some existing acceptance control charts when the underlying distribution is skew normal and gamma. Simulation results show that our exact skew normal acceptance control charts have better performance. Keywords: acceptance control chart, skew normal distribution, skew normal sample mean.. iii.

(6) 1. Introduction The control chart is one of the most powerful techniques for monitoring industrial. processes. It is used to determine whether a process is in statistical control state and to provide information in diagnosis and process capability. The most popular method to monitor the process is the conventional X control chart proposed by Shewhart (1931). However, when a high level of process capability has been achieved, acceptance control charts can be used to relax the level of surveillance. The acceptance control charts are generally used in situations where six-sigma spread of the process is considerably smaller than the spread in the specification limits. In these situations, the process can be allowed to shift mean over an interval, and will not product too much defect production. The control limits are usually designed under the assumption that the distribution of quality characteristic (also called process distribution) is normal or approximately normal. But many distributions of the industrial processes are not normal. There are some approaches to deal with non-normal underlying distributions. One approach is to increasing the sample size such that the sample mean becomes approximately normally distributed. However, large sample size may not be operationally feasible and may increase costs. Another approach is to transform the original data such that the transformed data are more closely modeled by a normal distribution, and then the control chart are established by the transformed data. The third approach is to establish the robust control limits based on heuristic method with no assumption on the form of the process distribution. For example, Bai and Choi (1995), Chang and Bai (2001) and Chan and Cui (2003) developed the weight variance, weighted standard deviation and skewness correction methods, respectively, to set up the control charts for the skewed underlying distributions. Tadikamulla and Popescu (2007) proposed the kurtosis correction methods to set up the control charts for the symmetric and leptokurtic distributions. Furthermore, if the form of the process distribution is assumed to be known, then the control charts can be constructed by using certain exact or approximate methods. Some literature fit the data for some specified distribution, and then constructed the appropriate control chart under non-normality. Let X1 , X2 , . . . , Xn be a random variables from a. 1.

(7) population. Also let X be the sample mean of X1 , X2 , . . . , Xn . Yourstone and Zimmer (1992) used the Burr distribution to design the control limits for X control chart. Corresponding to Burr X control chart, Chou et al. (2005) constructed a Burr acceptance control chart based on Burr distribution. Assume that the process distribution is a skew normal distribution. Tsai (2007) approximated the distribution of sample mean X by a new skew normal distribution, and constructed a skew normal X control chart. Using the same approximate approach, Tsai and Ching (2008) established a corresponding skew normal acceptance control chart for non-normal data. Su and Gupta (2014) derived the exact distribution of weighted function of independent skew normal random variables, which includes the sample mean. Li et al. (2014) constructed the skew normal X control chart by using the exact distribution of the sample mean X. In this paper, based on the exact distribution of the sample mean X, we will investigate a corresponding skew normal acceptance control charts for non-normal data. In Section 2, We will give some literature reviews about the acceptance control charts. In Section 3, the proposed exact acceptance control chart based on skew normal data will be developed. In Section 4, we will give an example to illustrate the computations of control limits. In Section 5, the performance of the proposed exact skew normal acceptance control chart will be compared with those of Conventional, Burr, and Tsai’s skew normal acceptance control charts. Finally, the discussions and conclusions are given in Section 6.. 2. Literature Reviews. 2.1. Conventional acceptance control chart Assume that the quality characteristic is normally distributed with mean µ and stan-. dard deviation σ. Also let USL and LSL be the upper and lower specification limits, respectively. There are three design approaches for the acceptance control charts, that is 1. Design based on a specified sample size n, an acceptable process level δ, and a type I error α. 2. Design based on a specified sample size n, a rejectable process level γ, and a type II error β. 2.

(8) 3. The sample size determination based on δ, α, γ and β. First, if a sample size n, an acceptable process level δ and a type I error α are given, it can be shown that the upper control limit (UCL) and lower control limit (LCL) of the acceptance control chart are given by. zα σ, UCL = USL − zδ − √ n zα LCL = LSL + zδ − √ σ. n. (1) (2). where σ is the standard deviation of the process distribution, and zp is the upper 100p percentage point of the standard normal distribution. Next, if a sample size n, a rejectable process level γ, and a type II error β are specified, the UCL and LCL of the acceptance control chart are given by zβ UCL = USL − zγ + √ σ, n zβ LCL = LSL + zγ + √ σ. n. (3) (4). Finally, assume that an acceptable process level δ, a rejectable process level γ, and type I and type II errors α and β are specified. Letting (1) = (3) and (2) = (4), the required sample size can be determined by n=. 2.2. zα + zβ zδ − zγ. 2 .. Burr acceptance control chart The cumulative distribution function (cdf) of the Burr distribution with parameters c. and k is F (x) = 1 −. 1 , x ≥ 0, (1 + xc )k. where c > 1 and k > 1. The Burr distributions cover a wide range of the skewness and kurtosis of various distributions. Burr (1942) tabulated the mean, standard deviation, skewness and kurtosis of the Burr distribution for various combinations of c and k, and 3.

(9) Burr (1973) tabulated the values of c and k, and corresponding mean and standard deviation for various combinations of skewness and kurtosis. For a random variable X with mean µ and standard deviation σ, it can be approximately expressed by the standardized transformation X −µ d Y −M = , σ S where Y is a Burr random variable, M and S are the mean and standard deviation of Y , respectively. From the tables proposed by Burr (1942) and (1973), Chou et al. (2005) used the skewness and kurtosis of X to determine the values of c, k, M and S, and denoted them by c1 , k1 , M1 and S1 , respectively. Also the skewness and kurtosis of X are used to determine the values of c, k, M and S, and denoted them by c2 , k2 , M2 and S2 , respectively. Then the UCL and LCL of the Burr acceptance control chart based on a specified sample size n, an acceptance process level δ, and a type I error α are UCL = USL −. σ −1/k1 [(δ − 1)1/c1 − M1 ] S1. σ √ [(α−1/k2 − 1)1/c2 − M2 ], S2 n σ LCL = LSL − [((1 − δ)−1/k1 − 1)1/c1 − M1 ] S1 σ + √ [((1 − α)−1/k2 − 1)1/c2 − M2 ]. S2 n +. (5). (6). Also the UCL and LCL of the acceptance control chart based on a specified sample size n, a rejectable process level γ, and a type II error β can be constructed as UCL = USL −. σ [(γ −1/k1 − 1)1/c1 − M1 ] S1. σ √ [((1 − β)−1/k2 − 1)1/c2 − M2 ]. S2 n σ LCL = LSL − [((1 − γ)−1/k1 − 1)1/c1 − M1 ] S1 σ + √ [(β −1/k2 − 1)1/c2 − M2 ]. S2 n +. (7). (8). Letting (5) = (7) and (6) = (8), the required sample size based on an acceptable process level δ, a rejectable level γ, and type I and type II errors α and β, is n = max{n1 , n2 }, 4.

(10) where i1/c2 1/c2 2 −1/k2 −1/k2 (1 − β) − 1 − α − 1     , 1/c 1/c 1 1 −1/k −1/k 1 − 1) 1 − 1) (γ − (δ h n1 =. S12 S22. and i1/c2 2 1/c2 h −1/k2 β −1 − (1 − α) −1 S2   n2 = 12  h i1/c1 h i1/c1  . S2 (1 − γ)−1/k1 − 1 − (1 − δ)−1/k1 − 1 . 2.3. −1/k2. Tsai’s skew normal acceptance control chart. The skew normal distribution proposed by Azzalini (1985) is suitable for the analysis of data exhibiting a unimodal empirical distribution function but having some skewness present. The probability density function (pdf) of the skew normal distribution, denoted by SN (ξ, σ, λ), is f (x; ξ, σ, λ) =. x−ξ 2 x−ξ φ( )Φ(λ ), −∞ < x < ∞, σ σ σ. where ξ and λ are real numbers, σ > 0, φ(·) and Φ(·) are the pdf and cdf of the standard normal distribution, respectively. The parameters ξ, σ and λ represent the location, scale and skewness parameters, respectively. If λ = 0, then the skew normal distribution reduces to a normal distribution with mean ξ and standard deviation σ. Also the distribution is skewed to the right if λ > 0; and skewed to the left if λ < 0. When ξ = 0 and σ = 1, the distribution is called the standard skew normal distribution, denoted by SN (λ). Note that if X has a SN (ξ, σ, λ) distribution, then Z = (X − ξ)/σ has a SN (λ) distribution, and X = (γ1 Z − γ2 )σX + µX ,. (9). p 2 where γ1 = 1/ 1 − a21 ρ2 , γ2 = γ1 a1 ρ, µX = ξ + a1 ρσ and σX = (1 − a21 ρ2 )σ 2 are the p √ mean and variance of X, respectively, also a1 = 2/π, ρ = λ/ 1 + λ2 . It can be easily seen that if X1 , X2 , . . . , Xn are a random sample from a skew normal distribution, then the distribution of the sample mean X is not skew normal. Tsai (2008). 5.

(11) approximated the distribution of X by a skew normal distribution SN (ξ1 , σ1 , λ1 ), for some ξ1 , σ1 , λ1 . Then X can be approximately expressed by σX X = (γ10 Z 0 − γ20 ) √ + µX , (10) n p p where ρ1 = λ1 / 1 + λ21 , γ10 = 1/ 1 − a21 ρ21 , γ20 = γ10 a1 ρ1 , and Z 0 has a SN (λ1 ) distribution. Using equations (9) and (10), the UCL and LCL of the acceptance control chart based on a specified sample size n, an acceptance process level δ, and a type I error α can be established by σX UCL = USL − (γ1 zδ,λ − γ2 )σX + (γ10 zα,λ1 − γ20 ) √ , n σX LCL = LSL − (γ1 z1−δ,λ − γ2 )σX + (γ10 z1−α,λ1 − γ20 ) √ , n. (11) (12). where zp,λ is the upper 100p percentage point of the SN (λ) distribution. The UCL and LCL of the acceptance control chart based on a specified sample size n, a rejectable process level γ, and a type II error β can also be established by σX UCL = USL − (γ1 zγ,λ − γ2 )σX + (γ10 z1−β,λ1 − γ20 ) √ , n σX LCL = LSL − (γ1 z1−γ,λ − γ2 )σX + (γ10 zβ,λ1 − γ20 ) √ . n. (13) (14). Letting (11) = (13) and (12) = (14), the required sample size based on an acceptable process level δ, a rejectable level γ, and type I and II errors α, β, should be taken as n = max{n1 , n2 }, where γ 0 (z1−β,λ1 − zα,λ1 ) n1 = 1 γ1 (zγ,λ − zδ,λ ). 2. γ 0 (zβ,λ1 − z1−α,λ1 ) n2 = 1 γ1 (z1−γ,λ − z1−δ,λ ). 2. . ,. and . 6. ..

(12) 3. Design of the Exact Skew Normal Acceptance Control Chart In this section, we will develop the exact skew normal acceptance control chart which. can be compared with that proposed by Tsai and Chiang (2008). Su and Gupta (2014) derived the exact distribution of weighted function of independent skew normal random variables. More precisely, let Z1 , Z2 , . . . , Zn be independently and identically SN (λ) distributed random variables, and ω1 , ω2 , . . . , ωn be non-zero real numbers. Then the cdf P of Y = ni=1 ωi Zi is given by     0 0 y ωω −λω  , y ∈ R, FY (y) = 2n Φn+1   ,  (15) 2 0 −λω (1 + λ )In where 0 = (0, 0, . . . , 0)0 , ω = (ω1 , ω2 , . . . , ωn )0 , In is the n × n identity matrix and Φn (·, Ω) denotes the cdf of Nn (0, Ω) distribution (the n-dimensional normal distribution with zero vector and covariance matrix Ω). Letting ω = (1/n, 1/n, . . . , 1/n)0 in P (15), it yields that the cdf of the sample mean Z = n1 ni=1 Zi is     1 λ 0 y −n1  , y ∈ R, Fλ (y) = 2n Φn+1   ,  n (16) 0 − nλ 1 (1 + λ2 )In where 1 = (1, 1, . . . , 1)0 . Now let X1 , X2 , . . . , Xn be a random sample from a population with SN (ξ, σ, λ) distribution. Assume that the acceptable process level is δ Let µL and µU be the smallest and largest permissable values of µX , respectively, consistent with producing a fraction nonconforming of at most δ, then P (Xi > USL|µX = µU ) = δ.. (17). From (9) and (17), we have P. 1 Zi > γ1. . USL − µU + γ2 σX. !

(13)

(14)

(15)

(16) µX = µU = δ.

(17). (18). Since Z has a SN (λ) distribution, from (18), it can be obtained that µU = USL − (γ1 zδ,λ − γ2 )σX , 7. (19).

(18) where zδ,λ is the upper 100δ percentage point of the SN (λ) distribution. Now assume that the type I error α is specified, then P (X > UCL|µX = µU ) = α.. (20). X = (γ1 Z − γ2 )σX + µX ,. (21). From (17), we have. Also from (20) and (21), we have P. 1 Z> γ1. . UCL − µX + γ2 σX. !

(19)

(20)

(21)

(22) µX = µU = α

(23). and it implies that UCL = (γ1 z n,α,λ − γ2 )σX + µU ,. (22). where z n,α,λ is the upper 100α percentage point of distribution of Z. Substituting (19) into (22), the UCL of the acceptance control chart is given as UCL = USL − γ1 (zδ,λ − z n,α,λ )σX .. (23). The LCL of the acceptance control chart can be established in a similar way as described above. First the equality P (X < LSL|µX = µL ) = δ. (24). µL = LSL − (γ1 z1−δ,λ − γ2 )σX .. (25). implies. Also from P (X < LCL|µX = µL ) = α, we have P. 1 Z< γ1. . LCL − µL + γ2 σX 8. !

(24)

(25)

(26)

(27) µX = µL = α

(28).

(29) and LCL = (γ1 z n,1−α,λ − γ2 )σX + µL .. (26). Substituting (25) into (26), the LCL of the acceptance control chart is given as LCL = LSL − γ1 (z1−δ,λ − z n,1−α,λ )σX .. (27). Furthermore, assume that a rejectable process level γ, and a type II error β are specified. Following the line of above derivation, the largest and smallest permissable values of µX are given by µU = USL − (γ1 zγ,λ − γ2 )σX ,. (28). µL = LSL − (γ1 z1−γ,λ − γ2 )σX ,. (29). and. respectively. Also the UCL and LCL of the acceptance control chart can be constructed as UCL = USL − γ1 (zγ,λ − z n,1−β,λ )σX ,. (30). LCL = LSL − γ1 (z1−γ,λ − z n,β,λ )σX ,. (31). and. respectively. Now assume that an acceptable process level δ, a rejectable level γ, and type I and type II errors α and β, are specified. Letting (23) = (30), it yields z n1 ,α,λ − z n1 ,1−β,λ = zδ,λ − zγ,λ .. (32). Also letting (27) = (31), it yields z n2 ,β,λ − z n2 ,1−α,λ = z1−γ,λ − z1−δ,λ . Thus the required sample size is n = max{n1 , n2 }. 9. (33).

(30) In practice, n1 and n2 are taken to be the smallest integers such that z n1 ,α,λ − z n1 ,1−β,λ ≤ zδ,λ − zγ,λ , and z n2 ,β,λ − z n2 ,1−α,λ ≤ z1−γ,λ − z1−δ,λ , respectively. In the following, we will derive an approximate closed form expression for the required sample size. Let µλ and σλ be the mean and standard deviation of SN (λ) distribution, respectively. It is known that as n is sufficiently large, the distribution of Z can be approximated by a normal distribution with mean µλ and standard deviation √ σλ / n. Hence it can be shown that σλ z n,α,λ ≈ µλ + √ zα , n and σλ z n,α,λ − z n,1−β,λ ≈ √ (zα − z1−β ). n. (34). Hence the equation (32) can be replaced by σλ zδ,λ − zγ,λ ≈ √ (zα − z1−β ) n1 and n1 ≈. σλ2. . zα − z1−β zδ,λ − zγ,λ. 2 .. (35). Similarly, the equation (33) can be replaced by σλ z1−γ,λ − z1−δ,λ ≈ √ (zβ − z1−α ) n2 and n2 ≈. σλ2. . zβ − z1−α z1−δ,λ − z1−γ,λ. 2 .. (36). Therefore, the required sample size based on normal approximation will be n = max{n1 , n2 }. The simulation results show that, in most situations, the required sample size based on (35) and (36) is almost the same as that based on (32) and (33). 10.

(31) Assume that the process parameters are unknown, m samples, each of size n units, are collected. For i = 1, 2, . . . , m and j = 1, 2, . . . , n, let Xij be the jth observation in the ith sample from SN (ξ, σ, λ) distribution. Let m. X= v u u SX = t. n. 1 XX Xij , nm i=1 j=1. m. n. XX 1 (Xij − X)2 nm − 1 i=1 j=1. and m. kˆ3 =. n. X X Xij − X 3 1 . nm − 3 i=1 j=1 S. As the skewness k3 of the SN (ξ, σ, λ) distribution satisfies that 3. k3 = b1 ρ3 (1 − a21 ρ2 )− 2 , where b1 = (4/π − 1)a1 . The skewness parameter λ can be estimated as  h i− 1   (b1 /kˆ3 )2/3 + a21 − 1 2 , kˆ3 ≥ 0, ˆ= λ i− 1 h   − (b1 /kˆ3 )2/3 + a2 − 1 2 , kˆ3 < 0, 1. (37). and γ1 can be estimated as 1. γˆ1 = (1 − a21 ρˆ2 )− 2 , ˆ where ρˆ = λ/. p. ˆ 2 . Hence if the standard deviation σX of X is estimated by the 1+λ. sample standard deviation SX , then the acceptance control chart based on an acceptable level δ and a type I error α can be established as UCL = USL − γˆ1 (zδ,λˆ − z n,α,λˆ )SX , LCL = LSL − γˆ1 (z1−δ,λˆ − z n,1−α,λˆ )SX , and the acceptance control chart based on a rejectable process level γ and a type II error β can be established as UCL = USL − γˆ1 (zγ,λˆ − z n,1−β,λˆ )SX , LCL = LSL − γˆ1 (z1−γ,λˆ − z n,β,λˆ )SX . 11.

(32) Some upper percentage points zp,λ and z n,p,λ are given in Tables 1 to 5. On the other hand, from (9) it can be seen that if R is the sample range of data from SN (ξ, σ, λ) distribution, then R = γ1 σX Rλ , where Rλ is the sample range of data from SN (λ) distribution. Let d∗2 = d∗2 (n, λ) = E(Rλ ) be the mean of the range RZ . The values of d∗2 are given in Table 6, which is obtained by simulations (200,000 random samples of size n are drawn). Now if Ri is the range of the ith sample, i = 1, 2, . . . , m, then the standard deviation σX can also be estimated by R. σ ˆX = where R =. 1 m. Pm. i=1. d∗2 γˆ1. ,. Ri . Hence the acceptance control chart based on an acceptable level. δ and a type I error α can also be constructed as UCL = USL − ( LCL = LSL − (. zδ,λˆ. z n,α,λˆ. −. d∗2 z1−δ,λˆ d∗2. −. )R, d∗2 z n,1−α,λˆ d∗2. (38) )R.. (39). Also the acceptance control chart based on a rejectable process level γ and a type II error β can also be constructed as UCL = USL − ( LCL = LSL − (. zγ,λˆ d∗2. −. z1−γ,λˆ d∗2. z n,1−β,λˆ −. d∗2 z n,β,λˆ d∗2. )R,. (40). )R.. (41). For providing tabulated constants that can be used the same way as Shewhart constants, some charting constants zα,λ /d∗2 and z n,α,λ /d∗2 are given in Tables 7 to 14. The values of charting constants are given for n = 2, 3, 5, 10, and for λ from 0 to 5.5 in increments of 0.5, and 6 to 10 in increments of 1. For λ < 0, the charting constants can also be obtained by Tables 7 to 14. First it is known that if X has a SN (λ) distribution, the −X has a SN (−λ) distribution. Thus, it can be seen that d∗2 (n, λ) = d∗2 (n, −λ) and zα,λ = −z1−α,−λ . In the following, we will show that z n,α,λ = −z n,1−α,−λ . Let Fλ be the 12.

(33) cdf of the sample mean of a sample from the SN (λ) distribution. Then from (16), we have Fλ (−z n,1−α,−λ )  = 2n Φn+1 . −z n,1−α,−λ 0.   ,. − nλ 10. 1 n. − nλ 1. 2. (1 + λ )In.  . = 2n P (Z1 ≤ −z n,1−α,−λ , Z2 ≤ 0, . . . , Zn+1 ≤ 0) = 2n [P (Z2 ≤ 0, . . . , Zn+1 ≤ 0) − P (−Z1 ≤ z n,1−α,−λ , Z2 ≤ 0, . . . , Zn+1 ≤ 0)] , where (Z1 , Z2 , . . . , Zn+1 ) has a multivariate normal distribution Nn+1 (0, Σ1 ), and   1 λ 0 − 1 n . Σ1 =  n λ 2 − n 1 (1 + λ )In. (42). It can be seen that Z2 , Z3 , . . . , Zn+1 are independent and identically distributed normal random variables with mean 0 and variance 1 + λ2 , and (−Z1 , Z2 , . . . , Zn+1 ) has a multivariate normal distribution Nn+1 (0, Σ2 ), where  Σ2 = . 1 n. λ 0 1 n. λ 1 n. (1 + λ2 )In.  .. Thus we have P (Z2 ≤ 0, . . . , Zn+1 ≤ 0) = 2−n , and    z n,1−α,−λ , Fλ (−z n,1−α,−λ ) =1 − 2n Φn+1  0. 1 n. λ 0 1 n. λ 1 n. (1 + λ2 )In.  . =1 − F−λ (z n,1−α,−λ ) =1 − α. The above equation implies that z n,α,λ = −z n,1−α,−λ . For example, if λ = −3, n = 5 and α = 0.01, then we have z0.01,−3 /d∗2 = −z0.99,3 /d∗2 = 0.29084 and z 5,0.01,−3 /d∗2 = −z 5,0.99,3 /d∗2 = −0.09608.. 4. An Illustrative Example In this section, we will give an example to demonstrate the computations of the control. limits for conventional acceptance control chart (CACC), Burr acceptance control chart 13.

(34) (BACC), Tsai’s skew normal acceptance control chart (TSNACC) and our exact skew normal acceptance control chart (ESNACC). We use the primer thickness data taken from Das and Bhattacharya (2008). The primer thickness data have 20 samples of size 10. From the histogram in Figure 1, it seems that the data are closer to the skew normal distribution. The normal Q-Q plot in Figure 2 also shows that the data depart from the fitted line most evidently in the extreme, or distribution tails. Using the Anderson-Darling test for normal distribution, the p-value is 0.005362. There is evidence that the data do not follow a normal distribution. The skew normal Q-Q plot in Figure 3 also shows that the skew normal model is visually good. Hence it is reasonable to assume that the data are drawn from some skew normal distribution SN (ξ, σ, λ). From the data, the estimated values of the skewness k3 and kurtosis k4 are kˆ3 = 0.3485 and kˆ4 = 2.6489, respectively. Thus the estimated skewness kˆ3 (X) and the estimated kurtosis kˆ4 (X) of the sample mean X are given by √ kˆ3 (X) = kˆ3 / 10 = 0.1102 and √ kˆ4 (X) = (kˆ4 − 3)/ 10 + 3 = 2.9649, respectively. Assume that the specification limits are 1.1 ± 0.7. From the table of Burr (1973), the estimated skewness kˆ3 and the estimated kurtosis kˆ4 determine that c1 = 2.876, k1 = 12.887, M1 = 0.373 and S1 = 0.147. Also the estimated skewness kˆ3 (X) and the estimated kurtosis kˆ4 (X) determine that c2 = 4.297, k2 = 6.283, M2 = 0.608 ˆ = 1.6482, λ ˆ 1 = 0.9080, and thus γˆ1 = 1.3676, and S2 = 0.171. Using(37), we can get λ γˆ2 = 0.9329, γˆ1 0 = 1.1849, γˆ2 0 = 0.6355 and d∗2 = 2.2460. Assume that the acceptable process level δ = 0.01 and the type I error α = 0.05 are specified, it can be obtained that zδ,λˆ = 2.5758, z1−δ,λˆ = −0.8461, zα,λˆ1 = 1.9510, z1−α,λˆ1 = −0.8242, z n,α,λˆ = 1.0696 and z n,1−α,λˆ = 0.3093. Also from the data, we have that the average of the sample ranges, R, is 0.348. Now based on the equations (1) and (2), (5) and (6), (11) and (12), (38) and (39), respectively, the control limits of the CACC, BACC, TSNACC and ESNACC can be derived 14.

(35) as below: UCLCACC = 1.5958. and. LCLCACC = 0.6042,. UCLBACC = 1.5720. and. LCLBACC = 0.5649,. UCLTSNACC = 1.5667 and LCLTSNACC = 0.5792, UCLESNACC = 1.5665 and LCLESNACC = 0.5791,. where the standard deviation σX in TSNACC is estimated by R/(d∗2 γˆ1 ). The control limits of the TSNACC and ESNACC are close. From the data, we obtain that the estimated mean µ ˆX = 1.1206 and the estimated standard deviation σ ˆX = 0.1135. As µX = ξ + a1 ρσ and 2 σX = (1 − a21 ρ2 )σ 2 , the moment estimators of ξ and σ are given by 1/2 2 σ ˆX σ ˆ= = 0.1552 1 − a21 ρˆ2. and ξˆ = µ ˆX − a1 ρˆσ ˆ = 1.0147, p ˆ ˆ 2 . If the underlying distribution is the SN (ξ, ˆσ ˆ distribution. where ρˆ = λ/ 1+λ ˆ , λ) Using equations (19) and (23), we have µL = 0.6368 and µU = 1.5066. Now if the non-normality of data is ignored and the CACC is used to monitor the process mean, then using (21), the real type I errors at µX = µL and µX = µU can be obtained as αL = P (X < LCL|µX = µL ) = P (X < 0.6042|µX = 0.6368) = P (Z < 0.4717|µX = 0.6368) = 0.1823 and αU = P (X > UCL|µX = µU ) = P (X > 1.5958|µX = 1.5066) = P (Z > 1.2603|µX = 1.5066) = 0.0082, 15.

(36) respectively. The values of αL and αU are considerably larger and smaller, respectively, than the nominal value α = 0.05. Furthermore, the real type I errors of BACC at µX = µL and µX = µU can also be obtained as αL = P (X < 0.5649|µX = 0.6368) = 0.0194 and αU = P (X > 1.5720|µX = 1.5066) = 0.0373, respectively. Both values of αL and αU are smaller than α = 0.05 On the other hand, if the rejectable process level γ = 0.05 and the type II error β = 0.1 are given. It can be shown that zγ,λˆ = 1.9599, z1−β,λˆ1 = −0.5325, z1−γ,λˆ = −0.4399, zβ,λˆ1 = 1.6261, z n,1−β,λˆ = 0.3888, z n,β,λˆ = 0.9810, then the control limits of CACC, BACC, TSNACC, and ESNACC, based on the equation (3) and (4), (7) and (8), (13) and (14), (40) and (41), can be construct as follows: UCLCACC = 1.5682. and. LCLCACC = 0.6318,. UCLBACC = 1.5571. and. LCLBACC = 0.62092,. UCLTSNACC = 1.5565 and LCLTSNACC = 0.6204, UCLESNACC = 1.5566 and LCLESNACC = 0.6201.. The control limits of the TSNACC and ESNACC are still closed. Again if the underlying ˆσ ˆ distribution, using the equations (28) and (29), we have distribution is the SN (ξ, ˆ , λ) µL = 0.5739 and µU = 1.6020. The real type II errors of CACC at µX = µL and µX = µU are given by βL = P (X > 0.6318|µX = 0.5739) = 0.0556 and βU = P (X < 1.5682|µX = 1.6020) = 0.1720, 16.

(37) respectively. The values of βL and βU are considerably smaller and larger, respectively, than the nominal value β = 0.1. Furthermore, the real type II errors of BACC at µX = µL and µX = µU can also be given by βL = P (X > 0.6210|µX = 0.5739) = 0.0959 and βU = P (X < 1.5571|µX = 1.6020) = 0.1022, respectively. In this case, both values of βL and βU are close to the nominal type II error β = 0.1. Finally, if an acceptable process level δ = 0.01, a rejectable process level γ = 0.05, and type I and type II errors α = 0.05 and β = 0.1, are specified, then the require sample sizes for constructing the CACC, BACC, TSNACC and ESNACC can be obtained as nCACC = max{19, 19} = 19, nBACC = max{14, 46} = 46, nTSNACC = max{13, 28} = 28, nESNACC = max{13, 28} = 28,. respectively. The TSNACC and ESNACC have the same required sample sizes, and the required sample sizes for constructing the CACC and BACC are, respectively, the smallest and the largest ones among four acceptance control charts. If we use the normal approximation for the ESNACC, the required sample size can be determined as nA = max{13, 28} = 28, which is the same as the required sample size for the ESNACC.. 5. Simulation Study and Comparisons A simulation study is constructed to evaluate the performance of the ESNACC. We. will compare the type I error of ESNACC with those of CACC, BACC and TSNACC. The 17.

(38) simulated data are generated from the standard skew normal and gamma distributions. Note that the pdf of the gamma distribution is f (x) =. 1 xk−1 e−x/θ , x > 0, Γ(k)θk. where θ > 0 and k > 0 are the scale and shape parameters, respectively. The skewness √ k3 = 2/ k and the kurtosis k4 = 6/k are independent of the scale parameter θ. In our simulation study, the observations are generated from the standard skew normal distributions with skewness parameter λ = 0 to 5.5 in increments of 0.5 and 6 to 10 in increments of 1, and gamma distribution with parameters θ = 1, k = 5 to 60 in increments of 5. For the standard skew normal distribution SN (λ), the specification limits will be set as µZ ± 6σZ = a1 ρ ± 6(1 − a21 ρ2 ). Also for the gamma distribution with parameters θ = 1 √ and k, we have the mean µG = k and the standard deviation σG = k, the specification √ limits will be set as µG ± 6σG = k ± 6 k. The steps of the simulations are described below. Step 1. Generate m=10,000 samples of size 5 from each specified distribution. Step 2. Assume that the parameters of the underlying distribution are unknown. Given the acceptable process level δ and the type I error α, the control limits of CACC, BACC, TSNACC and ESNACC are established based on the data from Step 1. Step 3. Calculate the type I errors αL = P (X < LCL|µX = µL ) and αU = P (X > UCL|µX = µU ). Tables 15 to 18 give the simulated type I errors of CACC, BACC, TSNACC and ESNACC under the standard skew normal distributions, where the nominal type I error α = 0.01 or 0.05. Also the acceptable level δ = 0.0027 in Tables 15 and 16, and δ = 0.01 in Tables 17 and 18. The corresponding simulated results for the gamma distributions are given in Tables 19 and 22. It is noted that all the acceptance control charts are designed based on the nominal type I error α = 0.01 or 0.05. Hence a performance is better if its estimated type I error is closer to α. From the simulation results, it can be found that. 1. The CACC performs well only for the normal distributions. 2. The performance of BACC is not good under the skew normal distributions. 18.

(39) 3. If the underlying distribution is a skew normal distribution, the performance of ESNACC is better than that of TSNACC, especially for large skewness parameter λ. 4. When the underlying distribution is a gamma distribution, the performances of BACC, TSNACC and ESNACC are close.. 6. Conclusions Non-normal data often occur in industrial processes. Using the conventional control. chart for non-normal data will not obtain the correct nominal type I and type II errors. Thus the development of designing the control charts under non-normality has become one of the important issues. In this paper, based on the exact distribution of sample mean for skew normal distributions, the ESNACC is proposed. The charting constants are also provided in tables for practical implementation. Simulation results show that if the underlying distribution is a skew normal distribution, our ESNACC performs significantly better than CACC, BACC and TSNACC. Also if the underlying distribution is a gamma distribution, the performances of BACC, TSNACC and ESNACC are close, and obviously better than that of CACC.. 19.

(40) References [1] Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian Journal of Statistics 12, 171-178. [2] Bai, D.S. and Choi, I.S. (1995). X and R control charts for skewed populations. Journal of Quality Technology 27, 120-131. [3] Burr, I.W. (1942). Cumulative frequency distribution. Annals of Mathematical Statistics 13, 215-232. [4] Burr, I.W. (1973). Parameters for a general system of distributions to match a grid of α3 and α4 , Communications in Statistics 2, 1-21. [5] Chan, L.K. and Cui, H.J. (2003). Skewness correction X and R charts for skewed distributions. Naval Reserach Logistic 50, 555-573. [6] Chang, Y.S. and Bai, D.S. (2001). Control charts for positively-skewed populations with weighed standard deviation. Quality and Reliability Engineering International 17, 397-406. [7] Chou, C.Y., Chen C.H. and Liu, H.R. (2005). Acceptance control charts for nonnormal data. Journal of Applied Statistics 32, 25-36. [8] Das, N. and Bhattacharya, A. (2008). A New non-parametric control chart for controlling variability. Quality Technology and Quantitative Management 4, 351-361. [9] Li, C.I., Su, N.C., Su, P.F. and Shyr, Y. (2014). The design of X and R control charts for skew normal distributed data. Accepted by Communication in Statistics-Theory & methods. [10] Shewhart, W.A. (1931). Economic control of quality of manufactured product, American Society for Quality Control, Milwaukee, WI. [11] Su, N.C. and Gupta A.K. (2014). One some sampling distribution for skew normal population. Submitted. 20.

(41) [12] Tadikamalla, P.R. and Popescu, D.G. (2007). Kurtosis correction method for X and R control charts for long-tailed symmetrical distributions. Naval Research Logistics 54, 371-383. [13] Tsai, T.R. (2007). Skew normal distribution and the design of control charts for averages. International Journal of Reliabilty, Quality and Safety Engineering 14, 49-63. [14] Tsai, T.R. and Chiang, J.Y. (2008). The design of acceptance control chart for nonnormal data. Journal of the Chinese Institute of Industrial Engineers 25, 127-135. [15] Yourstone, S.A. and Zimmer, W.J. (1992). Non-normality and the design of control charts for averages. Decision Sciences 23, 1099-1113.. 21.

(42) 5 0. 1. 2. Density. 3. 4. skew normal normal. 0.9. 1.0. 1.1. 1.2. 1.3. 1.4. Figure 1: The histogram of the primer thickness data taken from Das and Bhattacharya (2008). 22.

(43) 1.4. ● ● ●. 1.2 1.1 0.9. 1.0. Sample Quantiles. 1.3. ● ●● ● ●●●. ●. −3. ●. ●. ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ● ●●●● ●●. −2. −1. 0. 1. 2. 3. Theoretical Quantiles. Figure 2: Normal Q-Q plot of the primer thickness data taken from Das and Bhattacharya (2008). 23.

(44) 1.4. ● ● ●. 1.2 1.1 0.9. 1.0. Sample Quantiles. 1.3. ●● ● ●●●. ●. ●. ●. ●●● ● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●●● ●● ●. 0.9. 1.0. 1.1. 1.2. 1.3. 1.4. 1.5. Theoretical Quantiles. Figure 3: Skew Normal Q-Q plot of the primer thickness data taken from Das and Bhattacharya (2008). 24.

(45) 25. -2.21985. -1.62612. -1.20498. -0.92779. -0.74098. -0.60966. -0.51365. -0.44098. -0.38437. -0.33923. -0.30251. -0.27211. -0.22497. -0.19023. -0.16366. -0.14283 2.99998. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 5.5. 6. 7. 8. 9. 10. 2.99998. 2.99998. 2.99998. 2.99998. 2.99998. 2.99998. 2.99998. 2.99998. 2.99998. 2.99998. 2.99998. 2.99998. 2.99998. 2.99977. 2.98349. 2.78215. -2.78215. 0. 0.0027. 0.9973. λ. -0.07710. -0.09215. -0.11154. -0.13727. -0.17274. -0.19585. -0.22400. -0.25891. -0.30307. -0.36035. -0.43680. -0.54254. -0.69501. -0.92487. -1.28155. -1.80113. -2.32634. 0.99. 2.57583. 2.57583. 2.57583. 2.57583. 2.57583. 2.57583. 2.57583. 2.57583. 2.57583. 2.57583. 2.57583. 2.57583. 2.57583. 2.57582. 2.57496. 2.54762. 2.32634. 0.01. p. 0.03962. 0.03313. 0.02424. 0.01174. -0.00653. -0.01893. -0.03455. -0.05446. -0.08044. -0.11518. -0.16313. -0.23176. -0.33449. -0.49607. -0.76006. -1.17326. -1.64485. 0.95. 1.95996. 1.95996. 1.95996. 1.95996. 1.95996. 1.95996. 1.95996. 1.95996. 1.95996. 1.95996. 1.95996. 1.95996. 1.95996. 1.95974. 1.95451. 1.89966. 1.64485. 0.05. 0.12010. 0.11742. 0.11338. 0.10712. 0.09713. 0.08993. 0.08051. 0.06801. 0.05108. 0.02757. -0.00610. -0.05610. -0.13381. -0.26101. -0.47827. -0.83757. -1.28154. 0.9. Table 1: The upper 100p percentage point zp,λ of the SN (λ) distribution. 1.64485. 1.64485. 1.64485. 1.64485. 1.64485. 1.64485. 1.64485. 1.64485. 1.64485. 1.64485. 1.64485. 1.64485. 1.64480. 1.64377. 1.63222. 1.55604. 1.28155. 0.1.

(46) 26. -1.46935. -1.00318. -0.69322. -0.49781. -0.37079. -0.28435. -0.22231. -0.17680. -0.14217. -0.11506. -0.09343. -0.07606. -0.04922. -0.03119. -0.01724. -0.00658 2.23962. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 5.5. 6. 7. 8. 9. 10. 2.24964. 2.24221. 2.24543. 2.24110. 2.24110. 2.23660. 2.24803. 2.23373. 2.25249. 2.25400. 2.25416. 2.25851. 2.26241. 2.25528. 2.20778. 1.96728. -1.96728. 0. 0.0027. 0.9973. λ. 0.05884. 0.05228. 0.04329. 0.03001. 0.01117. -0.00139. -0.01745. -0.03778. -0.06440. -0.09987. -0.14874. -0.21857. -0.32314. -0.48731. -0.75500. -1.17206. -1.64497. 0.99. 1.98353. 1.97970. 1.98119. 1.98417. 1.98069. 1.97918. 1.98177. 1.98311. 1.98311. 1.98467. 1.97987. 1.97896. 1.98098. 1.97707. 1.96568. 1.90176. 1.64498. 0.01. p. 0.18834. 0.18530. 0.18115. 0.17494. 0.16534. 0.15853. 0.14976. 0.13812. 0.12239. 0.10050. 0.06911. 0.02224. -0.05099. -0.17168. -0.37940. -0.72647. -1.16309. 0.95. 1.58140. 1.58030. 1.58129. 1.58059. 1.58009. 1.57981. 1.58019. 1.58017. 1.57965. 1.57928. 1.57885. 1.57664. 1.57287. 1.56570. 1.54029. 1.44640. 1.16309. 0.05. Table 2: The upper 100p percentage point z n,p,λ with n = 2. 0.27963. 0.27774. 0.27493. 0.27090. 0.26446. 0.25992. 0.25391. 0.24581. 0.23455. 0.21862. 0.19508. 0.15898. 0.10067. 0.00119. -0.17675. -0.48837. -0.90616. 0.9. 1.37704. 1.37719. 1.37688. 1.37692. 1.37674. 1.37606. 1.37581. 1.37535. 1.37471. 1.37370. 1.37244. 1.36931. 1.36467. 1.35251. 1.31761. 1.20434. 0.90616. 0.1.

(47) 27. 0.9973. -1.60627. -1.13587. -0.72323. -0.45926. -0.29728. -0.19471. -0.12611. -0.07858. -0.04361. -0.01793. 0.00196. 0.01739. 0.03046. 0.04794. 0.06126. 0.07001. 0.07754. λ. 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 5.5. 6. 7. 8. 9. 10. 1.95344. 1.95956. 1.95608. 1.95834. 1.95643. 1.95420. 1.95187. 1.95290. 1.95733. 1.96105. 1.95365. 1.95701. 1.95421. 1.94950. 1.93301. 1.86597. 1.60627. 0.0027. 0.14660. 0.14075. 0.13566. 0.12609. 0.11371. 0.10467. 0.09294. 0.07764. 0.05767. 0.03036. -0.00792. -0.06384. -0.14944. -0.28718. -0.51840. -0.89261. -1.34311. 0.99. 1.73468. 1.73757. 1.74085. 1.73748. 1.74242. 1.73749. 1.73726. 1.73746. 1.73768. 1.73984. 1.73640. 1.73297. 1.73125. 1.72342. 1.70226. 1.61709. 1.34311. 0.01. p. 0.27959. 0.27723. 0.27362. 0.26873. 0.26087. 0.25556. 0.24846. 0.23915. 0.22637. 0.20844. 0.18256. 0.14331. 0.08103. -0.02381. -0.20886. -0.52818. -0.94966. 0.95. 1.42364. 1.42306. 1.42393. 1.42331. 1.42407. 1.42313. 1.42290. 1.42144. 1.42121. 1.41994. 1.41840. 1.41506. 1.40844. 1.39518. 1.35865. 1.24602. 0.94966. 0.05. Table 3: The upper 100p percentage point z n,p,λ of with n = 3. 0.36704. 0.36504. 0.36241. 0.35855. 0.35275. 0.34849. 0.34332. 0.33613. 0.32641. 0.31255. 0.29203. 0.26044. 0.20901. 0.12007. -0.04208. -0.33351. -0.73990. 0.9. 1.26425. 1.26453. 1.26349. 1.26355. 1.26295. 1.26279. 1.26244. 1.26080. 1.25976. 1.25823. 1.25571. 1.25113. 1.24264. 1.22451. 1.17862. 1.04879. 0.73990. 0.1.

(48) 28. 0.9973. -1.24420. -0.80070. -0.43926. -0.21859. -0.08816. -0.00775. 0.04455. 0.08015. 0.10609. 0.12423. 0.13767. 0.14841. 0.15622. 0.16909. 0.17735. 0.18312. 0.18724. λ. 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 5.5. 6. 7. 8. 9. 10. 1.67088. 1.66722. 1.66727. 1.66508. 1.66670. 1.66811. 1.66577. 1.66633. 1.66767. 1.66169. 1.66337. 1.66044. 1.65590. 1.64564. 1.61793. 1.52433. 1.24420. 0.0027. 0.25776. 0.25531. 0.24994. 0.24394. 0.23425. 0.22841. 0.22002. 0.20860. 0.19395. 0.17355. 0.14430. 0.10044. 0.03184. -0.08176. -0.27870. -0.61178. -1.04026. 0.99. 1.50422. 1.50344. 1.50577. 1.50439. 1.50450. 1.50420. 1.50295. 1.50167. 1.50196. 1.50062. 1.49823. 1.49490. 1.48841. 1.47578. 1.44003. 1.33199. 1.04026. 0.01. p. 0.38250. 0.38020. 0.37773. 0.37377. 0.36732. 0.36286. 0.35712. 0.34948. 0.33900. 0.32428. 0.30292. 0.27019. 0.21739. 0.12705. -0.03654. -0.32907. -0.73560. 0.95. 1.27365. 1.27307. 1.27263. 1.27258. 1.27143. 1.27045. 1.26966. 1.26839. 1.26726. 1.26517. 1.26224. 1.25691. 1.24690. 1.22652. 1.17778. 1.04527. 0.73560. 0.05. Table 4: The upper 100p percentage point z n,p,λ with n = 5. 0.45861. 0.45676. 0.45436. 0.45146. 0.44597. 0.44224. 0.43775. 0.43153. 0.42288. 0.41085. 0.39284. 0.36519. 0.31988. 0.24073. 0.09369. -0.17807. -0.57313. 0.9. 1.15477. 1.15467. 1.15451. 1.15329. 1.15246. 1.15143. 1.15049. 1.14950. 1.14733. 1.14473. 1.14067. 1.13379. 1.12140. 1.09693. 1.03955. 0.89267. 0.57313. 0.1.

(49) 29. 0.9973. -0.87979. -0.46263. -0.15004. 0.03041. 0.13267. 0.19375. 0.23248. 0.25791. 0.27595. 0.28826. 0.29812. 0.30553. 0.31174. 0.31929. 0.32423. 0.32793. 0.33089. λ. 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 5.5. 6. 7. 8. 9. 10. 1.38861. 1.38886. 1.38850. 1.38789. 1.38740. 1.38734. 1.38612. 1.38500. 1.38376. 1.38235. 1.37839. 1.37378. 1.36547. 1.34701. 1.30371. 1.18129. 0.87979. 0.0027. 0.39280. 0.39081. 0.38756. 0.38282. 0.37651. 0.37170. 0.36561. 0.35760. 0.34672. 0.33154. 0.30943. 0.27565. 0.22182. 0.13001. -0.03509. -0.32880. -0.73566. 0.99. 1.28128. 1.28117. 1.28073. 1.28032. 1.27923. 1.27878. 1.27776. 1.27617. 1.27512. 1.27235. 1.26898. 1.26302. 1.25228. 1.23069. 1.18000. 1.04573. 0.73566. 0.01. p. 0.49505. 0.49345. 0.49117. 0.48768. 0.48263. 0.47895. 0.47429. 0.46802. 0.45951. 0.44761. 0.43000. 0.40289. 0.35866. 0.28145. 0.13801. -0.12843. -0.52014. 0.95. 1.12620. 1.12581. 1.12532. 1.12449. 1.12315. 1.12232. 1.12099. 1.11945. 1.11721. 1.11396. 1.10899. 1.10099. 1.08695. 1.05918. 0.99682. 0.84335. 0.52014. 0.05. Table 5: The upper 100p percentage point z n,p,λ with n = 10. 0.55392. 0.55242. 0.55063. 0.54756. 0.54309. 0.54003. 0.53589. 0.53053. 0.52313. 0.51262. 0.49709. 0.47303. 0.43347. 0.36343. 0.23090. -0.02151. -0.40526. 0.9. 1.04656. 1.04592. 1.04539. 1.04452. 1.04297. 1.04181. 1.04045. 1.03845. 1.03597. 1.03210. 1.02642. 1.01701. 1.00077. 0.96921. 0.89992. 0.73563. 0.40526. 0.1.

(50) Table 6: The values of d∗2 = d∗2 (n, λ). n λ. 2. 3. 5. 10. 0. 1.12911. 1.69367. 2.32711. 3.08115. 0.5. 1.05192. 1.58093. 2.17469. 2.87833. 1. 0.92882. 1.39735. 1.91545. 2.54088. 1.5. 0.84086. 1.25966. 1.73410. 2.29813. 2. 0.78527. 1.17657. 1.62037. 2.14905. 2.5. 0.75200. 1.12598. 1.54586. 2.05087. 3. 0.73051. 1.09016. 1.50187. 1.98587. 3.5. 0.71187. 1.06904. 1.46737. 1.94245. 4. 0.70263. 1.05240. 1.44628. 1.90981. 4.5. 0.69443. 1.04051. 1.42725. 1.88546. 5. 0.68948. 1.03173. 1.41449. 1.86638. 5.5. 0.68484. 1.02533. 1.40719. 1.85437. 6. 0.68127. 1.01888. 1.39842. 1.84060. 7. 0.67686. 1.01191. 1.38780. 1.82527. 8. 0.67252. 1.00833. 1.38038. 1.81071. 9. 0.67035. 1.00566. 1.37539. 1.80380. 10. 0.66705. 1.00023. 1.36985. 1.79676. 30.

(51) 31. -2.11028. -1.75074. -1.43304. -1.18149. -0.98534. -0.83457. -0.72155. -0.62761. -0.55351. -0.49202. -0.44173. -0.39941. -0.33237. -0.28286. -0.24414. -0.21413 4.49738. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 5.5. 6. 7. 8. 9. 10. 4.47523. 4.46078. 4.43220. 4.40349. 4.38053. 4.35109. 4.32005. 4.26967. 4.21421. 4.10669. 3.98931. 3.82032. 3.56776. 3.22966. 2.83622. 2.46402. -2.46402. 0. 0.0027. 0.9973. λ. -0.11559. -0.13747. -0.16585. -0.20280. -0.25356. -0.28597. -0.32489. -0.37283. -0.43134. -0.50620. -0.59795. -0.72146. -0.88506. -1.09992. -1.37976. -1.71223. -2.06033. 0.99. 3.86152. 3.84250. 3.83009. 3.80555. 3.78090. 3.76119. 3.73591. 3.70926. 3.66600. 3.61838. 3.52607. 3.42528. 3.28019. 3.06333. 2.77229. 2.42187. 2.06033. 0.01. p. 0.05939. 0.04942. 0.03605. 0.01734. -0.00959. -0.02764. -0.05011. -0.07843. -0.11448. -0.16180. -0.22331. -0.30819. -0.42595. -0.58995. -0.81831. -1.11535. -1.45677. 0.95. 2.93826. 2.92379. 2.91434. 2.89567. 2.87691. 2.86192. 2.84268. 2.82240. 2.78949. 2.75325. 2.68301. 2.60632. 2.49591. 2.33064. 2.10429. 1.80589. 1.45677. 0.05. Table 7: The values of zp,λ /d∗2 with n = 2. 0.18004. 0.17517. 0.16858. 0.15825. 0.14257. 0.13131. 0.11677. 0.09794. 0.07269. 0.03872. -0.00835. -0.07460. -0.17040. -0.31040. -0.51492. -0.79623. -1.13500. 0.9. 2.46586. 2.45372. 2.44579. 2.43012. 2.41438. 2.40180. 2.38565. 2.36863. 2.34101. 2.31060. 2.25165. 2.18729. 2.09457. 1.95487. 1.75730. 1.47923. 1.13501. 0.1.




(55) 35. -1.39682. -1.08006. -0.82442. -0.63393. -0.49307. -0.38924. -0.31229. -0.25163. -0.20473. -0.16688. -0.13643. -0.11165. -0.07272. -0.04638. -0.02572. -0.00987 3.35750. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 5.5. 6. 7. 8. 9. 10. 3.35592. 3.33403. 3.31743. 3.28958. 3.27243. 3.24391. 3.23723. 3.17912. 3.16417. 3.08553. 2.99754. 2.87609. 2.69060. 2.42811. 2.09880. 1.74232. -1.74233. 0. 0.0027. 0.9973. λ. 0.08821. 0.07800. 0.06436. 0.04433. 0.01639. -0.00204. -0.02530. -0.05440. -0.09165. -0.14029. -0.20362. -0.29065. -0.41150. -0.57954. -0.81286. -1.11420. -1.45687. 0.99. 2.97359. 2.95323. 2.94591. 2.93143. 2.90733. 2.88998. 2.87430. 2.85573. 2.82243. 2.78795. 2.71026. 2.63158. 2.52267. 2.35125. 2.11631. 1.80788. 1.45688. 0.01. p. 0.28235. 0.27643. 0.26936. 0.25846. 0.24269. 0.23148. 0.21720. 0.19890. 0.17419. 0.14118. 0.09461. 0.02957. -0.06493. -0.20417. -0.40848. -0.69061. -1.03009. 0.95. 2.37073. 2.35742. 2.35128. 2.33519. 2.31931. 2.30682. 2.29187. 2.27549. 2.24821. 2.21849. 2.16130. 2.09659. 2.00298. 1.86203. 1.65833. 1.37501. 1.03009. 0.05. Table 11: The values of z n,p,λ /d∗2 with n = 2. 0.41921. 0.41432. 0.40880. 0.40023. 0.38819. 0.37953. 0.36827. 0.35397. 0.33382. 0.30711. 0.26704. 0.21140. 0.12819. 0.00142. -0.19030. -0.46427. -0.80254. 0.9. 2.06438. 2.05443. 2.04734. 2.03427. 2.02083. 2.00930. 1.99544. 1.98054. 1.95654. 1.92971. 1.87874. 1.82089. 1.73784. 1.60849. 1.41859. 1.14489. 0.80254. 0.1.

(56) 36. 0.9973. -0.94840. -0.71848. -0.51758. -0.36459. -0.25267. -0.17293. -0.11568. -0.07350. -0.04144. -0.01723. 0.00190. 0.01696. 0.02989. 0.04738. 0.06075. 0.06962. 0.07753. λ. 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 5.5. 6. 7. 8. 9. 10. 1.95299. 1.94852. 1.93991. 1.93530. 1.92017. 1.90593. 1.89185. 1.87686. 1.85986. 1.83440. 1.79207. 1.73806. 1.66094. 1.54764. 1.38334. 1.18030. 0.94840. 0.0027. 0.14656. 0.13995. 0.13454. 0.12460. 0.11160. 0.10209. 0.09008. 0.07462. 0.05480. 0.02840. -0.00726. -0.05670. -0.12702. -0.22798. -0.37099. -0.56461. -0.79302. 0.99. 1.73428. 1.72778. 1.72646. 1.71704. 1.71013. 1.69458. 1.68384. 1.66981. 1.65116. 1.62748. 1.59278. 1.53908. 1.47144. 1.36816. 1.21821. 1.02287. 0.79302. 0.01. p. 0.27953. 0.27567. 0.27136. 0.26557. 0.25603. 0.24925. 0.24082. 0.22984. 0.21510. 0.19498. 0.16746. 0.12727. 0.06887. -0.01890. -0.14947. -0.33409. -0.56071. 0.95. 1.42331. 1.41504. 1.41216. 1.40656. 1.39768. 1.38798. 1.37915. 1.36610. 1.35044. 1.32824. 1.30109. 1.25674. 1.19708. 1.10758. 0.97231. 0.78816. 0.56071. 0.05. Table 12: The values of z n,p,λ /d∗2 with n = 3. 0.36696. 0.36299. 0.35941. 0.35433. 0.34621. 0.33988. 0.33276. 0.32304. 0.31016. 0.29237. 0.26788. 0.23130. 0.17765. 0.09532. -0.03011. -0.21096. -0.43686. 0.9. 1.26396. 1.25740. 1.25305. 1.24868. 1.23955. 1.23160. 1.22362. 1.21171. 1.19703. 1.17697. 1.15185. 1.11115. 1.05616. 0.97210. 0.84347. 0.66340. 0.43686. 0.1.

(57) 37. 0.9973. -0.53466. -0.36819. -0.22933. -0.12605. -0.05440. -0.00501. 0.02966. 0.05462. 0.07335. 0.08704. 0.09733. 0.10546. 0.11171. 0.12184. 0.12848. 0.13314. 0.13668. λ. 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 5.5. 6. 7. 8. 9. 10. 1.21976. 1.21218. 1.20783. 1.19980. 1.19184. 1.18542. 1.17765. 1.16751. 1.15308. 1.13243. 1.10753. 1.07412. 1.02193. 0.94899. 0.84467. 0.70094. 0.53466. 0.0027. 0.18817. 0.18563. 0.18107. 0.17578. 0.16751. 0.16232. 0.15555. 0.14616. 0.13411. 0.11827. 0.09608. 0.06498. 0.01965. -0.04715. -0.14550. -0.28132. -0.44702. 0.99. 1.09810. 1.09310. 1.09084. 1.08401. 1.07585. 1.06894. 1.06254. 1.05214. 1.03850. 1.02267. 0.99758. 0.96703. 0.91856. 0.85103. 0.75180. 0.61250. 0.44702. 0.01. p. 0.27923. 0.27643. 0.27364. 0.26932. 0.26267. 0.25786. 0.25247. 0.24486. 0.23439. 0.22099. 0.20170. 0.17478. 0.13416. 0.07327. -0.01908. -0.15132. -0.31610. 0.95. 0.92977. 0.92561. 0.92194. 0.91697. 0.90919. 0.90282. 0.89761. 0.88870. 0.87622. 0.86220. 0.84044. 0.81308. 0.76952. 0.70729. 0.61489. 0.48065. 0.31610. 0.05. Table 13: The values of z n,p,λ /d∗2 with n = 5. 0.33479. 0.33209. 0.32916. 0.32531. 0.31891. 0.31427. 0.30947. 0.30235. 0.29239. 0.27999. 0.26157. 0.23624. 0.19741. 0.13882. 0.04891. -0.08188. -0.24628. 0.9. 0.84299. 0.83952. 0.83637. 0.83102. 0.82412. 0.81824. 0.81336. 0.80539. 0.79329. 0.78012. 0.75950. 0.73344. 0.69207. 0.63256. 0.54272. 0.41048. 0.24628. 0.1.

(58) 38. 0.9973. -0.28554. -0.16073. -0.05905. 0.01323. 0.06174. 0.09447. 0.11707. 0.13278. 0.14449. 0.15289. 0.15973. 0.16476. 0.16937. 0.17493. 0.17906. 0.18180. 0.18416. λ. 0. 0.5. 1. 1.5. 2. 2.5. 3. 3.5. 4. 4.5. 5. 5.5. 6. 7. 8. 9. 10. 0.77284. 0.76997. 0.76683. 0.76038. 0.75377. 0.74814. 0.74268. 0.73457. 0.72455. 0.71165. 0.69410. 0.66986. 0.63538. 0.58613. 0.51310. 0.41041. 0.28554. 0.0027. 0.21861. 0.21666. 0.21404. 0.20974. 0.20456. 0.20045. 0.19589. 0.18966. 0.18155. 0.17068. 0.15582. 0.13441. 0.10322. 0.05657. -0.01381. -0.11423. -0.23876. 0.99. 0.71311. 0.71026. 0.70731. 0.70144. 0.69501. 0.68960. 0.68462. 0.67685. 0.66767. 0.65502. 0.63900. 0.61585. 0.58271. 0.53552. 0.46441. 0.36331. 0.23876. 0.01. p. 0.27552. 0.27356. 0.27126. 0.26718. 0.26221. 0.25828. 0.25412. 0.24823. 0.24061. 0.23043. 0.21653. 0.19645. 0.16689. 0.12247. 0.05432. -0.04462. -0.16881. 0.95. 0.62680. 0.62413. 0.62148. 0.61607. 0.61021. 0.60523. 0.60062. 0.59373. 0.58499. 0.57348. 0.55844. 0.53684. 0.50578. 0.46089. 0.39231. 0.29300. 0.16881. 0.05. Table 14: The values of z n,p,λ /d∗2 with n = 10. 0.30829. 0.30626. 0.30409. 0.29999. 0.29506. 0.29122. 0.28713. 0.28138. 0.27392. 0.26390. 0.25031. 0.23065. 0.20170. 0.15814. 0.09087. -0.00747. -0.13153. 0.9. 0.58247. 0.57985. 0.57734. 0.57226. 0.56665. 0.56182. 0.55747. 0.55077. 0.54245. 0.53134. 0.51686. 0.49589. 0.46568. 0.42174. 0.35418. 0.25558. 0.13153. 0.1.

(59) Table 15: Type I error of acceptance control chart under the SN (λ) distribution, where δ = 0.0027, α = 0.01 method. CACC. BACC. TSNACC. ESNACC. λ. αL. αU. αL. αU. αL. αU. αL. αU. 0. 0.0096*. 0.0096. 0.0065. 0.0126. 0.0089. 0.0104*. 0.0089. 0.0104. 0.5. 0.0107*. 0.0081. 0.0051. 0.0147. 0.0092. 0.0101*. 0.0092. 0.0101. 1. 0.0188. 0.0041. 0.0038. 0.0112. 0.0098. 0.0098. 0.0098*. 0.0098*. 1.5. 0.0394. 0.0017. 0.0040. 0.0134. 0.0101*. 0.0092. 0.0103. 0.0094*. 2. 0.0793. 0.0009. 0.0028. 0.0124. 0.0096. 0.0095. 0.0101*. 0.0098*. 2.5. 0.1459. 0.0006. 0.0026. 0.0123. 0.0105*. 0.0103*. 0.0113. 0.0107. 3. 0.2083. 0.0004. 0.0020. 0.0119. 0.0099*. 0.0095. 0.0109. 0.0099*. 3.5. 0.2705. 0.0003. 0.0017. 0.0087. 0.0091. 0.0091. 0.0103*. 0.0094*. 4. 0.3476. 0.0003. 0.0000. 0.0000. 0.0079. 0.0108*. 0.0092*. 0.0114. 4.5. 0.3921. 0.0002. 0.0000. 0.0000. 0.0104*. 0.0091. 0.0121. 0.0096*. 5. 0.4444. 0.0002. 0.0042. 0.0181. 0.0070. 0.0104*. 0.0085*. 0.0108. 5.5. 0.4684. 0.0002. 0.0046. 0.0142. 0.0086. 0.0088. 0.0105*. 0.0094*. 6. 0.5081. 0.0002. 0.0000. 0.0000. 0.0107*. 0.0087. 0.0130. 0.0091*. 7. 0.5527. 0.0001. 0.0000. 0.0000. 0.0071. 0.0089. 0.0092*. 0.0093*. 8. 0.5981. 0.0001. 0.0000. 0.0000. 0.0071. 0.0092. 0.0093*. 0.0098*. 9. 0.6199. 0.0001. 0.0106*. 0.0167. 0.0062. 0.0087. 0.0080. 0.0092*. 10. 0.6430. 0.0001. 0.0133. 0.0154. 0.0081. 0.0087. 0.0105*. 0.0094*. ”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods. 39.

(60) Table 16: Type I error of acceptance control chart under the SN (λ) distribution, where δ = 0.0027, α = 0.05 method. CACC. BACC. TSNACC. ESNACC. λ. αL. αU. αL. αU. αL. αU. αL. αU. 0. 0.0482*. 0.0482. 0.0340. 0.0618. 0.0451. 0.0514*. 0.0451. 0.0514. 0.5. 0.0528*. 0.0417. 0.0280. 0.0688. 0.0466. 0.0502. 0.0466. 0.0501*. 1. 0.0844. 0.0228. 0.0224. 0.0558. 0.0493*. 0.0491*. 0.0493. 0.0491. 1.5. 0.1505. 0.0099. 0.0233. 0.0625. 0.0513. 0.0472. 0.0513*. 0.0472*. 2. 0.2484. 0.0051. 0.0177. 0.0620. 0.0503. 0.0489. 0.0502*. 0.0491*. 2.5. 0.3713. 0.0035. 0.0169. 0.0607. 0.0556. 0.0530*. 0.0554*. 0.0533. 3. 0.4582. 0.0024. 0.0132. 0.0584. 0.0538. 0.0490. 0.0534*. 0.0493*. 3.5. 0.5308. 0.0018. 0.0113. 0.0452. 0.0512. 0.0470. 0.0509*. 0.0473*. 4. 0.6123. 0.0017. 0.0000. 0.0000. 0.0478*. 0.0555*. 0.0474. 0.0558. 4.5. 0.6507. 0.0014. 0.0000. 0.0000. 0.0580. 0.0475. 0.0576*. 0.0478*. 5. 0.6954. 0.0014. 0.0219. 0.0808. 0.0452*. 0.0530*. 0.0446. 0.0541. 5.5. 0.7126. 0.0011. 0.0265. 0.0683. 0.0522. 0.0465. 0.0516*. 0.0475*. 6. 0.7425. 0.0010. 0.0000. 0.0000. 0.0608. 0.0451. 0.0601*. 0.0459*. 7. 0.7736. 0.0009. 0.0000. 0.0000. 0.0474*. 0.0462. 0.0471. 0.0465*. 8. 0.8049. 0.0009. 0.0000. 0.0000. 0.0482*. 0.0483. 0.0476. 0.0490*. 9. 0.8178. 0.0008. 0.0445*. 0.0769. 0.0435. 0.0456. 0.0427. 0.0463*. 10. 0.8324. 0.0008. 0.0522. 0.0723. 0.0521. 0.0452. 0.0513*. 0.0462*. ”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods. 40.

(61) Table 17: Type I error of acceptance control chart under the SN (λ) distribution, where δ = 0.01, α = 0.01 method. CACC. BACC. TSNACC. ESNACC. λ. αL. αU. αL. αU. αL. αU. αL. αU. 0. 0.0097. 0.0097. 0.0098*. 0.0098. 0.0092. 0.0101*. 0.0092. 0.0101. 0.5. 0.0104*. 0.0088. 0.0084. 0.0111. 0.0095. 0.0100. 0.0095. 0.0100*. 1. 0.0151. 0.0059. 0.0075. 0.0095. 0.0098. 0.0098. 0.0099*. 0.0099*. 1.5. 0.0256. 0.0034. 0.0077. 0.0096. 0.0100*. 0.0094. 0.0102. 0.0096*. 2. 0.0436. 0.0022. 0.0070. 0.0089. 0.0096. 0.0096. 0.0100*. 0.0098*. 2.5. 0.0727. 0.0017. 0.0072. 0.0090. 0.0102*. 0.0102*. 0.0109. 0.0106. 3. 0.1004. 0.0013. 0.0065. 0.0090. 0.0096. 0.0096. 0.0106. 0.0098*. 3.5. 0.1293. 0.0011. 0.0062. 0.0078. 0.0090. 0.0092. 0.0102*. 0.0097*. 4. 0.1687. 0.0011. 0.0000. 0.0000. 0.0081. 0.0104*. 0.0095*. 0.0109. 4.5. 0.1928. 0.0010. 0.0000. 0.0000. 0.0097*. 0.0093. 0.0114. 0.0097*. 5. 0.2225. 0.0009. 0.0106. 0.0119. 0.0074. 0.0101*. 0.0090*. 0.0105. 5.5. 0.2371. 0.0008. 0.0107. 0.0100*. 0.0085. 0.0091. 0.0103*. 0.0095. 6. 0.2616. 0.0008. 0.0000. 0.0000. 0.0098*. 0.0090. 0.0119. 0.0093*. 7. 0.2910. 0.0007. 0.0000. 0.0000. 0.0073. 0.0091. 0.0094*. 0.0096*. 8. 0.3226. 0.0007. 0.0000. 0.0000. 0.0074. 0.0094. 0.0094*. 0.0097*. 9. 0.3390. 0.0007. 0.0171. 0.0111. 0.0064. 0.0089. 0.0086*. 0.0095*. 10. 0.3561. 0.0008. 0.0196. 0.0107. 0.0078. 0.0089. 0.0101*. 0.0094*. ”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods. 41.

(62) Table 18: Type I error of acceptance control chart under the SN (λ) distribution, where δ = 0.01, α = 0.05 method. CACC. BACC. TSNACC. ESNACC. λ. αL. αU. αL. αU. αL. αU. αL. αU. 0. 0.0486*. 0.0486. 0.0467. 0.0511. 0.0465. 0.0505*. 0.0465. 0.0505. 0.5. 0.0515*. 0.0443. 0.0419. 0.0554. 0.0475. 0.0498*. 0.0475. 0.0498. 1. 0.0719. 0.0307. 0.0380. 0.0489. 0.0494*. 0.0493*. 0.0494. 0.0493. 1.5. 0.1110. 0.0179. 0.0390. 0.0479. 0.0509. 0.0479. 0.0508*. 0.0480*. 2. 0.1673. 0.0115. 0.0361. 0.0478. 0.0502. 0.0491. 0.0501*. 0.0493*. 2.5. 0.2412. 0.0090. 0.0363. 0.0472. 0.0543. 0.0521*. 0.0541*. 0.0525. 3. 0.2963. 0.0068. 0.0321. 0.0468. 0.0527. 0.0492. 0.0525*. 0.0496*. 3.5. 0.3461. 0.0056. 0.0294. 0.0410. 0.0508. 0.0477. 0.0506*. 0.0482*. 4. 0.4079. 0.0053. 0.0000. 0.0000. 0.0488*. 0.0539*. 0.0485. 0.0542. 4.5. 0.4386. 0.0048. 0.0000. 0.0000. 0.0558. 0.0478. 0.0554*. 0.0482*. 5. 0.4774. 0.0045. 0.0428. 0.0577. 0.0470*. 0.0519*. 0.0464. 0.0528. 5.5. 0.4913. 0.0041. 0.0470. 0.0510*. 0.0515. 0.0473. 0.0510*. 0.0480. 6. 0.5203. 0.0040. 0.0000. 0.0000. 0.0574. 0.0466. 0.0570*. 0.0466*. 7. 0.5504. 0.0036. 0.0000. 0.0000. 0.0482*. 0.0468. 0.0477. 0.0477*. 8. 0.5835. 0.0036. 0.0000. 0.0000. 0.0490*. 0.0487. 0.0480. 0.0494*. 9. 0.5972. 0.0032. 0.0617. 0.0558. 0.0452*. 0.0463. 0.0445. 0.0472*. 10. 0.6133. 0.0032. 0.0673. 0.0527*. 0.0513. 0.0465. 0.0508*. 0.0469. ”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods. 42.

(63) Table 19: Type I error of acceptance control chart under gamma distribution with parameters θ = 1 and k, where δ = 0.0027, α = 0.01 method. CACC. BACC. TSNACC. ESNACC. k. αL. αU. αL. αU. αL. αU. αL. αU. 5. 0.4267. 0.0001. 0.0050*. 0.0119*. 0.0029. 0.0045. 0.0038. 0.0047. 10. 0.2049. 0.0003. 0.0098*. 0.0108*. 0.0119. 0.0062. 0.0130. 0.0064. 15. 0.1417. 0.0006. 0.0042. 0.0140. 0.0142*. 0.0084. 0.0150. 0.0087*. 20. 0.1030. 0.0008. 0.0044. 0.0112*. 0.0153*. 0.0082. 0.0159. 0.0085. 25. 0.0817. 0.0010. 0.0059. 0.0116. 0.0131*. 0.0093. 0.0136. 0.0095*. 30. 0.0717. 0.0012. 0.0041. 0.0155. 0.0129*. 0.0102*. 0.0133. 0.0104. 35. 0.0605. 0.0013. 0.0049. 0.0129. 0.0129*. 0.0096. 0.0133. 0.0098*. 40. 0.0548. 0.0015. 0.0074*. 0.0117. 0.0130. 0.0097. 0.0133. 0.0099*. 45. 0.0541. 0.0019. 0.0073*. 0.0144. 0.0137. 0.0113*. 0.0140. 0.0115. 50. 0.0500. 0.0020. 0.0063. 0.0148. 0.0125*. 0.0117*. 0.0127. 0.0119. 55. 0.0441. 0.0020. 0.0060. 0.0110. 0.0140*. 0.0095. 0.0142. 0.0097*. 60. 0.0423. 0.0022. 0.0059. 0.0123. 0.0138*. 0.0098. 0.0141. 0.0100*. ”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods. 43.

(64) Table 20: Type I error of acceptance control chart under gamma distribution with parameters θ = 1 and k, where δ = 0.0027, α = 0.05 method. CACC. BACC. TSNACC. ESNACC. k. αL. αU. αL. αU. αL. αU. αL. αU. 5. 0.6831. 0.0005. 0.0282*. 0.0597*. 0.0238. 0.0257. 0.0234. 0.0261. 10. 0.4522. 0.0016. 0.0422*. 0.0534*. 0.0593. 0.0337. 0.0590. 0.0340. 15. 0.3620. 0.0033. 0.0235. 0.0682. 0.0674. 0.0446. 0.0672*. 0.0448*. 20. 0.2934. 0.0046. 0.0254. 0.0569. 0.0707. 0.0432. 0.0706*. 0.0433*. 25. 0.2504. 0.0057. 0.0307. 0.0597. 0.0626. 0.0479. 0.0625*. 0.0480*. 30. 0.2295. 0.0073. 0.0249. 0.0741. 0.0619. 0.0520*. 0.0619*. 0.0522. 35. 0.2031. 0.0080. 0.0261. 0.0617. 0.0616. 0.0492. 0.0615*. 0.0493*. 40. 0.1893. 0.0091. 0.0354. 0.0571. 0.0620. 0.0495. 0.0620*. 0.0495*. 45. 0.1887. 0.0111. 0.0353*. 0.0690. 0.0650. 0.0566*. 0.0650. 0.0567. 50. 0.1782. 0.0120. 0.0309. 0.0702. 0.0602. 0.0583*. 0.0601*. 0.0584. 55. 0.1620. 0.0120. 0.0323. 0.0543. 0.0657. 0.0485. 0.0657*. 0.0485*. 60. 0.1572. 0.0131. 0.0317. 0.0602. 0.0653. 0.0497. 0.0653*. 0.0497*. ”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods. 44.

(65) Table 21: Type I error of acceptance control chart under gamma distribution with parameters θ = 1 and k, where δ = 0.01, α = 0.01 method. CACC. BACC. TSNACC. ESNACC. k. αL. αU. αL. αU. αL. αU. αL. αU. 5. 0.1863. 0.0008. 0.0080*. 0.0098*. 0.0030. 0.0084. 0.0038. 0.0086. 10. 0.0859. 0.0013. 0.0131. 0.0083. 0.0088. 0.0089. 0.0096*. 0.0092*. 15. 0.0625. 0.0020. 0.0078. 0.0105. 0.0105*. 0.0103*. 0.0112. 0.0107. 20. 0.0481. 0.0023. 0.0080. 0.0094. 0.0114*. 0.0098. 0.0119. 0.0101*. 25. 0.0402. 0.0026. 0.0094. 0.0089. 0.0104*. 0.0104*. 0.0108. 0.0106. 30. 0.0366. 0.0030. 0.0072. 0.0110. 0.0104*. 0.0109*. 0.0108. 0.0111. 35. 0.0322. 0.0031. 0.0085. 0.0102*. 0.0105*. 0.0103. 0.0108. 0.0106. 40. 0.0300. 0.0034. 0.0109. 0.0091. 0.0107*. 0.0103*. 0.0110. 0.0105. 45. 0.0300. 0.0038. 0.0110*. 0.0105*. 0.0112. 0.0114. 0.0115. 0.0116. 50. 0.0284. 0.0040. 0.0099*. 0.0105*. 0.0106. 0.0116. 0.0108. 0.0117. 55. 0.0259. 0.0040. 0.0095. 0.0092. 0.0115*. 0.0100*. 0.0117. 0.0102. 60. 0.0252. 0.0042. 0.0095. 0.0099*. 0.0115*. 0.0102. 0.0116. 0.0103. ”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods. 45.

(66) Table 22: Type I error of acceptance control chart under gamma distribution with parameters θ = 1 and k, where δ = 0.01, α = 0.05 method. CACC. BACC. TSNACC. ESNACC. k. αL. αU. αL. αU. αL. αU. αL. αU. 5. 0.4338. 0.0035. 0.0393*. 0.0508*. 0.0239. 0.0433. 0.0235. 0.0439. 10. 0.2660. 0.0067. 0.0524. 0.0434. 0.0476*. 0.0456. 0.0474. 0.0459*. 15. 0.2155. 0.0102. 0.0374. 0.0542. 0.0543. 0.0527*. 0.0542*. 0.0529. 20. 0.1782. 0.0121. 0.0398. 0.0496. 0.0571. 0.0498. 0.0571*. 0.0500*. 25. 0.1557. 0.0137. 0.0439. 0.0484*. 0.0527. 0.0524. 0.0526*. 0.0525. 30. 0.1456. 0.0158. 0.0383. 0.0566. 0.0530. 0.0549*. 0.0529*. 0.0550. 35. 0.1316. 0.0165. 0.0402. 0.0512*. 0.0530. 0.0521. 0.0529*. 0.0522. 40. 0.1248. 0.0178. 0.0477*. 0.0467. 0.0536. 0.0519*. 0.0536. 0.0519. 45. 0.1256. 0.0203. 0.0483*. 0.0535*. 0.0562. 0.0569. 0.0561. 0.0570. 50. 0.1202. 0.0212. 0.0442. 0.0537*. 0.0532. 0.0577. 0.0532*. 0.0578. 55. 0.1112. 0.0210. 0.0457*. 0.0472. 0.0567. 0.0504*. 0.0567. 0.0504. 60. 0.1090. 0.0222. 0.0455*. 0.0508*. 0.0567. 0.0511. 0.0567. 0.0511. ”*” denotes the type I error which is closet to the nominal value 0.01 among the four methods. 46.

(67)