Nirpeksh Kumar
email: nirpeksh@gmail.com

Subha Chakraborti

Philippe Castagliola
email: philippe.castagliola@univ-nantes.fr

Phase II Exponential Charts for Monitoring Time Between Events Data: Performance Analysis Using Exact Conditional Average Time to Signal Distribution

Keywords: Conditional and Unconditional perspectives, Equal-tailed and ATS-unbiased exponential chart, Exact distribution, Guaranteed IC performance, Quality control

In this paper, we study the performance properties of the phase II exponential chart with an unknown in-control (IC) rate parameter, used to monitor high-yield processes. The average time to signal (ATS) is used as the chart performance criterion instead of the usual average run length (ARL). Based on the IC conditional ATS (CATS) distribution, we examine the properties of both the equal-tailed and the ATS-unbiased exponential charts with estimated control limits and adjust the limits so that a nominal IC ATS performance is achieved. Two perspectives are investigated: the unconditional, under which the average of IC CATS distribution is set equal to a nominal ATS and, the conditional so that the IC CATS is set to or exceed a nominal ATS with a high probability. It is shown that the ATS-unbiased exponential chart under the conditional perspective has a better IC performance.

Introduction

Processes with a very low rate of nonconforming items, say, parts per million (ppm) are termed as high-yield processes. High-yield processes can be encountered in the manufacturing of integrated circuits, automated manufacturing processes, in health care surveillance, in the monitoring of earthquake occurrences [START_REF] Xie | Some effective control chart procedures for reliability monitoring[END_REF][START_REF] Xie | Monitoring time-between-events for health management[END_REF][START_REF] Santiago | Control charts based on the exponential distribution: adapting runs rules for the t chart[END_REF]. Monitoring such processes with conventional attribute control charts such as the 𝑝-, 𝑛𝑝-charts is known to be not as effective due to several reasons such as increased FAR (false alarm rate), negative lower control limits, etc. [START_REF] Xie | Monitoring time-between-events for health management[END_REF]. Instead, for such processes, the time between two consecutive nonconforming items (or events of interest) is recommended as the quality characteristic to be monitored. The control charts which consider the times between nonconforming items (events) are known as TBE (Time Between Events) control charts and they have several advantages over the conventional attribute charts [START_REF] Montgomery | Introduction to statistical quality control[END_REF]. In such situations, the occurrences of nonconforming items may be modeled by a homogeneous Poisson process with a constant rate of nonconforming, say, 𝜆, and therefore, the TBE, which is also known as the inter-arrival time, follows an exponential distribution with mean 1/𝜆. Several control charts (Shewhart-type or not) have been proposed based on this exponential assumption [START_REF] Xie | Some effective control chart procedures for reliability monitoring[END_REF][START_REF] Santiago | Control charts based on the exponential distribution: adapting runs rules for the t chart[END_REF][START_REF] Gan | Exact run length distributions for one-sided exponential CUSUM schemes[END_REF][START_REF] Gan | Designs of one-and two-sided exponential EWMA charts[END_REF][START_REF] Gan | Computing average run lengths of exponential EWMA charts[END_REF][START_REF] Liu | A study of EWMA chart with transformed exponential data[END_REF][START_REF] Vardeman | Average run lengths for CUSUM schemes when observations are exponentially distributed[END_REF][START_REF] Zhang | Design of exponential control charts using a sequential sampling scheme[END_REF][START_REF] Zhang | A control chart for the gamma distribution as a model of time between events[END_REF][START_REF] Rakitzis | Monitoring exponential data using two-sided control charts with runs rules[END_REF][START_REF] Guo | ARL-unbiased control charts for the monitoring of exponentially distributed characteristics based on type-II censored samples[END_REF]. The readers may also refer to the review papers by Liu et al. [START_REF] Liu | A comparative study of exponential time between events charts[END_REF] and Ali et al. [START_REF] Ali | An overview of control charts for high-quality processes[END_REF]. Recently, Qu et al. [START_REF] Qu | Exponential cumulative sums chart for detecting shifts in time-between-events[END_REF] proposed a weighted cumulative sum chart based on the exponential distribution using the power parameter of sample mean shifts i.e. using (𝑥 𝑖 - ) in the CUSUM statistic where 𝑥 𝑖 is the 𝑖-th observation (TBE data) and 𝑤 > 0 is the power parameter. In order to improve the sensitivity of the exponential chart against small shifts in the process parameter, Cheng et al. [START_REF] Shah | Time between events control charts for gamma distribution[END_REF] proposed a synthetic exponential chart consisting of an exponential sub-chart and a conforming run length sub-chart. Recently, Shah et al. [START_REF] Cheng | Phase II synthetic exponential charts and effect of parameter estimation[END_REF] discussed the TBE chart for the gamma distribution using an exact distribution of the sample TBE mean.

Most often the control charts are evaluated by their run length and associated statistical properties such as the mean, standard deviation and percentiles. The run length may be defined as the total number of charting points to be plotted on the chart to get the first OOC signal. We should mention here that the run length only considers the plotted points on the chart, but, it completely ignores the time needed to obtain these points, which is an important factor in the context of the TBE monitoring. Therefore, in the Statistical Process Control (SPC) literature it is recommended that the times needed to obtain the charting points must be considered in evaluating the chart's performance [START_REF] Montgomery | Introduction to statistical quality control[END_REF][START_REF] Zhang | A control chart for the gamma distribution as a model of time between events[END_REF]. We denote the time to signal by TS, as the waiting time to get the first OOC signal. Because the TS conveys more information about the process than the run length, its average i.e. the average time to signal (denoted ATS) may be a preferred metric of chart performance over the traditional average of run length i.e. the ARL, which is more meaningful when the plotting points are obtained at equal time intervals. Several authors including Zhang et al. [START_REF] Zhang | A control chart for the gamma distribution as a model of time between events[END_REF], Bourke [START_REF] Bourke | Performance comparisons for the synthetic control chart for detecting increases in fraction nonconforming[END_REF], Yang et al. [START_REF] Yang | Design of exponential control charts based on average time to signal using a sequential sampling scheme[END_REF] have used the ATS to evaluate the TBE chart's performance.

It is well accepted that when a chart parameter is unknown, the control limits for monitoring (in phase II) may be constructed by replacing this parameter in the control limits for the known parameter case, by a suitable estimator using an IC retrospective (phase I or a reference) sample. It has been established by several studies that parameter estimation can drastically affect the control chart's properties in a negative way and the resulting chart performs much differently (many more false alarms, for example) from what it is nominally expected [START_REF] Chakraborti | Run length distribution and percentiles: the Shewhart chart with unknown parameters[END_REF][START_REF] Epprecht | Effect of the amount of Phase I data on the Phase II performance of S 2 and Scontrol charts[END_REF][START_REF] Kumar | Design and implementation of qth quantile-unbiased t r -chart for monitoring times between events[END_REF][START_REF] Kumar | Phase II Shewhart-type control charts for monitoring times between events and effects of parameter estimation[END_REF][START_REF] Kumar | Bayesian monitoring of times between events: The Shewhart t r -chart[END_REF][START_REF] Zhang | Exponential CUSUM charts with estimated control limits[END_REF]. Note that most of the studies evaluate the performance of phase II control charts using the unconditional run length distribution (that is averaging over the distribution of the estimator) and some associated characteristics. However, it is now well recognized that the unconditional analysis may not provide a very realistic picture about the control chart's properties for a given estimate from a given phase I sample [START_REF] Chakraborti | Run length distribution and percentiles: the Shewhart chart with unknown parameters[END_REF][START_REF] Epprecht | Effect of the amount of Phase I data on the Phase II performance of S 2 and Scontrol charts[END_REF][START_REF] Saleh | The difficulty in designing Shewhart X-bar and X control charts with estimated parameters[END_REF]. The recent papers by Diko et al. [START_REF] Diko | Guaranteed in-control performance of the EWMA chart for monitoring the mean[END_REF]; Guo and Wang [START_REF] Guo | The design of the S 2 control charts based on conditional performance via exact methods[END_REF]; Kumar [START_REF] Kumar | Conditional analysis of phase II exponential chart for monitoring times to an event[END_REF] highlighted the importance of the conditional performance analysis and advocated the use of the conditional run length (CRL) distribution and the conditional average run length CARL distribution and its associated properties to evaluate the chart's performance for a given phase I sample. As noted before, for TBE data, the TS is the more suitable random variable of interest and the average time to signal (ATS) has been the chosen performance metric. Thus, following the literature, with estimated parameters from a phase I sample, the phase II control limits of TBE control charts are mainly constructed and examined under the unconditional perspective, that is based on the expected value of conditional ATS (CATS) distribution [START_REF] Zhang | A control chart for the gamma distribution as a model of time between events[END_REF][START_REF] Yang | Design of gamma charts based on average time to signal[END_REF].

Note that the CATS is a random variable in the unknown parameter case, whereas in the known parameter case, it is a constant. In this paper, we derive the exact distribution of CATS and investigate chart properties under both conditional and unconditional perspectives using various characteristics of the CATS distribution such as the average, standard deviation and percentiles. Note that the conditional approach is advocated to take into account the practitioner-to-practitioner variability in evaluating the chart's performance [START_REF] Saleh | The difficulty in designing Shewhart X-bar and X control charts with estimated parameters[END_REF][START_REF] Goedhart | Shewhart control charts for dispersion adjusted for parameter estimation[END_REF].

When the plotting statistic has a skewed distribution, the common design to construct the control limits of two-sided control charts in the known parameter case is based on the conventional equaltailed probability limits (henceforth called the equal-tailed chart). However, it is found that the equal-tailed chart based on the ATS leads to an ATS-biased chart [START_REF] Yang | Design of exponential control charts based on average time to signal using a sequential sampling scheme[END_REF][START_REF] Yang | Design of gamma charts based on average time to signal[END_REF]. A control chart is said to be ATS-biased chart if its ATS function does not achieve the maximum when the process is IC. This implies that the chart takes on average, longer time to raise an OOC signal, than it takes to raise a false alarm. This phenomenon has been considered as a highly undesirable property of a control chart [START_REF] Zhang | A control chart for the gamma distribution as a model of time between events[END_REF][START_REF] Yang | Design of gamma charts based on average time to signal[END_REF]. To overcome this undesirable situation, ATS-unbiased charts are proposed in the literature [START_REF] Yang | Design of exponential control charts based on average time to signal using a sequential sampling scheme[END_REF][START_REF] Yang | Design of gamma charts based on average time to signal[END_REF] which do not have this shortcoming and provide a more balanced guard against both process deterioration and improvement.

Many control charts are designed and evaluated on the basis of their ARL. It has been acknowledged in the literature that when control charts for the known parameters are used with estimated parameters, the variability in the run length distribution increases which shortens the IC ARL, which increases the number of false alarms, and thus reduces the value of process monitoring.

In order to rectify the situation, many authors including Aly et al. [START_REF] Aly | A re-evaluation of the adaptive exponentially weighted moving average control chart when parameters are estimated[END_REF]; Diko et al. [START_REF] Diko | Guaranteed in-control performance of the EWMA chart for monitoring the mean[END_REF]; Faraz et al. [START_REF] Faraz | Guaranteed conditional performance of the S 2 control chart with estimated parameters[END_REF][START_REF] Faraz | The np chart with guaranteed in-control average run lengths[END_REF]; Gandy and Kvaloy [START_REF] Gandy | Guaranteed conditional performance of control charts via bootstrap methods[END_REF]; Goedhart et al. [START_REF] Goedhart | Shewhart control charts for dispersion adjusted for parameter estimation[END_REF][START_REF] Goedhart | On guaranteed in-control performance for the Shewhart 𝑋 and 𝑋 ̅ control charts[END_REF]; Hu and Castagliola [START_REF] Hu | Guaranteed conditional design of the median chart with estimated parameters[END_REF]; Jones and Steiner [START_REF] Jones | Assessing the effect of estimation error on the risk-adjusted CUSUM chart performance[END_REF]; Zhao and Driscoll [START_REF] Zhao | The c-chart with bootstrap adjusted control limits to improve conditional performance[END_REF] have considered adjusting the control limits. In this paper, we consider adjusting control limits of both equal-tailed and ATS-unbiased exponential charts under both unconditional and conditional perspectives. Under the unconditional perspective, the control limits will be adjusted so that the mean of IC CATS is equal to a nominal ATS 0 value whereas under the conditional perspective, they will be adjusted so that the IC CATS meets or exceeds the nominal ATS 0 value with a high pre-fixed probability.

The rest of this paper is organized as follows. In section 2, the phase II control limits of the exponential chart are constructed and the distribution of the CATS is derived. Section 3 examines the conditional performance of the phase II equal-tailed and ATS-unbiased exponential charts for a given nominal ATS 0 value. The control limits of these charts are then adjusted to guarantee the IC performance in Section 4 and their conditional IC and OOC performances are evaluated. An example is given for illustration in Section 5 and some concluding remarks are summarized in Section 6.

The Exponential chart and the CATS distribution

2.1.

Case of the known parameter

Let 𝑋 𝑖 , 𝑖 = 1,2, … denote the time between two consecutive nonconforming items in a process which follows an exponential distribution with density function 𝑓(𝑥) = 𝜆 exp(-𝜆𝑥) , 𝑥 > 0, 𝜆 > 0. The process is said to be IC when 𝜆 = 𝜆 0 , otherwise it is OOC, that is when 𝜆 = 𝜆 1 (𝜆 1 ≠ 𝜆 0 ). Suppose that LCL and UCL denote the lower and upper control limits of the exponential chart for individual data, respectively. For equal-tailed limits with known parameter case, the control limits LCL and UCL are obtained such that 𝑃(𝑋 < LCL|IC) = 𝛼 0 /2 and 𝑃(𝑋 > UCL|IC) = 𝛼 0 /2 , where 𝛼 0 is the nominal false alarm rate (FAR). However, in general setting of unequal-tailed limits that is to have flexibility in the unequal-tailed exponential charts and to attain the desired IC performance, we introduce two design constants such that the probability of an observation X plotting below the LCL is 𝜉𝑝 and above the UCL is (1 -𝜉)𝑝 that is

𝑃(𝑋 < LCL|IC) = ξ𝑝 and 𝑃(𝑋 > UCL|IC) = (1 -ξ)𝑝
where and 0 < 𝑝 < 1 and 0 < ξ < 1 is a constant (to be determined) that is used to split the FAR towards finding the LCL and the UCL, respectively, in order to maintain the IC performance at a desired level. Thus, for the equal-tailed exponential chart in the known parameter case, 𝜉 = 0.5 and 𝑝 = 𝛼 0 for a fixed FAR, 𝛼 0 .

The control limits, the UCL and LCL, can be expressed as

LCL = -ln(1-ξ𝑝) λ 0 = 𝐴 L λ 0 and UCL = -ln((1-ξ)𝑝) λ 0 = 𝐴 U λ 0 (1)
where

𝐴 L = -ln(1 -ξ𝑝) and 𝐴 U = -ln((1 -ξ)𝑝). (2)
The center line (CL) of the exponential chart can be considered as the median of the charting statistic 𝑋 and is given by CL = -ln(0.5) λ 0 .

Let δ = λ 1 λ 0
quantify the standardized shift of the parameter λ from λ = λ 0 (in-control) to some λ = λ 1 (out-of-control). Clearly, 𝛿 < 1 i.e. 𝜆 1 < 𝜆 0 represents an improvement in the process as the mean time between events increases while, 𝛿 > 1 i.e. 𝜆 1 > 𝜆 0 represents a deterioration in the process as the mean time between events decreases. For the IC process, we have 𝛿 = 1. Let 𝐸 𝑖 , 𝑖 = 1,2, … denote the event 𝐸 𝑖 = (𝑋 𝑖 < LCL) ∪ (UCL < 𝑋 𝑖 ) that a charting statistic lies outside the control limits LCL and UCL. This is called a signalling event. When the parameter λ 0 is known, the run length RL of the exponential control chart is a geometric random variable with parameter

β(δ) = 𝑃(𝐸 𝑖 ) equal to β(δ) = 𝑃[𝑋 < LCL or 𝑋 > UCL] = 1 -exp(-δ𝐴 L ) + exp(-δ𝐴 U )
It is shown in Appendix A that when the parameter λ 0 is known, the random variable TS follows an exponential distribution with parameter δλ 0 β(δ). Concerning a change in the process parameter it can be assumed to occur either (i) at the beginning of the monitoring period or, more generally, (ii) at some random point in time (this case is known as the random shift model [START_REF] Szarka | A review and perspective on surveillance of Bernoulli processes[END_REF]). See also Schuh et al.

[42] and Zwetsloot and Woodall [START_REF] Zwetsloot | A review of some sampling and aggregation strategies for basic statistical process monitoring[END_REF] for more details. In this paper, in order to simplify the model, we assume that any change in the process parameter occurs at the beginning of the monitoring period and we leave the more general random shift model for future developments.

Case of the Unknown parameter

When the parameter λ 0 is unknown, the estimated control limits are established by replacing , where 𝑇 = ∑ 𝑌 𝑖 is a sufficient statistic that follows a gamma distribution with parameters 𝑚 and λ 0 . Thus, the estimated control limits can be re-written as

λ 0 into
LCL ̂= 𝐴 L 𝑇 𝑚-1 and UCL ̂= 𝐴 U 𝑇 𝑚-1 (3)
We mention here that when the parameter λ 0 is estimated by λ ̂0, the signaling events 𝐸 𝑖 are dependent and hence, the TS no longer follows an exponential distribution. However, for a given value of λ ̂0, the events 𝐸 𝑖 are independent and consequently, the CTS (conditional time to signal) (given λ ̂0) follows an exponential distribution with parameter 𝛿λ ̂0𝛽 ̂(𝛿) where

β ̂(δ) = 𝑃[𝑋 < LCL ̂} or 𝑋 > UCL ̂ ] = 1 -exp (- δλ 0 𝐴 𝐿 𝑇 𝑚-1
) + exp (-

δλ 0 𝐴 𝑈 𝑇 𝑚-1 ) (4)
Thus, the conditional ATS (denoted CATS) which is the mean of the CTS distribution, for the control limits in Equation ( 3) is given by

CATS(𝛿) = 1 𝛿𝜆 ̂0 𝛽 ̂(𝛿) (5)
Using Equation (4), we can re-express the CATS in Equation (5) as follows.

CATS(δ) = 𝑇 δ(𝑚 -1) [1 -exp (- δλ 0 𝑇𝐴 L 𝑚 -1 ) + exp (- δλ 0 𝑇𝐴 U 𝑚 -1 )] -1 = φ(𝑇) δ(𝑚-1) (6)
where

φ(𝑇) = 𝑇 [1 -exp (- δλ 0 𝑇𝐴 L 𝑚-1
) + exp (-

δλ 0 𝑇𝐴 U 𝑚-1 )] -1 (7)
For the known parameter case , the ATS = 1/(𝛿𝜆 0 𝛽(𝛿) is a constant, but, for the unknown parameter case, the CATS(δ) is a random variable since it's a function of the random variable 𝑇.

Because the conditional performance analysis heavily depends on the CATS(δ) and its distributional characteristics, we derive the exact distribution of CATS(δ) in the following theorem.

Distribution of CATS(𝛅)

To obtain the exact distribution of CATS(𝛿) in Equation ( 6), first we prove the following lemma.

Lemma 1. For given 0 < ξ, 𝑝 < 1, λ 0 >0, δ >0 and 𝑚 > 1, the function φ(𝑇) is an increasing function of 𝑇 in (0, ∞).

The proof is provided in Appendix B.

Theorem 1. For the two-sided exponential chart with control limits defined as in Equation ( 2), the cumulative distribution function (c.d.f.) of the conditional average time to signal, CATS(𝛿) is given by

𝐺(𝑧) = 𝐹 Γ (φ -1 (δ(𝑚 -1)𝑧)), 0 < 𝑧 < ∞, δ > 0, 𝑚 > 1 (8)
where the function 𝜑 -1 (⋅) is the inverse function of 𝜑(⋅) shown in Equation [START_REF] Gan | Computing average run lengths of exponential EWMA charts[END_REF]. The probability density function (p.d.f.) of CATS(𝛿) is given by

𝑔(𝑧) = 𝑓 Γ (𝜑 -1 (𝛿(𝑚 -1)𝑧)) × δ(𝑚-1) φ ′ (𝜑 -1 (𝛿(𝑚-1)𝑧)) , 0 < 𝑧 < ∞ (9)
where 𝐹 Γ (⋅) and 𝑓 Γ (⋅) are the c.d.f. and p.d.f. of the gamma distribution with shape parameter 𝑚 and scale parameter 𝜆 0 , respectively. The function 𝜑 ′ (𝑧) is the first derivative of 𝜑(𝑧) with respect to 𝑧.

The proof is provided in Appendix C.

Using this theorem, some properties of the CATS distribution can be obtained as follows. These results play important roles in the developments that follow.

Theorem 2. The 𝑞-th quantile of the CATS distribution is given by

CATS 𝑞 (𝛿) = 1 δ(𝑚-1) φ(𝐹 Γ -1 (𝑞)) (10)
where 𝐹 Γ -1 (⋅) is the 𝑞-th quantile of a gamma distribution with parameters 𝑚 and 𝜆 0 . The proof is given in Appendix D.

Theorem 3. The 𝑟 𝑡ℎ moment about origin of the CAT𝑆(𝛿) distribution is given by

μ 𝑟 ′ (δ) = ( 1 δλ 0 (𝑚-1) ) 𝑟 Γ(𝑚+𝑟) Γ(𝑚) ∑ ∑ (-1) 𝑗+𝑖 ( -𝑟 𝑗 ) ( 𝑗 𝑖 ) [1 + δ 𝑚-1 {𝐴 L (𝑗 -𝑖) + 𝐴 U 𝑖}] -𝑚 𝑗 𝑖=0 ∞ 𝑗=0 (11)
The constants 𝐴 L and 𝐴 U are defined in Equation ( 2). The proof is provided in Appendix E.

Thus, the mean of the CATS(𝛿), i.e. ACATS(𝛿) = 𝜇 1 ′ (𝛿) is given by putting 𝑟 = 1 in Equation [START_REF] Zhang | A control chart for the gamma distribution as a model of time between events[END_REF] as follows.

μ 1 ′ (δ) = ( 𝑚 δλ 0 (𝑚-1) ) ∑ ∑ (-1) 𝑖 ( 𝑗 𝑖 ) [1 + δ 𝑚-1 {𝐴 L (𝑗 -𝑖) + 𝐴 𝑈 𝑖}] -𝑚 𝑗 𝑖=0 ∞ 𝑗=0
We mention here that in the context of the ARL, Kumar [START_REF] Kumar | Conditional analysis of phase II exponential chart for monitoring times to an event[END_REF] and Kumar et al. [START_REF] Kumar | Shewhart-type phase II control charts for monitoring times to an event with a guaranteed in-control and good out-of-control performance[END_REF] suggested two IC performance metrics: the LPB (lower Prediction Bound) which is defined as the smallest IC CARL value that can be attained by the control chart with high probability, say, 1 -γ, and the probability that the IC CARL is greater than or equal to a nominal ARL value. The latter is referred to as the exceedance probability criterion (EPC) [START_REF] Jardim | Two perspectives for designing a phase II control chart with estimated parameters: The case of the Shewhart Chart[END_REF] and has been investigated by several authors including Albers et al. [START_REF] Albers | Exceedance probabilities for parametric control charts[END_REF]; Gandy and Kvaløy [START_REF] Gandy | Guaranteed conditional performance of control charts via bootstrap methods[END_REF], Faraz et al. [START_REF] Faraz | Guaranteed conditional performance of the S 2 control chart with estimated parameters[END_REF]; Goedhart et al. [START_REF] Goedhart | Shewhart control charts for dispersion adjusted for parameter estimation[END_REF][START_REF] Goedhart | On guaranteed in-control performance for the Shewhart 𝑋 and 𝑋 ̅ control charts[END_REF]; Guo and Wang [START_REF] Guo | The design of the S 2 control charts based on conditional performance via exact methods[END_REF] to design the phase II control limits for fixed probability, say, 0.90 to guaranteeing that the IC CARL will meet or exceed the nominal ARL 0 .

Motivated by these, we define the metrics LPB and the exceedance probability (EP) in terms of IC CATS i.e. CATS(1) (thereafter we will denote it by CATS IC by first defining function 𝜑(𝑇) for the IC situation as follows.

φ IC (𝑇) = 𝑇 [1 -exp (- λ 0 𝑇𝐴 L 𝑚 -1 ) + exp (- λ 0 𝑇𝐴 U 𝑚 -1 )] -1
Now we define the (1 -γ)% LPB for the CATS IC , denoted by LPB 1-γ as

𝑃[CATS IC ≥ 𝐿𝑃𝐵 1-γ ] = 1 -γ which gives LPB 1-γ = 1 (𝑚 -1) φ IC (𝐹 Γ -1 (γ))
For the control chart with the LPB equal to LPB 1-γ , the user has a usually high confidence (1 -γ)% that his CATS IC will be at least as much as LPB 1-γ . Note that the LPB 1-γ is in fact the γ-th quantile of the CATS IC distribution.

On the other hand, the exceedance probability (EP) that the CATS IC is at least as much as a nominal ATS 0 value is defined as

EP = 𝑃[CATS IC ≥ ATS 0 ] = 1 -𝐹 Γ (φ IC -1 ((𝑚 -1)ATS 0 ))
Note that it is desirable that both LPB 1-γ and EP be high for a good chart.

Phase II Exponential chart under the unconditional perspective

We should mention here that the exponential charts with known parameter 𝜆 0 are often constructed using an equal-tailed probability approach by assigning the half of the FAR, α 0 to the both tails i.e. In order to construct the equal-tailed exponential chart with estimated parameters under the unconditional perspective for a fixed nominal ATS 0 , we determine the unique pair (ξ, 𝑝) that gives a pair of control limits satisfying the following two equations:

μ 1 ′ (1) = ATS 0 (15) 1 -(1 + 𝐴 L 𝑚-1 ) -𝑚 = (1 + 𝐴 U 𝑚-1 ) -𝑚 (16)
Clearly, Equation [START_REF] Ali | An overview of control charts for high-quality processes[END_REF] ensures that the expected value of CATS IC i.e. the 𝐴CATS IC (given by 𝜇′(1))

is equal to the nominal ATS 0 , whereas Equation ( 16) ensures that the unconditional tail probabilities

𝐸 [𝑃[𝑋 < LCL ̂|IC]] and 𝐸 [𝑃[𝑋 > UCL ̂|IC]

] are equal i.e. a charting statistic has on average an equal chance of lying below the lower control limit and above the upper control limit. As noted above, this is one analogue of the equal-tailed probability approach used in the known parameter case. Once we find the required pair of (ξ, 𝑝) satisfying Equations ( 15) and ( 16), the corresponding 𝑃[𝑋 < LCL ̂|IC]

and 𝑃[𝑋 < UCL ̂|IC] can be obtained from Equation (2) for the phase II equal-tailed exponential chart. Note that the unconditional perspective is defined here in terms of the ATS 0 rather than in terms of the ARL 0 as in [START_REF] Jardim | Two perspectives for designing a phase II control chart with estimated parameters: The case of the Shewhart Chart[END_REF].

It is worth mentioning that the equal-tailed exponential chart under the unconditional perspective is ATS-biased as its ATS function does not achieve its maximum at the IC state (i.e. δ = 1), which causes the control chart take more time to give a signal in the OOC state than in the IC state of the process. This is considered undesirable and hence, in this situation, it is worth using the ATSunbiased exponential charts (see [START_REF] Zhang | A control chart for the gamma distribution as a model of time between events[END_REF]). In order to design the ATS-unbiased chart with the estimated control limits under the unconditional perspective, we first set the μ 1 ′ (1) (the mean of the CATS IC )

is equal to the nominal ATS 0 (see, Equation ( 17)) and then we set the derivative of ACATS(δ), with respect to δ, that is μ 1 ′ (δ) evaluated at δ = 1 equal to zero (see, for example, [START_REF] Guo | The design of the S 2 control charts based on conditional performance via exact methods[END_REF]) (see, Equation ( 18)).Thus, the control limits of the ATS-unbiased exponential chart with estimated parameters under the unconditional perspective can be obtained from first solving the following the two equations

μ 1 ′ (1) = ATS 0 ( 17
) 𝑑 𝑑δ μ 1 ′ (δ)| δ=1 = 0 (18)
for the (𝜉, 𝑝) pair and then using Equations ( 8) and ( 9) respectively by plugging in the values of the pair (𝜉, 𝑝) of the corresponding chart. Note that Equation [START_REF] Shah | Time between events control charts for gamma distribution[END_REF] ensures that the mean of the CATS IC is equal to the nominal ATS 0 whereas Equation ( 18) ensures the unbiasedness that is the maximum of the ACATS(δ) is at 𝛿 = 1. The chart design parameters are calculated and reported in Table 1 for both the phase II equal-tailed and the phase II ATS-unbiased exponential charts corresponding to different values of 𝑚. The control limits of the phase II equal-tailed and the ATS-unbiased exponential charts with unknown parameter can be obtained from Equation (3) by plugging in the appropriate values of the pair (𝜉, 𝑝) of the corresponding chart.

In-control Performance of the phase II exponential charts under the unconditional perspective

The c.d.f. and p.d.f. of the phase II equal-tailed and the ATS-unbiased exponential charts with unknown parameter can be obtained from Equations ( 8) and ( 9) respectively by plugging in the appropriate values of the pair (𝜉, 𝑝) of the corresponding chart. Figures 1 and2 show the p.d.f.s of the CATS IC of the phase II equal-tailed and ATS-unbiased exponential charts, respectively, with nominal ATS 0 =370.4, 𝑚=20, 50, 100, 500 and λ 0 = 1. We should mention here that we consider small sample sizes, 𝑚 ≤ 30, moderate sample sizes, 30 < 𝑚 ≤ 100 and large sample sizes, 𝑚 > 100. It can be seen from Figure 1 that for smaller values of 𝑚, the p.d.f. of the CATS IC for the phase II equal-tailed exponential chart is skewed to the left of ATS 0 that is the higher values of the CATS IC are located to the right of ATS 0 which produces a high EP value. However, as 𝑚 increases, the distribution of the CATS IC becomes more symmetric about ATS 0 . Thus, even when 𝑚 is large, the EP values tend to 0.50 which means that there is only about a 50% chance that the CATS IC will be greater than or equal to ATS 0 . On the other hand, the CATS IC distribution of the ATS-unbiased exponential chart is more left skewed than the corresponding equal-tailed chart for moderate to large sample sizes (Figure 2). Note that for smaller values of 𝑚, the p.d.f. of the CATS IC distributions for both control charts reaches its maximum, and then decreases rapidly tending to zero.

Insert Figures 1 and 2 only a 50% probability that the CATS IC value will be greater than or equal to the nominal ATS 0 .

However, the ATS-unbiased chart has higher EP values than the equal-tailed chart except for smaller values of 𝑚 ≤ 30.

Insert Table 1 Note that in the context of monitoring rare events, it can take a long time to even collect a small number of phase I observations. The study suggests the use of an equal-tailed exponential chart when a small number of phase I observations is available to estimate the parameter. However, under the unconditional perspective neither chart accounts for practitioner-to-practitioner variability which is reflected by their respective low EP values. This provides a motivation for designing the control chart under the conditional perspective using the exceedance probability criterion (EPC), which ensures a guaranteed IC performance in the sense that the CATS IC meets or exceeds (is at least equal to) a nominal ATS 0 value with a prespecified (high) probability.

Out-of-control Performance of the phase II exponential charts under the unconditional perspective

In order to examine the conditional OOC performance of the equal-tailed and ATS-unbiased charts under the unconditional perspective, we consider δ = 2.0, 4.0 to reflect upward shifts and δ = 0.25, 0.5 for downward shifts and different values of 𝑚 = 20, 30, 50, 100, 200, 500. Table 2 provides the mean, standard deviation and percentiles of the conditional OOC CATS, denoted by CATS OOC , distributions of the equal-tailed and ATS-unbiased exponential charts under the unconditional perspective when ATS 0 = 370.4, λ 0 = 1.

Insert Table 2

It can be observed from Table 2 that when the process improves i.e. 𝛿 < 1 and the phase I sample size 𝑚 is small, the equal-tailed exponential chart under the unconditional perspective has larger OOC mean and percentiles of the CATS OOC distribution than those for the ATS-unbiased exponential chart under the unconditional perspective whereas for the moderate to large values of 𝑚, the opposite pattern is observed i.e. the OOC mean and percentiles of the equal-tailed exponential chart under the unconditional perspective are smaller than the corresponding values of the ATSunbiased exponential chart under the unconditional perspective.

However, when the process deteriorates i.e. 𝛿 > 1 and the phase I sample size is small, the OOC mean and percentiles of CATS OOC distribution for the equal-tailed chart under the unconditional perspective are smaller than those for the ATS-unbiased exponential chart under the unconditional perspective, however, for moderate to large values of 𝑚, these values for the former chart are larger than the latter.

Overall, for a moderate to large phase I sample size, the ATS-unbiased exponential chart under the unconditional perspective has better OOC performance in terms of lower ACATS OOC values than the equal-tailed exponential chart when the process deteriorates. Note that generally speaking, in typical applications, the deterioration case is considered to be more serious than the improvement case in practice. On the other hand, when 𝑚 is small the latter performs better than the former. For the improvement case i.e. 𝛿 > 1, the ATS-unbiased exponential chart under the unconditional perspective has better (worse) performance than the equal-tailed exponential chart under the unconditional perspective for small (moderate to large) 𝑚.

Phase II exponential charts under the conditional perspective

The study in Section 3 shows that both the equal-tailed and ATS-unbiased phase II exponential charts adjusted under the unconditional perspective have poor IC conditional performance since the variation in the CATS IC distribution is not accounted for. Noting that the SDCATS IC value decreases as the phase I sample size, 𝑚 (See Table 1) increases, the SDCATS IC can be reduced to reasonable degree, i.e., say within 10% of the desired ATS 0 , by increasing 𝑚. On the other hand, increasing 𝑚 reduces the confidence that the CATS IC will meet or exceed the nominal ATS 0 since the CATS IC distribution tends to become symmetric around ATS 0 . Thus, when 𝑚 is large, there is about 50% chance that the CATS IC will be less than the ATS 0 which will cause more frequent false alarms and more unwanted process interruptions than desired [START_REF] Guo | The design of the S 2 control charts based on conditional performance via exact methods[END_REF]. We now consider adjusting the control limits of both the phase II equal-tailed and ATS-unbiased exponential charts under the conditional perspective to achieve a fixed EP value, say, 1 -𝛾 (usually high, say, .90) i.e. CATS IC meets or exceeds the desired nominal ATS 0 with a certain probability, 1 -𝛾. In terms of the average run length ARL, this criterion is known as the exceedance probability criterion (see for example, [START_REF] Faraz | Guaranteed conditional performance of the S 2 control chart with estimated parameters[END_REF][START_REF] Gandy | Guaranteed conditional performance of control charts via bootstrap methods[END_REF]).

First consider the Phase II exponential chart.

Let LCL ̂ * and UCL ̂ * be the lower and upper control limits of the phase II exponential chart under conditional perspective such that the resulting CATS IC is greater than or equal to the nominal ATS 0 with a high probability, say, 1 -𝛾. The adjusted control limits are given by

LCL ̂ * = 𝐴 L * λ ̂ and UCL ̂ * = 𝐴 U * λ ̂ (19) where A
and for the ATS-unbiased exponential chart from the following equations The c.d.f. and p.d.f. of CATS(δ) of the respective charts can be obtained from ( 8) and ( 9), respectively by plugging in the corresponding design parameters.

𝑃[CATS IC ≥ ATS 0 ] = 1 -𝛾 (22)

Conditional In-control Performance of phase II exponential charts

The p.d.f.s of the proposed phase II equal-tailed and ATS-unbiased exponential charts with fixed EP=0.90 i.e. 𝑃[CATS IC ≥ ATS 0 ] = 0.90 and nominal ATS 0 = 370.4 are shown in Figures 3 and4, respectively for 𝑚 = 20, 50, 100, 500 and 𝜆 0 = 1. It can be seen from these figures that the distribution of CATS IC of the ATS-unbiased exponential chart adjusted under the conditional perspective is more skewed to the left than that of the corresponding equal-tailed exponential chart, except for smaller values of 𝑚. Note that the CATS IC distribution of both exponential charts under the conditional perspective becomes more symmetrical as 𝑚 gets larger, however, the distribution of the ATS-unbiased exponential chart under the conditional perspective is more skewed than the equal-tailed chart under the conditional perspective, even for large 𝑚.

Insert Figures 3 and4 In order to examine the IC performance of the phase II exponential charts under the conditional perspective, various metrics are calculated from the exact distribution of CATS IC . These metrics are reported in Table 3. The following observations can be made.

1. The mean of CATS IC i.e. 𝐴CATS IC values of both equal-tailed and ATS-unbiased exponential charts under the conditional perspective are much higher than the nominal ATS 0 . It is the price to be paid to ensure that the CATS IC will meet or exceed the nominal ATS 0 with a guarantee of 90% probability. Similar observations have been made by Jardim et al. [START_REF] Jardim | Two perspectives for designing a phase II control chart with estimated parameters: The case of the Shewhart Chart[END_REF] for the Shewhart chart for the mean. It is observed that the 𝐴CATS IC values of the adjusted ATSunbiased exponential chart under the conditional perspective are closer to ATS 0 than the adjusted equal-tailed exponential chart under the conditional perspective except for smaller values of 𝑚 < 30. The difference between the ACATS IC values of both charts under the conditional perspective becomes smaller with the increase in 𝑚.

2. There is a significant difference in the conditional IC performances of the two types of charts. The study reveals that for smaller values of 𝑚, there is large variability in the CATS IC values of the ATS-unbiased exponential chart under the conditional perspective relative to the corresponding equal-tailed chart. However, for 𝑚 ≥ 30, the former has lower SDCATS IC values than the latter. For example, when 𝑚 = 100, the SDCATS IC is 108.67 for the ATS-unbiased exponential chart under the conditional perspective whereas it is 214.18

for the equal-tailed exponential chart under the conditional perspective.

3. Due to a lower variability in the IC CATS values, the ATS-unbiased exponential chart under the conditional perspective needs less phase I observations than the corresponding adjusted equal-tailed chart to keep SDCATS IC value under the conditional perspective within 10% of the ATS 0 . It can be observed from the last column of Table 3 that about 500 phase I observations are required for the ATS-unbiased exponential chart under the conditional perspective whereas more than 1000 observations are required for the equal-tailed chart under the conditional perspective to keep SDCATS IC value below 10% of the ATS 0 .

However, as it was stated earlier, collecting such a large number of phase I observations may not always be possible and therefore, one may need a trade off with the IC performance. In this scenario (less phase I observations), the equal-tailed ATS-unbiased exponential chart under the conditional perspective is preferred to the corresponding ATS-unbiased chart.

Insert Table 3 4.

Conditional Out-of-control performance of the phase II exponential charts

In this section, we examine the conditional performance of the proposed equal-tailed and ATSunbiased charts under the conditional perspective in an OOC situation. For this, we consider the same 𝛿 values and the 𝑚 values as in the OOC study of the charts under the unconditional perspective. The various performance metrics of the proposed charts are provided in Table 4 when 𝐴𝑇𝑆 0 = 370.4, 𝜆 0 = 1 and EP=0.90.

It can be observed from Table 4 that the findings are same as found in the OOC study of the equaltailed and ATS-unbiased exponential charts under the unconditional perspective. For example, when 𝛿 < 1 (𝛿 > 1) and the phase I sample size 𝑚 is small, the equal-tailed exponential chart under the conditional perspective has larger (smaller) OOC mean and percentiles of the CATS OOC distribution than those for the ATS-unbiased exponential chart under the conditional perspective whereas for the moderate to large values of 𝑚, the opposite pattern is observed i.e. the OOC mean and percentiles of the equal-tailed exponential chart under the conditional perspective are smaller (larger) than the corresponding values of the ATS-unbiased exponential chart under the conditional perspective.

Like the ATS-unbiased exponential chart under the unconditional perspective, for a moderate to large phase I sample size, the ATS-unbiased exponential chart under the conditional perspective has better OOC performance in terms of less ACATS OOC values than the equal-tailed exponential chart when the process deteriorates.

Overall, for moderate to large phase I sample size, the ATS-unbiased exponential chart under both perspectives outperforms than the corresponding equal-tailed exponential chart in the deterioration case, however, for the improvement case, the former has worse performance than the latter. We should mention here that though the performance of the phase II (equal-tailed or ATS-unbiased) exponential chart under the conditional perspective seems to be worse than the corresponding exponential chart under the unconditional perspective. However, one should keep in mind that the IC performance of the charts under both perspectives are not comparable in the sense that the ACATS IC of the exponential charts under the conditional perspective are larger than the corresponding charts under the unconditional perspectives. Note that these observations are similar to the those of [START_REF] Jardim | Two perspectives for designing a phase II control chart with estimated parameters: The case of the Shewhart Chart[END_REF] that the chart under the conditional perspective neither control the unconditional performance (ACATS IC ) nor its variability (SDCATS IC ) and produces larger ACATS OOC values than the chart under the unconditional perspective.

Insert Table 4 Finally, in order to help practitioners in implementing the exponential chart with estimated parameters, we provide in Tables 5 and6 the required charting constants of both the equal-tailed and ATS-unbiased exponential charts for different 𝜆 0 . A program to obtain the charting constants, written in MATLAB, is available from the authors on request.

Insert Tables 5,6 5

. Example

In order to illustrate the application of the proposed phase II equal-tailed and ATS-unbiased exponential charts, adjusted under the conditional perspective, we first use the data given by Jarrett [START_REF] Jarrett | A note on the intervals between coal-mining disasters[END_REF]. The data comprises 190 recorded time intervals (in days) between coal-mining disasters in England, from 15 March, 1851 to 22 March, 1962 and have been extensively used in the literature by a number of authors for illustration of various procedures. The data are shown in Table 7 for convenience. It has been tested that the time intervals between successive explosions follow an exponential distribution and we monitor the unknown rate parameter λ of this distribution [START_REF] Jarrett | A note on the intervals between coal-mining disasters[END_REF]. We should mention here that according to Jarrett [START_REF] Jarrett | A note on the intervals between coal-mining disasters[END_REF], the first 125 time intervals between the explosions follow an exponential distribution with a mean rate of (an explosion) one every 106 days while for the next 65 explosions, the mean rate is about one every 388 days.

In order to examine the ability of the proposed charts under both perspectives to trace the improvement in the process with small phase I sample, we consider the first 𝑚 = 15 observations to be from an IC process and construct the proposed exponential charts under both perspective, from which we estimate that 𝜆 ̂0 = 0.0072 (or, equivalently, the sum of 15 TBEs is 1937 days). The control limits of the charts under the conditional perspective are constructed for the nominal ATS 0 =40,000 days for which the design parameters 𝜉 * and 𝑝 * are obtained solving Equations ( 20)-( 21)

and ( 22)-( 23), respectively. Note that the ATS 0 =40,000 days means a false alarm is expected on the average, every 40,000/106=377 explosions. The lower and upper control limits of the equal-tailed exponential chart under the conditional perspective are found to be 0.0839, 1222.4406 respectively whereas these are found to be 0.0331, 1191.3600 respectively for the ATS-unbiased exponential chart under the conditional perspective. For the unconditional perspective, the lower and upper control limits of the equal-tailed exponential chart are found to be 0.2527 and 998.7904, respectively, whereas these limits are 0.2084 and 904.6048, respectively, for the ATS-unbiased exponential chart. The four pairs of the control limits are depicted in Figure 5. It follows from Figure 5 that except the equal-tailed chart under the conditional perspective, all the other chars show an OOC alarm at the 119th time point of the phase II monitoring (corresponds to the 134th observation )

whereas the equal-tailed exponential chart under the conditional perspective does not signal at this point. This may be explained by the fact that the conditional perspective produces wider limits which leads to poor OOC performance (see Table 4).

We mention here that the OOC signals indicated by the charts are on the upper side which indicates a decrease in the mean rate of explosions per day. This is consistent with the fact that with more technological advances, coal-mine explosions occur less frequently nowadays. As was stated earlier, Jarrett [START_REF] Jarrett | A note on the intervals between coal-mining disasters[END_REF] showed that the last 65 time intervals had an decreased mean rate of explosion per day than the earlier 125 time intervals. The proposed ATS-unbiased exponential chart under the conditional perspective detects the change in the mean rate of explosions earlier, after the 9th time interval since starting the improvement in the process whereas the equal-tailed exponential chart under the conditional perspective detects this change after the 28th time interval of the starting the improvement.

Insert Table 7 Insert Figure 5

Concluding Remarks

In this paper, we consider the phase II exponential chart, which is one of the most widely used control chart for monitoring TBE data from high-yield processes. To understand the impact of parameter estimation, and to help the user better implement the chart in practice, we study the conditional chart performance in terms of the time to signal and the metric ATS, which is known to be more appropriate than the traditional run length and the ARL in case of TBE data. To study the chart's properties, the exact distribution of the CATS was derived, and the (adjusted) control limits of both the equal-tailed and the ATS-unbiased exponential charts fixed nominal ATS 0 were found under two perspectives for a fixed nominal ATS 0 . Under the unconditional perspective, which sets the average of the IC CATS distribution to a fixed nominal ATS 0 value while finding the adjusted control limits, the study showed that both charts have poor IC conditional performance in terms of very high practitioner-to-practitioner variability (SDCATS IC values), low LPB and low EP values.

Alternatively, under the conditional perspective, the control limits of both charts were calculated so that the CATS IC is at least equal to a specified value i.e. ATS 0 value with a high probability. For these charts, the performance study revealed good IC conditional performance, even though they have increased mean CATS IC than the nominal ATS 0 , this may be considered as the price to pay for a guaranteed and improved IC performance. It was observed that for moderate to large phase I sample sizes 𝑚 ≥ 30, the adjusted ATS-unbiased exponential chart under the conditional perspective has less variability in CATS IC values than the corresponding equal-tailed chart. Nevertheless, if a practitioner wishes to control both the metrics, for example, to keep 𝑆𝐷CATS IC within 10% of ATS 0 and EP=0.90, i.e., a guarantee that the CATS IC will meet or exceed ATS 0 with 90% probability, then about 400 phase I observations are required for the adjusted ATS-unbiased exponential chart whereas more than 1000 observations are needed for the adjusted equal-tailed exponential chart under the conditional perspective. Moreover, the adjusted ATS-unbiased exponential chart under the conditional perspective has a good OOC performance in the deterioration case for moderate to large values of 𝑚. Thus, the phase II ATS-unbiased exponential chart adjusted under the conditional perspective is recommended to monitor TBE individual data from high-yield processes following an exponential distribution with an unknown rate parameter.

Finally, note that we have only considered the simpler Shewhart-type charts; however, the same principles can be extended to a more general TBE chart such as the 𝑡 𝑟 -chart [START_REF] Xie | Some effective control chart procedures for reliability monitoring[END_REF][START_REF] Epprecht | Effect of the amount of Phase I data on the Phase II performance of S 2 and Scontrol charts[END_REF], which monitors the time to 𝑟 th event/failure. However, there may be some concern about loss of information with 𝑡 𝑟type charts since data collection at the end of an aggregation period (say at the 𝑟 th event) may have an adverse effect on the chart's performance [START_REF] Kumar | Improved phase I control charts for monitoring times between events[END_REF][START_REF] Kumar | Shewhart-type phase II control charts for monitoring times to an event with a guaranteed in-control and good out-of-control performance[END_REF]. It will be interesting to examine the effect of aggregation on the 𝑡 𝑟 -chart's performance with the estimated control limits constructed under the ATS criteria. Note also that when the failures in a process occur due to sudden shock instead of a slow wear and tear, the exponential distribution is a more suitable distribution to model time between events/failures. However, when the failure rate changes over time, then the Weibull distributions is more appropriate and can be considered as a probability model for TBE data. Other non-Shewharttype charts such as the CUSUM exponential and the EWMA exponential charts can be considered in a future study. Let us define 𝑃(𝑇) = 𝜁(𝑇) -𝑇𝜁 ′ (𝑇). It can be easily shown that 𝜁(𝑇) attains its minimum at

𝑃[TS ≤ 𝑠

] = 𝑃[∑ 𝑋 𝑖 𝑅 𝑖=1 ≤ 𝑠] = ∑ 𝑃[∑ 𝑋 𝑖 𝑗 𝑖=1 ≤ 𝑠|𝑅 = 𝑗]𝑃[𝑅 = 𝑗] ∞ 𝑗=1 = ∑ (∫ λ 1 𝑗 Γ(𝑗) 𝑡 𝑗-1 𝑒 -λ 1 𝑡 𝑑𝑡 𝑠 0 ) β(δ)(1 -β(δ)) 𝑗-1 ∞ 𝑗=1 = ∫ λ 1 β(δ)𝑒 -λ 1 𝑡 𝑠 0 (∑ (λ 1 (1-β(δ))𝑡) 𝑗-1 Γ(𝑗) ∞ 𝑗=1 ) 𝑑𝑡 = ∫ λ 1 β𝑒 -λ 1 𝑡 𝑒 λ 1 (1-β(δ))𝑡 𝑑𝑡 𝑠 0 = 1 -𝑒 -λ 1 β(δ)𝑠 ,
𝑇 = 𝑀 1 = ln(𝐴 U /𝐴 L )
𝜅(𝐴 U -𝐴 L ) so that 𝜁 ′ (𝑀 1 ) = 0 and 𝜁 ′′ (𝑀 1 ) > 0. Further, 𝜁 ′ (𝑇) also attains its maximum at 𝑇 = 𝑀 2 so that 𝜁 ′′ (𝑀 2 ) = 0 and 𝜁 ′′′ (𝑀 2 ) < 0 where 𝑀 2 = 2 ln(𝐴 U /𝐴 L ) 𝜅(𝐴 U -𝐴 L ) .

Since, 𝑃 ′ (𝑇) = -𝑇ζ ′′ (𝑇) and 𝑃 ′ (𝑇) = 0 implies either 𝑇 = 0 or ζ ′′ (𝑇) = 0 which yields solution 𝑇 = 𝑀 2 in (0, ∞). Again, 𝑃 ′′ (𝑇) = -𝑇ζ ′′′ (𝑇) -ζ ′′ (𝑇) and hence 𝑃′′(0) < 0 and 𝑃 ′′ (𝑀 2 ) = -𝑀 2 ζ ′′′ (𝑀 2 ) > 0.

This implies that the function 𝑃(𝑇) attains its minimum at 𝑇 = 𝑀 2 in (0, ∞). Thus the minimum value of 𝑃(𝑇) is 𝑃(𝑀 2 ) > 0. It implies that φ ′ (𝑇) > 0 for all 𝑇 ∈ (0, ∞) and consequently, the function φ(𝑇) is an increasing on (0, ∞).

C. Proof of Theorem 1

By definition, the c.d.f. of the CATS(𝛿) is

𝐺(𝑧) = 𝑃[CATS(δ) ≤ 𝑧] = 𝑃[φ(𝑇) ≤ δ(𝑚 -1)𝑧]
Using Lemma 1, we can write 𝐺(𝑧) = 𝑃[𝑇 ≤ φ -1 (δ(𝑚 -1)𝑧)] = 𝐹 Γ (φ -1 (δ(𝑚 -1)𝑧))

The p.d.f. of the CATS(𝛿) in Equation ( 9) can be obtained by differentiating the c.d.f. in Equation ( 8) with respect to 𝑧. Further note that the function 𝜑 (-1) (⋅) has no closed form expression, so must be computed numerically.

D. Proof of Theorem 2

The 𝑞 𝑡ℎ (0 < 𝑞 < 1) quantile of CATS(𝛿) distribution, denoted by CATS 𝑞 (𝛿), can be obtained by the equation 𝐺 (CATS q (𝛿)) = 𝑞, that is.

𝐹 Γ (φ -1 (δ(𝑚 -1)CATS 𝑞 (δ))) = 𝑞 which gives φ -1 (δ(𝑚 -1)CATS 𝑞 (δ)) = 𝐹 Γ -1 (𝑞),

where 𝐹 Γ -1 (⋅) is the 𝑞-th quantile of a gamma distribution with parameters 𝑚 and 𝜆 0 . Using Lemma 1, we have δ(𝑚 -1)CATS 𝑞 (δ) = 𝜑(𝐹 Γ -1 (𝑞))

which gives the CATS 𝑞 (δ) in Equation [START_REF] Zhang | Design of exponential control charts using a sequential sampling scheme[END_REF].

E. Proof of Theorem 3

Proof. The 𝑟 𝑡ℎ moment of CAT𝑆(𝛿) is )| < 1. Thus, using the negative binomial series expansion in the above expression, we get

μ 𝑟 ′ (δ) = 1 (δ(𝑚 -1)) 𝑟 ∫ ∑(-1) 𝑗 ( -𝑟 𝑗 ) [exp (- δλ 0 𝐴 L 𝑡 𝑚 -1 ) -exp (- δλ 0 𝐴 𝑈 𝑡 𝑚 -1 )] 𝑗 ∞ 𝑗=0 ∞ 0 × λ 0 𝑚 Γ(𝑚) 𝑡 𝑚+𝑟-1 𝑒 -λ 0 𝑡 𝑑𝑡
Again, using the binomial theorem, we can re-express μ 𝑟 ′ (δ) as follows.

𝜇 𝑟 ' (𝛿) = 1/(𝛿(𝑚 -1)) 𝑟 ∫ ∑ ∑ (-1) 𝑗+𝑖 ( -𝑟 𝑗 ) ( 𝑗 𝑖 ) exp (-

𝛿𝜆 0 𝑡 𝑚-1 [(𝑗 -𝑖)𝐴 L + 𝐴 U 𝑖]) 𝑗 𝑖=0 ∞ 𝑗=0 ∞ 0 × (𝜆 0 𝑚 ) Γ(𝑚)
𝑡 𝑚+𝑟-1 𝑒 -𝜆 0 𝑡 𝑑𝑡 Since the summation is absolutely convergent, we can interchange the summation and the integral.

We then have We mention here that the infinite series in Equation ( 11) is bounded that is

μ 𝑟 ′ (δ) = 1 (δ(𝑚-1)) 𝑟 ∑ ∑ (-1) 𝑗+𝑖 ( -𝑟 𝑗 ) ( 𝑗 𝑖 ) ∫ 𝜆 0 𝑚 Γ(𝑚)2 𝑚 𝑡 𝑚+𝑟-1 ∞ 0 𝑗 𝑖=0 ∞ 𝑗=0 × exp (-λ 0 𝑡 [1 +
Γ(𝑚+𝑟) (δλ 0 (𝑚-1)) 𝑟 Γ(𝑚) [1 -(1 + 𝛿𝐴 1 𝑚-1 ) -𝑚 {1 -(1 + 𝛿(𝐴 2 -𝐴 1 ) 𝑚-1 ) -𝑚 }] -𝑟 ≤ 𝜇 𝑟 ′ (𝛿) ≤ Γ(𝑚+𝑟) (δλ 0 (𝑚-1)) 𝑟 Γ(𝑚) min {(1 - δA 2 𝑟 m -1 ) -m , ( m A 1
)

r Γ(m-r) Γ(m)
} and hence, it always converges. The reader may refer to Appendix B of [START_REF] Diko | Guaranteed in-control performance of the EWMA chart for monitoring the mean[END_REF] to obtain bounds of series [START_REF] Zhang | A control chart for the gamma distribution as a model of time between events[END_REF].

F. Unconditional probabilities

The unconditional probability that a charting point lies below the lower control limits LCL ̂ is

∫ 𝑃[𝑋 < LCL ̂]𝑓 Γ (𝑡)𝑑𝑡 ∞ 0 = ∫ (1 -𝑒 -δλ 0 𝐴 𝐿 𝑚-1 𝑡 ) λ 0 𝑚 Γ(𝑚) 𝑡 𝑚-1 𝑒 -λ 0 𝑡 𝑑𝑡 ∞ 0 = 1 -∫ λ 0 𝑚 Γ(𝑚) 𝑡 𝑚-1 𝑒 -λ 0 𝑡(1+ δ𝐴 𝐿 𝑚-1 ) 𝑑𝑡 ∞ 0 = 1 -(1 + δ𝐴 𝐿 𝑚-1 ) -𝑚 (E.1)
In like manner, we can obtain the unconditional probability that a charting point lies above the upper control limits UCL ̂ which is

∫ 𝑃[𝑋 > UCL ̂]𝑓 Γ (𝑡)𝑑𝑡 ∞ 0 = (1 + δ𝐴 𝑈 𝑚-1 ) -𝑚 (E.2)
For the IC process, the unconditional probabilities can be obtained from (C.1)-(C.2) respectively by letting δ = 1.

  using 𝑃[𝑋 < LCL|IC] = 𝑃[𝑋 > UCL|IC] = α 0 /2. However, when the parameter is unknown, the control limits LCL ̂ and UCL ̂, being the function of λ ̂ (or 𝑇), become random variables. As a result, we set 𝐸 [𝑃[𝑋 < LCL ̂|IC]] = 𝐸 [𝑃[𝑋 > UCL ̂|IC]] for the equal-tailed exponential chart in the unknown parameter case. The exact expressions for the unconditional probabilities are provided in Appendix F.

  L * = -ln(1 -ξ * 𝑝 * )) and 𝐴 U * = -ln((1 -ξ * )𝑝 * ). The constants ξ * and p * for the phase II equal-tailed exponential chart under the conditional perspective can be obtained from the following equations 𝑃[CATS IC ≥ ATS 0 ] = 1 -γ

  𝑠 > 0 B. Proof of Lemma 1 To show φ(𝑇) = 𝑇 [1 -exp (an increasing function of 𝑇 ∈ (0, ∞), we must have the first derivative φ ′ (𝑇) > 0 for all 𝑇 ∈ (0, ∞). Let ζ(𝑇) = [1 -) -𝑇ζ ′ (𝑇) [ζ(𝑇)] 2 where ζ ′ (𝑇) is the first derivative of 𝜁(𝑇). Because 𝜁(𝑇) > 0 for all 𝑇, then φ ′ (𝑇) > 0 implies ζ(𝑇) -𝑇ζ ′ (𝑇) > 0.

1 ) 0 For

 10 𝑚-1)) 𝑟 ∫ 𝑡 𝑟 [1 -exp (-δλ 0 𝐴 L 𝑡 𝑚-)𝑡 𝑚-1 𝑒 -λ 0 𝑡 𝑑𝑡 ∞ 𝑚 > 1, δ > 0, λ 0 > 0 and 𝑡 > 0, we have |exp (-

δ 𝑚- 1 {

 1 (𝑗 -𝑖)𝐴 L + 𝑟𝐴 U }]) 𝑑𝑦On simplification, we have

Figure 1 .

 1 Figure 1. The p.d.f. of the CATS IC distribution for the equal-tailed exponential chart under the unconditional perspective for 𝑚 = 20, 50, 100 and 500 for nominal ATS 0 = 370.4, 𝜆 0 = 1.

Figure 2 .Figure 3 .

 23 Figure 2. The p.d.f.s of the CATS IC distribution of the ATS-unbiased exponential chart under the unconditional perspective for 𝑚 = 20, 50, 100 and 500 for nominal ATS 0 = 370.4, 𝜆 0 = 1.

Figure 4 .Figure 5 .

 45 Figure 4. The p.d.f.s of the CATS IC distribution of the ATS-unbiased exponential chart under conditional perspective for 𝑚 = 20, 50, 100 and 500 for EP=0.90 with nominal ATS 0 = 370.4, 𝜆 0 = 1.

  Equation (1) by its estimator 𝜆 ̂0, obtained from a phase I sample, say, 𝑌 1 , … , 𝑌 𝑚 of size 𝑚.

	See Kumar and Chakraborti [44] for a phase I analysis of the exponential chart. Thus, the
	estimated (phase II) control limits become
		LCL ̂= 𝐴 L 𝜆 ̂0	and UCL ̂= 𝐴 U 𝜆 ̂0
	Usually, the UMVUE (uniformly minimum variance unbiased estimator) is used to estimate the IC
	rate parameter λ 0 which is 𝜆 ̂0 =	𝑚-1 𝑇

Table 1

 1

presents the values of various performance metrics of the of CATS IC distribution for both the phase II equal-tailed and ATS-unbiased exponential charts obtained under the unconditional perspective, such as the mean (ACATS IC ), standard deviation SDCATS IC , some percentiles, coefficient of variation (CV=( SDCATS IC /ACATS IC ) × 100) for different values of 𝑚 with ATS 0 =370.4. The following observations can be made from

Table 1 . 1 .

 11 Though, the ACATS IC values are kept fixed at nominal ATS 0 =370.4, the corresponding SDCATS IC are quite high as compared to ACATS IC especially for small sample sizes for both the charts, for example, when 𝑚 = 20, the CV values are 42.24 and 47.18 for the phase II equaltailed and ATS-unbiased chart, respectively. Thus, looking at the ACATS IC only conveys half the story and this raises a question about the unconditional perspective. In order to reduce the practitioner-to-practitioner variability, i.e., to reduce 𝑆DCATS IC value to a reasonable level, a huge number of phase I observations are required. Zhang et al.[START_REF] Zhang | Exponential CUSUM charts with estimated control limits[END_REF] suggested that the variability for the ATS-unbiased exponential chart. Thus, the ATS-unbiased exponential chart needs substantially lower number of observations. 2. It is worth noting that for smaller values of 𝑚, the equal-tailed exponential chart has smaller SDCATS IC values than the ATS-unbiased exponential chart. For example, when 𝑚 = 20, the SDCATS IC value for the equal-tailed chart is 156.46, whereas it is 174.76 for the ATS-unbiased chart about 12% lower. However, for moderate to large sample sizes 𝑚, the 𝑆𝐷CATS IC values of the ATS-unbiased exponential chart are smaller than that of the equal-tailed exponential chart.For example, for 𝑚 = 150, the SDCATS IC values are 99.80 and 61.86 (about 38% less) for the equal-tailed and the ATS-unbiased exponential charts, respectively.

in IC CARL values should be within 10% of the nominal ARL value. Following them, to keep the SDCATS IC value within 10% of the desired ATS 0 , it is observed that more than 1000 phase I observations are needed for the equal-tailed exponential chart whereas about 400 phase I observations are required 3. The EP values for both the charts are not particularly appealing, especially, for larger sample sizes. This happens because as 𝑚 becomes larger, the CATS IC distribution for both the charts tends to become symmetric about the ATS 0 which produces EP values close to 0.50. This gives.

  Using these, the control limits of the equal-tailed and the ATS-unbiased exponential charts under the conditional perspective can be obtained from Equation (3) by replacing 𝐴 L and 𝐴 U by 𝐴 L * and 𝐴 U

	𝑑 𝑑𝛿	𝜇 1 ′ (𝛿)| 𝛿=1	= 0	(23)
	respectively where 𝑃[CATS IC ≥ ATS 0 ] can be obtained from Equation (14) by replacing 𝐴 L and 𝐴 U
	by 𝐴 L * and 𝐴 U * , respectively in 𝜑 IC (⋅). Equation (21) is obtained by equating the two unconditional
	probabilities 𝐸 [𝑃[𝑋 < LCL ̂ * |IC]] and 𝐸 [𝑃[𝑋 > UCL ̂ * |IC]] which can be obtained from Equations

(E.1) and (E.2) by replacing 𝐴 L and 𝐴 U by 𝐴 L * and 𝐴 U * . The 𝜇 1 ′ (𝛿) can be obtained from Equation (11) using 𝐴 1 * and 𝐴 2 * instead of 𝐴 L * and 𝐴 U * , respectively. The design parameters ξ * and 𝑝 * are calculated and reported in Table 3 for both phase II equal-tailed and ATS-unbiased exponential charts under the conditional perspective corresponding to different values of 𝑚. * , respectively.

Table 1 .

 1 The performance metrics of phase II equal-tailed and ATS-unbiased exponential chart under unconditional perspective with fixed nominal ATS 0 =

	370.4 and 𝜆 0 = 1
		Design Parameters	Percentiles
	𝑚	𝜉	𝑝	ACATS IC SDCATS IC 10% 25% 50% 75%	90%	EP	CV
	Equal-tailed

Table 2 .

 2 OOC performance of the equal-tailed and ATS-unbiased exponential chart under the unconditional perspective with nominal ATS0 = 370.4

	Equal-tailed

Table 3 .

 3 The performance metrics of phase II equal-tailed and ATS-unbiased exponential chart under conditional perspective with fixed 𝑃[𝐶𝐴𝑇𝑆 𝐼𝐶 ≥ 𝐴𝑇𝑆 0 ] = 0.90, nominal ATS 0 = 370.4 and 𝜆 0 = 1

	Design parameters

Table 4 .

 4 OOC performance of the equal-tailed and ATS-unbiased exponential chart under the conditional perspective with nominal ATS0 = 370.4 and

	𝑃[CATS IC ≥ ATS 0 ] = 0.90
	Equal-tailed

Table 5 .

 5 The design constants of the phase II equal-tailed and ATS-unbiased exponential chart under unconditional perspective with fixed nominal ATS 0 =

	370.4 and different 𝜆 0
							𝜆 0
		0.01		0.1		0.5		2		5		10
	𝑚	𝜉	𝑝	𝜉	𝑝	𝜉	𝑝	𝜉	𝑝	𝜉	𝑝	𝜉	𝑝
	Equal-tailed

Table 6 .

 6 The design constants of the phase II equal-tailed and ATS-unbiased exponential chart under conditional perspective with fixed 𝑃[𝐶𝐴𝑇𝑆 𝐼𝐶 ≥ 𝐴𝑇𝑆 0 ] = 0.90, nominal ATS 0 = 370.4 and different 𝜆 0

	𝜆 0

Acknowledgement

The authors thank the anonymous referees and the editor for their valuable comments.

Appendices

A. Distribution of time to signal (TS)

For the known parameter case, the run length variable, 𝑅 follows a geometric distribution with parameter β(δ), thus the probability mass function of 𝑅 is given by

Thus, using the fact that sum of 𝑗 independent and identically distributed exponential variables with mean λ 1 = δλ 0 follows a gamma distribution with PDF

the probability distribution of time to signal, TS is given by