ARL-unbiased control charts for the monitoring of ...

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ARL-unbiased control charts for the monitoring of exponentially distributed characteristics based on type-II censored samples a

a

b

Baocai Guo , Bing Xing Wang & Min Xie a

School of Statistics, Zhejiang Gongshang University, Hangzhou, People's Republic of China b

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Department of Systems Engineering and Engineering Management, City University of Hong Kong, Hong Kong, People's Republic of China Published online: 21 Mar 2014.

To cite this article: Baocai Guo, Bing Xing Wang & Min Xie (2014) ARL-unbiased control charts for the monitoring of exponentially distributed characteristics based on type-II censored samples, Journal of Statistical Computation and Simulation, 84:12, 2734-2747, DOI: 10.1080/00949655.2014.898766 To link to this article: http://dx.doi.org/10.1080/00949655.2014.898766

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Journal of Statistical Computation and Simulation, 2014 Vol. 84, No. 12, 2734–2747, http://dx.doi.org/10.1080/00949655.2014.898766


ARL-unbiased control charts for the monitoring of exponentially distributed characteristics based on type-II censored samples Baocai Guoa , Bing Xing Wanga∗ and Min Xieb a School of Statistics, Zhejiang Gongshang University, Hangzhou, People’s Republic of China; b Department of Systems Engineering and Engineering Management, City University of Hong Kong,

Hong Kong, People’s Republic of China (Received 27 August 2013; accepted 24 February 2014) In this paper, the problem of monitoring process data that can be modelled by exponential distribution is considered when observations are from type-II censoring. Such data are common in many practical inspection environment.An average run length unbiased (ARL-unbiased) control scheme is developed when the in-control scale parameter is known. The performance of the proposed control charts are investigated in terms of the ARL and standard deviation of the run length. The effects of parameter estimation on the proposed control charts are also evaluated. Then, we consider the design of the ARL-unbiased control charts when the in-control scale parameter is estimated. Finally, an example is used to illustrate the implementation of the proposed control charts. Keywords: exponential distribution; average run length; process control; unbiased control chart; type-II censoring AMS Subject Classification: Primary: 62P30; Secondary: 62F10

1.

Introduction

Control charts for the monitoring of exponential distribution has attracted much attention recently, as statistical process control procedures are used in many non-manufacturing environments. Recently, there have been some studies on the monitoring of time-between-events (TBE) and exponential distribution is the most common type of TBE distribution. Jones and Champ [1] provided methods for computing the control limits of Phase I control charts for time between events and evaluated the overall type-I error rates of these charts. Scariano and Calzada [2] derived the lower-sided synthetic chart for exponentials. Zhang et al. [3] derived economic design of exponential charts for time between events monitoring, and evaluated the performances of statistical design, economic design and economic-statistical design. Zhang et al. [4] considered exponential control charts using a sequential sampling scheme and proposed an average run length unbiased (ARL-unbiased) design approach. A control chart for the Gamma distribution as a model of time between events was studied by Zhang et al. [5] Wu et al. [6] proposed a control scheme for monitoring the frequency and magnitude of an event. Ozsan et al. [7] studied the effects of parameter estimation on the performance of the exponential EWMA control chart. Kao [8] investigated how a location parameter influences the transformation results for exponentially distributed data to ∗ Corresponding

author. Email: [email protected]

© 2014 Taylor & Francis


Journal of Statistical Computation and Simulation

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allow approximately normally distribution. Hsu and Shu [9] proposed a new two-phase controlling method for monitoring an Erlang-failure process. Zhang et al. [10] proposed an economic model of the exponential chart for monitoring TBEs data under random process shift. Cheng and Chen [11] considered an ARL-unbiased design of TBEs control charts with runs rules, and proposed a simple and effective procedure to design a cumulative quantity control chart. Dovoedo and Chakraborti [12] proposed Boxplot-based phase I control charts for time between events. Qu et al. [13] proposed a single CUSUM scheme for simultaneously monitoring the time interval and magnitude of an event. For the design of control charts, it is often assumed that we have complete observations. In many practical applications such as reliability and survival analysis, however, observations are often censored because of the cost or time limitations.[14–18] Type-II censoring is an important form of censoring. It happens when testing is terminated at the time of the rth unit failure. One particular situation in which type-II censored sample can be obtained is that for a k-out-of-n: F system consisting of n independent and identical components, if the lifetimes of r failure components are recorded, then we have a type-II censored sample. Moreover, when the distribution of the components has exponential, the control charts for the exponential distribution based on the typeII censored sample can be used to monitor the k-out-of-n: F system. Meanwhile, since the TBE is actually failure times of 1-out-of-1 system: F, this study can also be considered as an extension of the basic TBE-chart. Many studies mentioned above are based on probability limits or 3-sigma limits. They are not ARL unbiased. A control chart is said to be ARL-unbiased if its ARL curve achieves its maximum when the process is in-control [19–21]. The ARL-biased problem is highly undesirable in practice, since it takes a longer time on average to signal the out-of-control process than that when the process is in-control. This paper only considers ARL-unbiased control charts. This paper is organized as follows. In Section 2, we propose the monitoring statistic to monitor the scale parameter, and develop the control charts based on type-II censored sample when the in-control scale parameter is known. In Section 3, we evaluate the effects of parameter estimation on the proposed control charts. In Section 4, we provide new control charts with the adjusted control limits when the in-control scale parameter is estimated. Finally, an example is shown to illustrate the proposed control charts. The design and performance of the proposed control charts are obtained by developing computer programs using Matlab R2011b.

2.

Design of the control charts with known parameter

2.1. Monitoring statistic The probability density function (p.d.f.) of the exponential distribution is given by f (x) =

x 1 exp − , θ θ

x ≥ 0,

(1)

where θ > 0 is the scale parameter. Suppose that X1 ≤ X2 ≤ · · · ≤ Xr is a type-II censored sample with size n from the exponential distribution (1). Let 2[ ri=1 Xi + (n − r)Xr ] T (θ ) = . θ Then, T (θ ) ∼ χ 2 (2r).[22] Let θ0 be the target value of the scale parameter θ when the process is in control, and θ1 = ρθ0 is the true value of θ . If ρ = 1, the parameter θ is in control, otherwise the parameter θ has shifted.

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Notice that r

r


T (θ0 ) =

2[

i=1

Xi + (n − r)Xr ] θ1 2[ = θ0 θ0

i=1

Xi + (n − r)Xr ] = ρT (θ1 ), θ1

thus the value of the statistic T (θ0 ) becomes larger on average when θ shifts from θ0 up to θ1 (> θ0 , or equivalently ρ > 1), as well as the value of T (θ0 ) becomes smaller on average when θ shifts from θ0 down to θ1 (< θ0 , or equivalently 0 < ρ < 1). Thus, T (θ0 ) can be used to monitor the changes in θ. In particular, the upper one-sided control chart can be used to monitor the upward shifts in the scale parameter, the lower one-sided control chart is used to monitor the downward shifts, and the two-sided control chart is used to monitor the upward and downward shifts. 2.2.

Control chart with equal-tailed probability limits

When the in-control scale parameter θ0 is known, for a given false alarm rate α, let UCL and LCL be the upper and lower control limits of the two-sided control chart based on the monitoring statistic T (θ0 ) and the traditional equal-tailed probability limits are given by 2 LCL = χα/2 (2r),

2 (2r), UCL = χ1−α/2

respectively. Where χα2 (k) is α percentile of the χ 2 (k) distribution. For the two-sided control chart, the probability that a point falls between the control limits is given by β(ρ) = P(LCL ≤ T (θ0 ) ≤ UCL | θ = θ1 ) LCL UCL =P ≤ T (θ1 ) ≤ θ = θ 1 ρ ρ UCL UCL = Fχ2r2 − Fχ2r2 , ρ ρ

(2)

where Fχk2 (x) is the cumulative distribution function of the χ 2 (k) distribution. Since the distribution of the run length of the two-sided control chart is geometric, the ARL function of this control chart is given by ARL(ρ) =

1 . 1 − β(ρ)

Figure 1 gives the ARL performance for the two-sided control charts based on the traditional equal-tailed probability limits for r = 3 and 6 when α = 0.0027. It is observed from Figure 1 that the control chart with the traditional equal-tailed probability limits is not ARL-unbiased. This is an undesirable result as it means that when the process deteriorates as θ decreases from a nominal value, it will take a longer time to give an alarm than the in-control process. 2.3. Design of the ARL-unbiased control charts Since the control chart with the equal-tailed probability limits is not ARL-unbiased, we need to look for UCL and LCL which can result in an ARL-unbiased control chart based on T (θ0 ). The following theorem provides a procedure how to obtain the control limits of the ARL-unbiased two-sided control chart.


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450 r=3 r=6

400 370.37 350

ARL(ρ)


300 250 200 150 100 50 0

0

0.5

1

1.5

ρ

2

2.5

3

3.5

Figure 1. ARL curves of the control charts based on the equal-tailed probability limits for r = 3, 6 when α = 0.0027.

Theorem 2.1 For a given false alarm rate α, the upper and lower control limits of the ARLunbiased two-sided control chart for θ , UCL and LCL are, respectively, given by 2 (2r), UCL = χ1−α+β

LCL = χβ2 (2r),

(3)

in which β ∈ (0, α) is determined by the following equation 2 2 2 (χβ2 (2r)) = fχ2r+2 (χ1−α+β (2r)), fχ2r+2

(4)

where fχk2 (x) is the p.d.f. of the χ 2 (k) distribution. Proof Suppose that θ0 and θ1 = ρθ0 are the in-control and true values of θ , respectively. Notice that the probability that a point falls beyond the control limits of the proposed two-sided control chart is given by p1 (ρ) = 1 − P(LCL ≤ T (θ0 ) ≤ UCL | θ = θ1 ) LCL UCL =1−P ≤ T (θ1 ) ≤ θ = θ1 ρ ρ UCL/ρ =1− fχ2r2 (x) dx,

(5)

LCL/ρ

the control limits of the ARL-unbiased control chart should then satisfy the conditions: p1 (1) = α, and p1 (ρ) > p1 (1), for ∀ρ = 1. 2 Thus, we need to look for β ∈ (0, α) such that LCL = χβ2 (2r), UCL = χ1−α+β (2r) can lead to that p1 (ρ) reaches its minimum at ρ = 1. As a consequence, p1 (1) = 0 or equivalently UCL · fχ2r2 (UCL) = LCL · fχ2r2 (LCL).

(6)

2 Notice that, for the χ 2 (2r) distribution, it always has that xfχ2r2 (x) = 2rfχ2r+2 (x). Therefore, 2 2 the Equation (6) is rewritten as fχ2r+2 (UCL) = fχ2r+2 (LCL). So the value of β leading to the

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Table 1. Values of β for the ARL-unbiased control chart. α 0.0050 0.0027 0.0020 0.0010

r=3

r=4

r=5

r=6

r=7

r=8

r=9

0.00384531 0.00210746 0.00157149 0.00079690

0.00369651 0.00202596 0.00151083 0.00076636

0.00358734 0.00196559 0.00146568 0.00074338

0.00350313 0.00191872 0.00143052 0.00072537

0.00343570 0.00188104 0.00140220 0.00071079

0.00338018 0.00184992 0.00137877 0.00069868

0.00333344 0.00182366 0.00135898 0.00068843


ARL-unbiased control chart satisfies the following equation: 2 2 2 fχ2r+2 (χ1−α+β (2r)) = fχ2r+2 (χβ2 (2r)).

Notice that the p.d.f. of the χ 2 (2r + 2) distribution is unimodal, the above equation has unique solution. The proof is completed. It is observed from Equations (3) and (4) that the control limits of the ARL-unbiased control chart depend on the failure number r and the false alarm rate α, not on the sample size n. Table 1 gives the values of β which are obtained numerically. Since the monitoring statistic T (θ0 ) is an increasing function of ρ, the upper and lower one-sided control charts are ARL-unbiased. For a given false alarm rate α, the UCL and LCL of the upper 2 and lower one-sided charts are, respectively, given by UCL = χ1−α (2r) and LCL = χα2 (2r). For the upper and lower one-sided control charts, we have UCL LCL 2 2 p2 (ρ) = 1 − Fχ2r and p3 (ρ) = Fχ2r , ρ ρ respectively. Note that the run lengths of the proposed two-sided and one-sided control charts follow the geometric distributions with success probabilities pi , the means and standard deviations of the run lengths of these charts are thus given by ARLi (ρ) = and

1 , pi (ρ)

√ 1 − pi (ρ) SDRLi (ρ) = , pi (ρ)

(7)

i = 1, 2, 3,

(8)

respectively. It is observed from Equations (7) and (8) that the out-of-control ARLs and standard deviation of the run lengths (SDRLs) of the proposed two-sided and one-sided control charts depend only on ρ, the failure number r and the false alarm rate α, but not on the sample size n. It can also be observed that the ARL is larger than the corresponding SDRL because of the fact that 1 − pi (ρ) < 1 for all ρ > 0, regardless of the one-sided or two-sided control charts.

3. The performance of the control charts with the estimated parameter The discussion above assumes that the target parameter θ0 is known. In most applications, however, the target parameter θ0 is unknown and has to be estimated by Phase I samples. It is well known that the performance of the control charts with estimated parameters are different from that of



2739

the control charts with known parameters due to the variability of the estimated parameters. So it is essential to study the effects of parameter estimation on the performance of the control charts. Quesenberry,[23] Jones et al. [24,25] studied the effects of the parameter estimation on the properties of the traditional control charts. Jensen et al. [26] gave a review of literature on this issue. In this section, we consider the effects of the parameter estimation on the performance of the proposed control charts. Suppose that m type-II censored samples with size n and the failure number r are obtained from a stable exponential process. Let Yi,1 ≤ Yi,2 ≤ · · · ≤ Yi,r is the ith type-II censored sample. Then, the unbiased estimator of θ0 based on the ith sample is given by r j=1 Yi,j + (n − r)Yi,r θˆ0,i = , i = 1, . . . , m. r Thus, the unbiased estimator of θ is given by θ¯ˆ = (1/m) m θˆ , and 2mr θ¯ˆ /θ ∼ χ 2 (2mr). 0

0,m

i=1 0,i

0,m

0

For simplicity, when the target value θ0 is replaced by its estimator θ¯ˆ0,m , the control charts in Section 2 are called the control charts with the estimated parameter. When θ0 is unknown, it is natural that the monitoring statistic T (θ0 ) is replaced by T (θ¯ˆ0,m ). Since T (θ¯ˆ0,m ) = 2mrρT (θ1 )/(2mr θ¯ˆ0,m /θ0 ), given Zm = 2mr θ¯ˆ0,m /θ0 = z, the conditional probability that a point falls beyond the control limits of the proposed two-sided control chart is given by q1,r,m (ρ, z) = 1 − P(LCL ≤ T (θ¯ˆ0,m ) ≤ UCL | Zm = z, θ = θ1 ) zUCL zLCL = 1 − Fχ2r2 + Fχ2r2 . (2mrρ) (2mrρ) For the upper and lower one-sided control charts, we have q2,r,m (ρ, z) = P(T (θ¯ˆ0,m ) > UCL | Zm = z, θ = θ1 ) zUCL = 1 − Fχ2r2 , (2mrρ) and q3,r,m (ρ, z) = P(T (θ¯ˆ0,m ) < LCL | Zm = z, θ = θ1 ) zLCL = Fχ2r2 , (2mrρ) respectively. Let RLi , i = 1, 2, 3 be the run lengths of the proposed two-sided, upper and lower one-sided control charts with the estimated parameter , respectively. According to Quesenberry,[23] different alarm events are not independent, but the conditional distributions of the run lengths under Zm = z are geometric. Thus, the first two moments of RLi of the proposed control charts with the estimated parameter are given by ∞ 1 (9) f 2 (z) dz, ARLi,r,m (ρ) = E(RLi ) = qi,r,m (ρ, z) χ2mr 0 and

E(RL2i )

respectively.

= 0

∞

2 − qi,r,m (ρ, z) 2 (z) dz, fχ2mr 2 qi,r,m (ρ, z)

i = 1, 2, 3,

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B. Guo et al.

Table 2. The ARLs and SDRLs of the proposed two-sided control charts with the estimated parameter when α = 0.0027. ρ (r, m)

6.0

4.0

2.0

1.5

1.3

1.1

1.0

0.9

0.8

0.6

0.5

0.3

0.2

0.1

(3, 25)

1.5 (0.9) 1.5 (0.8) 1.5 (0.8) 1.5 (0.8) 1.5 (0.8) 1.2 (0.6) 1.2 (0.5) 1.2 (0.5) 1.2 (0.5) 1.2 (0.5) 1.1 (0.4) 1.1 (0.4) 1.1 (0.4) 1.1 (0.4) 1.1 (0.4)

2.4 (1.9) 2.3 (1.8) 2.3 (1.8) 2.3 (1.8) 2.3 (1.7) 1.8 (1.3) 1.8 (1.2) 1.8 (1.2) 1.8 (1.2) 1.8 (1.2) 1.5 (0.9) 1.5 (0.9) 1.5 (0.9) 1.5 (0.9) 1.5 (0.9)

17.4 (21.8) 15.6 (15.7) 15.5 (15.4) 15.3 (15.0) 15.2 (14.7) 11.8 (13.8) 10.8 (10.6) 10.8 (10.5) 10.7 (10.3) 10.6 (10.1) 8.7 (9.7) 8.1 (7.8) 8.1 (7.7) 8.0 (7.6) 8.0 (7.5)

77.5 (107.0) 66.7 (71.1) 66.1 (69.3) 65.1 (66.0) 64.4 (63.9) 56.9 (78.2) 48.9 (51.6) 48.6 (50.4) 47.9 (48.3) 47.4 (46.9) 43.8 (59.1) 38.0 (39.6) 37.7 (38.8) 37.2 (37.3) 36.9 (36.4)

162.9 (205.0) 153.1 (163.1) 152.3 (159.8) 150.8 (153.6) 149.7 (149.2) 136.4 (176.2) 123.9 (132.1) 123.0 (129.1) 121.5 (123.7) 120.4 (119.9) 115.8 (152.2) 102.9 (109.6) 102.2 (107.0) 100.8 (102.4) 99.8 (99.3)

287.9 (309.6) 317.3 (321.8) 319.2 (322.6) 322.9 (324.0) 325.5 (325.0) 279.6 (303.2) 304.6 (310.2) 306.1 (310.3) 309.1 (310.6) 311.0 (310.6) 271.8 (297.1) 292.7 (299.1) 293.9 (298.8) 296.2 (298.0) 297.7 (297.2)

323.3 (332.0) 360.4 (360.4) 362.8 (362.6) 367.2 (366.8) 370.37 369.9 322.6 (331.2) 360.2 (360.2) 362.6 (362.4) 367.2 (366.7) 370.37 369.9 322.1 (330.7) 360.0 (360.0) 362.5 (362.3) 367.1 (366.7) 370.37 369.9

311.2 (319.6) 328.2 (329.9) 328.8 (330.1) 329.9 (330.2) 330.5 (330.0) 302.0 (312.3) 315.8 (318.2) 316.2 (318.0) 317.0 (317.4) 317.3 (316.9) 293.1 (305.5) 304.0 (307.0) 304.3 (306.6) 304.7 (305.4) 304.9 (304.4)

256.7 (271.7) 253.1 (256.2) 252.7 (254.9) 251.9 (252.5) 251.4 (250.9) 230.6 (248.9) 223.9 (227.4) 223.4 (225.9) 222.4 (223.2) 221.8 (221.3) 208.1 (228.3) 199.5 (203.1) 198.9 (201.5) 197.9 (198.7) 197.2 (196.7)

123.9 (135.9) 117.1 (118.5) 116.8 (117.7) 116.2 (116.2) 115.8 (115.3) 90.5 (100.6) 84.9 (86.0) 84.7 (85.3) 84.2 (84.2) 83.8 (83.4) 68.3 (76.4) 63.8 (64.6) 63.6 (64.1) 63.3 (63.1) 63.0 (62.5)

76.1 (83.2) 72.0 (72.6) 71.8 (72.1) 71.4 (71.2) 71.2 (70.7) 49.7 (54.7) 46.8 (47.1) 46.7 (46.8) 46.4 (46.2) 46.2 (45.7) 34.3 (37.7) 32.2 (32.3) 32.1 (32.1) 32.0 (31.6) 31.9 (31.3)

20.7 (22.0) 19.7 (19.5) 19.7 (19.4) 19.6 (19.2) 19.5 (19.0) 10.7 (11.1) 10.2 (9.9) 10.2 (9.8) 10.2 (9.7) 10.1 (9.6) 6.4 (6.3) 6.1 (5.7) 6.1 (5.6) 6.1 (5.6) 6.1 (5.5)

8.2 (8.2) 7.8 (7.4) 7.8 (7.3) 7.8 (7.3) 7.7 (7.2) 3.9 (3.6) 3.8 (3.3) 3.8 (3.3) 3.8 (3.3) 3.8 (3.2) 2.4 (1.9) 2.3 (1.8) 2.3 (1.8) 2.3 (1.7) 2.3 (1.7)

2.3 (1.8) 2.2 (1.7) 2.2 (1.7) 2.2 (1.7) 2.2 (1.6) 1.3 (0.7) 1.3 (0.6) 1.3 (0.6) 1.3 (0.6) 1.3 (0.6) 1.1 (0.3) 1.1 (0.3) 1.1 (0.3) 1.1 (0.3) 1.1 (0.3)

(3, 150) (3, 200)


(3, 500) (3, ∞) (4, 25) (4, 150) (4, 200) (4, 500) (4, ∞) (5, 25) (5, 150) (5, 200) (5, 500) (5, ∞)

Thus, the SDRLs of the proposed control charts with the estimated parameter are given by SDRLi,r,m (ρ) =

E(RL2i ) − [E(RLi )]2 ,

i = 1, 2, 3,

(10)

respectively. In particular, ARLi,r,m (1) is the in-control ARL, i = 1, 2, 3. It is observed from Equations (9) and (10) that the performance of the proposed control charts with the estimated parameter are independent of the sample size n and the target value θ0 . Tables 2–4 report the ARLs and SDRLs of the proposed two-sided and one-sided control charts with the estimated parameter when α = 0.0027, respectively. m = ∞ corresponds to the known parameter case. It is observed from Tables 2–4 that the proposed two-sided control chart with the estimated parameter can result in more false alarms than that with the known parameter, but the one-sided control charts with the estimated parameter have less false alarms than that with the known parameter. Similar to the known parameter case, the one-sided control charts can detect the shifts in θ quicker than the corresponding two-sided control chart. It is also observed that when m ≥ 500, the performance of the proposed two-sided control chart with the estimated parameter is similar to that with the known parameter. When m ≥ 500, the performance of the upper one-sided control chart with the estimated parameter is similar to that with the known parameter. When m ≥ 200,


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Table 3. The ARLs and SDRLs of upper one-sided control charts with the estimated parameter when α = 0.0027. ρ (r, m)

1.0

1.1

1.3

1.5

2.0

4.0

6.0

(r, m)

1.0

1.1

1.3

1.5

2.0

4.0

6.0

(3, 25)

616.8 (1499.8) 434.2 (580.1) 400.5 (463.5) 392.7 (438.2) 379.1 (395.9) 370.37 (369.9) 587.4 (1304.1) 428.1 (557.1) 397.8 (454.3)

264.4 (541.5) 200.4 (252.9) 187.9 (211.0) 185.0 (201.7) 180.0 (185.9) 176.6 (176.1) 235.7 (445.5) 184.1 (226.8) 173.8 (192.8)

76.1 (123.8) 63.5 (73.9) 60.8 (65.3) 60.2 (63.4) 59.0 (60.0) 58.3 (57.8) 62.3 (94.9) 53.2 (60.6) 51.2 (54.4)

32.2 (45.1) 28.3 (31.2) 27.5 (28.6) 27.3 (28.0) 26.9 (26.9) 26.7 (26.2) 25.1 (33.3) 22.6 (24.3) 22.0 (22.6)

8.9 (10.0) 8.3 (8.3) 8.2 (7.9) 8.2 (7.8) 8.1 (7.7) 8.1 (7.6) 6.7 (7.1) 6.4 (6.1) 6.3 (5.9)

1.9 (1.3) 1.9 (1.3) 1.9 (1.3) 1.8 (1.3) 2.8 (1.3) 1.8 (1.2) 1.5 (0.9) 1.5 (0.9) 1.5 (0.9)

1.3 (0.7) 1.3 (0.6) 1.3 (0.6) 1.3 (0.6) 1.3 (0.6) 1.3 (0.6) 1.2 (0.4) 1.2 (0.4) 1.2 (0.4)

(4, 200)

390.7 (431.7) 378.3 (393.6) 370.37 (369.9) 569.2 (330.7)) 424.2 (542.5) 396.0 (448.3) 389.4 (427.4) 377.8 (392.0) 370.37 (369.9)

171.4 (185.1) 167.1 (172.0) 164.3 (163.8) 215.2 (297.1) 171.5 (207.7) 162.6 (178.8) 160.5 (172.2) 156.8 (161.0) 154.3 (153.8)

50.8 (53.0) 50.0 (50.5) 49.4 (48.9) 52.8 (152.2 45.8 (51.4) 44.3 (46.6) 43.9 (45.5) 43.3 (43.6) 42.8 (42.3)

21.9 (22.1) 21.6 (21.4) 21.4 (20.9) 20.5 (59.1) 18.6 (19.8) 18.2 (18.5) 18.1 (18.2) 17.9 (17.7) 17.8 (17.3)

6.3 (5.8) 6.2 (5.8) 6.2 (5.7) 5.3 (9.7) 5.1 (4.8) 5.1 (4.6) 5.0 (4.6) 5.0 (4.5) 5.0 (4.5)

1.5 (0.9) 1.5 (0.9) 1.5 (0.9) 1.3 (0.9) 1.3 (0.7) 1.3 (0.7) 1.3 (0.7) 1.3 (0.7) 1.3 (0.7)

1.2 (0.4) 1.2 (0.4) 1.2 (0.4) 1.1 (0.4) 1.1 (0.3) 1.1 (0.3) 1.1 (0.3) 1.1 (0.3) 1.1 (0.3)

(3, 75) (3, 150)


ρ

(3, 200) (3, 500) (3, ∞) (4, 25) (4, 75) (4, 150)

(4, 500) (4, ∞) (5, 25) (5, 75) (5, 150) (5, 200) (5, 500) (5, ∞)

Table 4. The ARLs and SDRLs of lower one-sided control charts with the estimated parameter when α = 0.0027. ρ

ρ

(r, m)

1.0

0.9

0.6

0.5

0.3

0.2

0.1

(r, m)

1.0

0.9

0.6

0.5

0.3

0.2

0.1

(3, 25)

398.6 (442.2) 379.4 (392.4) 374.8 (381.0) 373.7 (378.2) 371.7 (373.2) 370.37 (369.9) 403.1 (456.8) 380.9 (396.7) 375.6 (383.1)

296.9 (328.8) 282.8 (292.2) 279.4 (283.7) 278.5 (281.7) 277.1 (278.0) 276.1 (275.6) 277.3 (313.1) 262.3 (272.8) 258.7 (263.6)

97.9 (107.3) 93.5 (96.0) 92.4 (93.4) 92.1 (92.7) 91.7 (91.6) 91.4 (90.9) 69.3 (76.8) 65.9 (67.9) 65.1 (65.8)

60.3 (65.7) 57.7 (59.0) 57.1 (57.5) 56.9 (57.1) 56.7 (56.4) 56.5 (56.0) 38.5 (42.1) 36.7 (37.4) 36.3 (36.4)

16.8 (17.6) 16.1 (16.0) 16.0 (15.7) 15.9 (15.6) 15.9 (15.4) 15.8 (15.3) 8.7 (8.8) 8.3 (8.0) 8.3 (7.9)

6.8 (6.7) 6.5 (6.2) 6.5 (6.0) 6.5 (6.0) 6.5 (6.0) 6.4 (5.9) 3.3 (3.0) 3.2 (2.7) 3.2 (2.7)

2.0 (1.5) 2.0 (1.4) 2.0 (1.4) 2.0 (1.4) 2.0 (1.4) 2.0 (1.4) 1.2 (0.6) 1.2 (0.5) 1.2 (0.5)

(4, 200)

374.3 (379.7) 371.9 (373.8) 370.37 (369.9) 406.7 (468.6) 382.0 (400.2) 376.1 (384.7) 374.7 (381.0) 372.1 (374.3) 370.37 (369.9)

257.8 (261.4) 256.3 (257.4) 255.2 (254.7) 260.5 (298.5) 245.1 (256.1) 241.4 (246.6) 240.5 (244.2) 238.9 (240.1) 237.9 (237.4)

64.9 (65.3) 64.6 (64.4) 64.3 (63.8) 51.3 (57.1) 48.7 (50.1) 48.1 (48.5) 47.9 (48.1) 47.7 (47.4) 47.5 (47.0)

36.2 (36.1) 36.0 (35.7) 35.9 (35.4) 26.2 (28.6) 25.0 (25.3) 24.7 (24.6) 24.6 (24.4) 24.5 (24.1) 24.4 (23.9)

8.3 (7.8) 8.2 (7.7) 8.2 (7.7) 5.2 (5.0) 5.0 (4.6) 5.0 (4.5) 5.0 (4.5) 5.0 (4.4) 4.9 (4.4)

3.2 (2.7) 3.2 (2.7) 3.2 (2.7) 2.1 (1.6) 2.0 (1.5) 2.0 (1.5) 2.0 (1.4) 2.0 (1.4) 2.0 (1.4)

1.2 (0.5) 1.2 (0.5) 1.2 (0.5) 1.1 (0.2) 1.1 (0.2) 1.0 (0.2) 1.0 (0.2) 1.0 (0.2) 1.0 (0.2)

(3, 75) (3, 150) (3, 200) (3, 500) (3, ∞) (4, 25) (4, 75) (4, 150)

(4, 500) (4, ∞) (5, 25) (5, 75) (5, 150) (5, 200) (5, 500) (5, ∞)

the performance of the lower one-sided control chart with the estimated parameter is similar to that with the known parameter. It is important to note from Tables 2–4 that SDRL can exceed the corresponding ARL especially when m is not large, which is in contrast to the case of known parameter where SDRL < ARL, regardless of the one-sided or two-sided control charts. Considering the nature of the run length distribution, it can be observed that when the SDRL exceeds the ARL, a larger number of short runs and long runs would be expected, when compared with the standard geometric distribution for the known parameter case. This is an undesirable phenomenon of which user should be aware.

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Table 5. Values of UCLr,m and LCLr,m of the proposed ARL-unbiased two-sided control charts when the parameter is estimated and ARL0 = 370.37. m

r=3

r=4

r=5

r=6

m

r=3

r=4

r=5

r=6

10

25.64552 0.47562 25.16911 0.48118 24.89007 0.48432 24.70431 0.48634 24.57078 0.48776 24.39041 0.48961 24.27343 0.49076

28.91161 1.00412 28.47475 1.01433 28.21810 1.02013 28.04694 1.02388 27.92379 1.02651 27.75726 1.02996 27.64918 1.03212

32.12042 1.67133 31.70380 1.68660 31.45866 1.69529 31.29505 1.70094 31.17727 1.70491 31.01795 1.71012 30.91451 1.71340

35.26299 2.44588 34.85750 2.46636 34.61870 2.47805 34.45926 2.48565 34.34445 2.49101 34.18913 2.49806 34.08829 2.50250

60

24.19106 0.49155 24.12975 0.49211 24.08228 0.49254 24.04439 0.49287 24.01343 0.49314 23.91670 0.49395 23.70161 0.49552

27.57303 1.03360 27.51634 1.03467 27.47242 1.03548 27.43737 1.03612 27.40873 1.03663 27.31921 1.03818 27.12000 1.04126

30.84162 1.71565 30.78736 1.71728 30.74532 1.71852 30.71177 1.71950 30.68435 1.72028 30.59866 1.72266 30.40796 1.72747

34.01723 2.50555 33.96434 2.50778 33.92336 2.50947 33.89066 2.51080 33.86393 2.51187 33.78041 2.51514 33.59460 2.52179

15 20


25 30 40 50

4.

70 80 90 100 150 ∞

Design of the control charts when the parameter is estimated

The results of Section 3 indicate that only when the number of Phase I samples is large, the performance of the control charts with the estimated parameter is similar to that with the known parameter. In most cases, however, it may not be practical to wait for the accumulation of such large Phase I data set because of the cost or time limitations.[23,27,28] Thus, when the number of Phase I samples is small or moderate, one may consider adjusting the control limits of the control charts to obtain the desired in-control ARL.[26,29,30] In this section, we shall consider the design of the ARL-unbiased control charts with the desired in-control ARL, ARL0 , when the parameter is estimated. Suppose that a Phase I data set consisting of m type-II censored samples with size n and the failure number r is available. When the scale parameter is estimated, notice from Equation (9) that the ARLs of the proposed control charts are independent of the sample size n, regardless of whether the process is in-control or out-of-control, let UCLr,m and LCLr,m be thus the new upper and lower control limits of the ARL-unbiased two-sided control chart with the desired ARL0 , which is called the control chart with the adjusted control limits, based on the monitoring statistic T (θ¯ˆ0,m ), respectively. Since the control chart is ARL-unbiased, ARL1,r,m (ρ) attains the maximum value ARL0 at ρ = 1. Hence, the control limits UCLr,m and LCLr,m should satisfy the following two equations: ARL1,r,m (1) = ARL0 (11) and

0

∞

2 2 fχ2r+2 (zUCLr,m /(2mr)) − fχ2r+2 (zLCLr,m /(2mr))

[1 − Fχ2r2 (zUCLr,m /(2mr)) + Fχ2r2 (zLCLr,m /(2mr))]2

2 (z) dz = 0, fχ2mr

(12)

where Equation (12) is derived by (dARL1,r,m (ρ)/dρ)|ρ=1 = 0. The proof is similar to Theorem 2.1. Since the control limits UCLr,m and LCLr,m cannot be computed by analytical means, they must be computed by solving numerically Equations (11) and (12). Table 5 reports the values of UCLr,m and LCLr,m for m = 10, 15, . . . , 150 and r = 3, 4, 5, 6 when ARL0 = 370.37, where m = ∞ represents the known θ0 case. It is observed from Table 5 that, as m increases, the values of UCLr,m decrease and eventually converge to the values corresponding to the known parameter case, however, the values of LCLr,m increase and eventually converge to the values corresponding to the known parameter case.



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Figure 2. Comparison of the control charts with the known parameter, the estimated parameter and the adjusted control limits when r = 3 and ARL0 = 370.37.

In order to assess the performance of the proposed two-sided control chart with the adjusted control limits, Figure 2 gives the ARL curves for the control charts for r = 3 and m = 10, 20, 30 and 80 when ARL0 = 370.37. For comparison purpose, the corresponding ARL curves for the control charts with the known and estimated parameter are given. The other r cases are similar in nature and not provided for saving space. It is observed from Figure 2 that using the control chart with the adjusted control limits UCLr,m , LCLr,m has two advantages: the in-control ARL is the desired ARL0 and the control chart is ARL-unbiased. It is interesting to observe that when r ≥ 3 and m ≥ 80, the performance of the two-sided control chart with the adjusted control limits is almost the same as that with the known parameter. It is also observed that the two-sided control chart with the estimated parameter is ARL-biased and results in more false alarms than expected in the in-control situation. Let UCLr,m (LCLr,m ) be the upper (lower) control limit of the upper (lower) one-sided control chart with the desired ARL0 based on the monitoring statistic T (θ¯ˆ0,m ). Hence, the control limit UCLr,m should satisfy the following equation: ARL2,r,m (1) =

∞

0

1 f 2 (z) dz = ARL0 , 1 − Fχ2r2 (zUCLr,m /(2mr)) χ2mr

(13)

and the control limit LCLr,m should satisfy the following equation: ARL3,r,m (1) = 0

∞

1 f 2 (z) dz = ARL0 . Fχ2r2 (zLCLr,m /(2mr)) χ2mr

(14)

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Table 6. Values of UCLr,m and LCLr,m of the one-sided control charts when the parameter is estimated and ARL0 = 370.37. UCLr,m


m 10 15 20 25 30 40 50 60 70 80 90 100 150 ∞

LCLr,m

r=3

r=4

r=5

r=6

r=3

r=4

r=5

r=6

17.46310 18.28285 18.70966 18.97136 19.14821 19.37199 19.50773 19.59884 19.66422 19.71342 19.75179 19.78255 19.87516 20.06190

20.99448 21.81805 22.24318 22.50264 22.67747 22.89812 23.03164 23.12114 23.18529 23.23354 23.27115 23.30128 23.39193 23.57439

24.31638 25.14781 25.57460 25.83429 26.00894 26.22900 26.36196 26.45100 26.514787 26.56273 26.60009 26.63001 26.71999 26.90091

27.49586 28.33723 28.76739 29.02858 29.20401 29.42478 29.55803 29.64719 29.71105 29.75903 29.79640 29.82633 29.91630 30.09705

0.57881 0.56566 0.55933 0.55560 0.55315 0.55011 0.54831 0.54712 0.54627 0.54563 0.54514 0.54475 0.54357 0.54123

1.19905 1.17440 1.16250 1.15549 1.15087 1.14515 1.14175 1.13949 1.13789 1.13669 1.13576 1.13502 1.13279 1.12837

1.96834 1.93092 1.91282 1.90214 1.89510 1.88639 1.88120 1.87777 1.87532 1.87349 1.87207 1.87093 1.86754 1.86079

2.84962 2.79886 2.77427 2.75976 2.75019 2.73833 2.73128 2.72660 2.72327 2.72077 2.71884 2.71729 2.71267 2.70347

It is obvious that the one-sided control chart determined by Equation (13) or (14) is ARLunbiased. Table 6 reports the values of UCLr,m and LCLr,m of the one-sided control charts for m = 10, 15, . . . , 150 and r = 3, 4, 5, 6 when ARL0 = 370.37, where m = ∞ represents the known θ0 case. It is observed from Table 6 that, as m increases, the values of UCLr,m increase and eventually converge to the values corresponding to the known parameter case, however, the values of LCLr,m decrease and eventually converge to the values corresponding to the known parameter case.

5. An illustrative example In this section, we present a simulated example to illustrate the use of our proposed control charts. Assume that the target value of θ is θ0 = 1. Table 7 presents 30 type-II censored samples with size n = 10 and the failure number r = 3. The first 10 samples were drawn from the exponential distribution with θ0 = 1, which represents the Phase I data set. The second 10 were also drawn from the exponential process with θ0 = 1, which represents the in-control process. The last 10 were drawn from a different exponential process with θ1 = 2, which represents the out-of-control process. The two-sided control charts are used to detect the shifts in θ . The estimate of θ0 is θ¯ˆ0,10 = 0.96322 by the first 10 samples. Thus, the monitoring statistic is T (0.96322). Since r =3 and m =10, for the desired in-controlARL, ARL0 =370.37, from Table 5, the adjusted control limits of the control chart are UCL3,10 = 25.64552 and LCL3,10 = 0.47562, respectively. For comparison, we also give the performance of the control chart with the estimated parameter. The control limits 2 are UCL = χ1−0.0027+β (6) = 23.70161 and LCL = χβ2 (6) = 0.49552, in which β can be obtained from Table 1. It can be observed from Figure 3 that, when the values of the monitoring statistic T (0.96322) for the 11–20th samples, which come from the in-control process, are plotted against the control limits, sample #16, falling beyond the upper control limit of the control chart with the estimated parameter, is out-of-control. However, sample #16, falling between the control limits of the control chart with the adjusted control limits, is in-control. This is due to the fact that the control chart with the adjusted control limits takes the variability of the parameter estimate into account and

Journal of Statistical Computation and Simulation Table 7.

Simulated data from the exponential distribution. Sample no.

Censored data 0.00833 0.01158 0.30389 0.03996 0.41509 0.10739 0.10212 0.12701 0.00467 0.00227

0.09318 0.09531 0.51695 0.04136 0.51015 0.22432 0.44493 0.21559 0.06396 0.00419

0.09722 0.35362 0.60616 0.05489 0.52976 0.24631 0.58787 0.23503 0.13884 0.34586

11 12 13 14 15 16 17 18 19 20

Censored data 0.03536 0.03733 0.36287 0.01205 0.16929 0.11898 0.00207 0.38124 0.06791 0.01471

0.04840 0.20164 0.40848 0.11184 0.23037 0.61867 0.04646 0.45122 0.19186 0.06792

0.34015 0.24939 0.43086 0.21553 0.23525 1.34811 0.45416 0.50236 0.37298 0.18245

Sample no. 21 22 23 24 25 26 27 28 29 30

Censored data 0.10918 0.22793 0.19566 0.10705 0.13831 0.15592 0.23519 0.12787 0.02215 0.09172

0.19656 0.28666 0.60892 0.54131 0.20106 0.16638 0.68664 0.24539 0.27677 0.22189

0.36218 0.43910 0.66217 0.72961 0.23297 0.46248 1.50679 0.60279 0.36814 0.30227

UCL3,10=25.64552 UCL=23.70161 101 T(0.96322)


Sample No. 1 2 3 4 5 6 7 8 9 10

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100 LCL=0.49552 LCL3,10= 0.47562 10

15

20

25

30

35

Sample No. Figure 3. The performance of the control charts with the adjusted control limits/the estimated parameter.

has wider control limits, which results in an improved in-control performance. For the last 10 samples drawn from the out-of-control process, the control charts with both the adjusted control limits and the estimated parameter have successfully detected the shift.

6.

Conclusions

In this paper, we develop the ARL-unbiased one-sided and two-sided control charts to monitor the shifts in the scale parameter of the exponential distribution under type-II censoring when the in-control scale parameter is known. The one-sided control charts have better detection ability than the corresponding two-sided control charts, as expected. We also study the effects of the parameter estimation on the performance of the proposed control charts. It shows that the parameter estimation seriously affects the performance of the proposed control charts, that is, for the upper one-sided, lower one-sided and two-sided control charts, the number of Phase I samples m of at least 500, 200 and 500, respectively, can ensure that the difference between the known parameter case and the estimated parameter case can be negligible. Since taking more than 200 samples is often impractical, we develop the ARL-unbiased control charts with the adjusted control limits which have the desired in-control ARL when the number of Phase I samples available to estimate the in-control parameter is small. We observe that when

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r ≥ 3 and m ≥ 80, the performance of the two-sided control chart with the adjusted control limits is almost the same as that with the known parameter. Acknowledgements The authors thank the Editor, the Associate Editor and the referees for their detailed comments and suggestions, which helped improve the manuscript.


Funding The work of Guo is sponsored by ‘Foundation of Ministry of Education of China [Grant No. 13YJC910005 and No. 13YJC910010]’, ‘Philosophy and Social Science Research Project in Zhejiang Province of China [Grant No. 13NDJC055YB]’, ‘National Statistical Research Plan Project [Grant No. 2012LY163]’, and ‘Zhejiang Provincial Key Research Base for Humanities and Social Science Research (Statistics)’. The work of Wang is supported by ‘the National Natural Science Foundation of China under the contract number [11371322]’ and ‘Zhejiang Provincial Key Research Base for Humanities and Social Science Research (Statistics)’. The work by M Xie was supported by ‘National Natural Science Foundation of China under the contract number [71371163]’

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