Univariate fast initial response statistical process control with taut strings

Abstract

We present a novel real-time univariate monitoring scheme for detecting a sustained departure of a process mean from some given standard assuming a constant variance. Our proposed stopping rule is based on the total variation of a nonparametric taut string estimator of the process mean and is designed to provide a desired average run length for an in-control situation. Compared to the more prominent CUSUM fast initial response (FIR) methodology and allowing for a restart following a false alarm, the proposed two-sided taut string (TS) scheme produces a significant reduction in average run length for a wide range of changes in the mean that occur at or immediately after process monitoring begins. A decision rule for when to choose our proposed TS chart compared to the CUSUM FIR chart that takes into account both false alarm rate and average run length to detect a shift in the mean is proposed and implemented. Supplementary materials are available online.

1. Introduction

Let $(X_t{)}_{t\in T}$ be a univariate random process with independent $X_{t_i}$'s indexed by a discrete index set $T=\{t_1,t_2,\dots ,$ $t_n,\dots \}$ with uniform spacing $\mathrm{\Delta }t\equiv t_{i+1}-t_i$, $i=1,2,\dots ,\mathrm{\infty }$, and $x_1,x_2,\dots ,x_n,\dots$ be a realization of $(X_t{)}_{t\in T}$. Our focus is on process data where each $x_t$ is a characteristic of interest from a sequence of single independent items. In statistical process control, one is interested in detecting whether the process mean departs from a given standard ${\mu }_0$ assuming no change in the variance. Putting this into a formal framework, two different approaches can be adopted. Truncating the process to the first n time periods, one typically performs a test of size α based on these data, where $\alpha \in (0,1)$ is chosen small. Letting $\mu (t)=\mathbf{E}[X_t]$, the decision structure for this test can be formulated as

$$\begin{array}{rl}& H_0:\text{For all}t\in \{t_1,t_2,\dots ,t_n\},\mu (t)\equiv {\mu }_0\mathrm{vs}.\\ & H_1:\text{There exists}{t}^{\mathrm{\prime }}\in \{t_1,t_2,\dots ,t_n\}\text{ such that }\mu (t)\equiv {\mu }_0\text{ for}t<t^{\mathrm{\prime }}\text{ and }\\ & \phantom{H_1:\text{There exists}{t}^{\mathrm{\prime }}\in \{t_1,t_2,\dots ,t_n\}\text{ such that }}\mu (t)\equiv {\mu }_1\ne {\mu }_0\text{ for}t\ge t^{\mathrm{\prime }}.\end{array}$$ (1)

An alternative framework adopted in statistical process control [27, pp. 146, 148] is given by a ‘sequential testing’ or ‘on-line monitoring’ approach. With a slight modification of the hypotheses in Equation (1)

$$\begin{array}{rl}& H_0:\mathrm{For}\mathrm{all}t\in \{t_1,t_2,\dots ,t_n,\dots \},\mu (t)\equiv {\mu }_0\mathrm{vs}.\\ & H_1(t_n):\mu (t)\equiv {\mu }_0\mathrm{for}t<t_n\mathrm{and}\mu (t)\equiv {\mu }_1\ne {\mu }_0\mathrm{for}t\ge t_n,\end{array}$$ (2)

this methodology implements a stopping rule producing a stopping time τ for the process $(X_t{)}_{t\in T}$ such that the average run length (measured in time periods)

$${\mathbf{E}}_{\mathrm{\infty }}[N]\equiv \mathbf{E}[N|H_0]\text{where }\tau \equiv t_N$$ (3)

under the $H_0$ hypothesis is maintained at a given level $\vartheta >0$. The expectation in Equation (3) is taken with respect to the probability measure ${\mathbf{P}}_{\mathrm{\infty }}$ induced by $H_0$ from Equation (2) and typically referred to as the in-control (IC) average run length (ARL). In this paper, we refer to the independent Gaussian process $(X_t{)}_{t\in T}$ as stable or in-control if $H_0$ from Equation (2) is true assuming constant variance, i.e. if all $X_{t_i}$'s are independent and have the same prescribed mean ${\mu }_0$ and variance ${\sigma }_0^2$.

The alternative hypothesis $H_1(t_n)$ in Equation (2) assumes a sustained shift of size $\delta ={\mu }_1-{\mu }_0$ happens at time period $t_n$. In this situation, the process $(X_t{)}_{t\in T}$ is referred to as unstable or out-of-control (OC). With ${\mathbf{P}}_{t_n,\delta }$ denoting the associated probability measure, the OC run length is defined as

$${\mathbf{E}}_{t_n,\delta }[N-(n-1)|\tau \ge t_n]\mathrm{with}\tau \equiv t_N.$$

Given two stopping rules with identical IC ARLs, that rule with a shorter OC ARL ${\mathbf{E}}_{t_n,\delta }[N-(n-1)|\tau \ge t_n]$ is preferred for detecting shifts of size δ at time $t_n$.

Throughout this article, we restrict our focus to a ‘two-sided’ detection of a change in the mean and changes in the mean that occur before or shortly after process monitoring begins. In a practical sense, it is not uncommon for analysts to be particularly interested in whether the mean has shifted by an amount δ at or shortly after starting a process. A head start adaptation referred to as a fast initial response (FIR) can be applied to the classical CUSUM chart that reduces detection time for this scenario. The CUSUM FIR sets itself apart from the CUSUM by boosting the initial signals, the upper and lower, with a head start value of $\pm h/2,$ where h is the control limit.

Important advancements in statistical process control have recently been made by combining statistical techniques known to be effective in other applications to fundamental methodologies of statistical process control. For example, Zou and Qiu [29] incorporated aspects of the powerful statistical technique LASSO into multivariate process control methods. In this paper, the taut string techniques of Davies and Kovac [5] are implemented to construct a new control chart for detecting a change in the process mean. We refer to our new chart as the Taut String (TS) chart for the mean. Barlow et al. [1] introduced the taut string technique in the context of isotonic function estimation. Mammen and van de Geer [20] and Dümbgen and Kovac [6] used the taut string methodology for minimizing various functionals, in particular, the sum of squares, penalized by the total variation term. Overgaard [22] investigated the connection between the taut string and the one-dimensional version of the well-known Rudin–Osher–Fatemi model, Makovetskii et al. [19] presented a tube-based taut string algorithm for total variation regularization. Kim et al. [13] implemented the taut string approach in the context of statistical inverse problems. As for the present paper, we prove an asymptotic consistency result for the taut string estimator in the total variation norm and develop an asymptotic theory for the TS chart signal process in the IC situation. The latter plays a crucial role in computing control limits for the TS chart as it helps to significantly reduce simulation time which otherwise would have been infeasible.

Lucas and Crosier [18] noted that when a process is about to shift or perhaps has already shifted, applying the FIR produces a quicker shift detection on average than using the optimally tuned CUSUM without FIR. A downside of the FIR is the in-control (IC) ARL is somewhat reduced. This drawback is usually accepted by practitioners in exchange for quicker detection of the highly suspected immediate change in the process mean. Sometimes, however, the process is known or believed to be stable or in a steady state when the CUSUM method is applied. Lucas and Crosier [18] stated in these situations the head start adaptation does not in any meaningful way reduce the ARL for a shift of interest and the IC ARL is about the same as the IC ARL for a CUSUM without a head start. Hence, the CUSUM without FIR is recommended when a stable process is assumed or the process status is unknown. Because the CUSUM FIR chart focuses on processes highly suspected of a current or soon-to-be shift in the mean, to compare our proposed TS chart and CUSUM FIR chart performance, we selected four configurations of process mean shifts beginning at time $t_{n_{\mathrm{lead}}+1}$ in a discretized sense (steady-state size $n_{\mathrm{lead}}$), $n_{\mathrm{lead}}=0,1,3,5$.

The comparison focus of our paper is the TS chart versus the CUSUM FIR chart. Knoth [14] pointed out the typical exponentially weighted moving average (EWMA) chart with asymptotic control limits performs very poorly for small steady-state sizes. Further, Knoth [14] proposed EWMA charts with modified control limits more sensitive to shifts that occur early in the process. Results for steady-state size 0 are not competitive with our TS chart for small shifts in the mean. Averaging ARLs over all shifts considered shows the TS chart to be superior. Accounting for false alarms for larger steady-state sizes and incorporating a cost analysis is needed for TS chart versus EWMA chart comparison. One further reason we focus on TS versus CUSUM FIR charts instead of TV versus EWMA charts follows. Vardeman and Jobe [26, p. 142] pointed out under worst-case scenarios where the EWMA signal is drifting towards one control limit and a process shift in μ in the direction of the other control limit occurs, the run length required to reach the opposite control limit will be long. CUSUM and Shewhart charts do not share this worst-case inertia problem. Hence, we restrict the rest of this paper to TS and CUSUM FIR charts.

Page [23] and Brook and Evans [3] established the IC run length for the typical CUSUM can be reasonably modeled as a geometric distribution after the first several time periods. In contrast, the IC distribution for the CUSUM FIR chart has a much higher number of short run lengths and then becomes more like the IC distribution for CUSUM as run lengths get larger. This difference in the IC distribution reflects the higher sensitivity of the CUSUM FIR chart to a shift in the mean occurring at or shortly after process monitoring begins. Lucas and Crosier [18] illustrated this higher sensitivity to a process mean shift that occurs before or at the beginning of process monitoring with several overlaid run length distributions from the CUSUM FIR and CUSUM (see [18, Figures 1 and 2]). It is clear that the CUSUM FIR method detects this type of shift earlier than what the typical CUSUM chart would produce. IC run length distributions for the proposed TS chart are displayed in Figure 1 (IC ARL 370) and supplemental Figures I(2) and I(4) (IC ARLs 100 and 750). These figures reveal the sensitive feature of the TS chart which is an advantage for detecting shifts that occur at or shortly after monitoring begins but a possible disadvantage for shifts that occur much later than the start of process monitoring. This sensitivity results in an increased probability of false alarms associated with short run lengths. Thus, we chose to compare the shift-detection performance of our TS chart to the CUSUM FIR chart for process shifts occurring at or immediately after process monitoring begins. Further, false alarms will have some effect on our choice of the chart for steady-state sizes (1, 3 and 5). For that reason, we develop a decision rule for choosing the TS or CUSUM FIR chart that takes into account both false alarm rate and average run length to first detection of a process shift that occurs at some time $t^{\mathrm{\prime }}$ such that $t_1<t^{\mathrm{\prime }}\le t_2$, $t_3<t^{\mathrm{\prime }}\le t_4$, or $t_5<t^{\mathrm{\prime }}\le t_6$.

Figure 1.

TS chart ( $\alpha =0.60$) estimated run length cdf and pdf for IC ARL 370.

A two-sided control chart involving the non-Markovian generalized likelihood ratio statistic from Barnard [2] was investigated by Siegmund and Venkatraman [25]. The primary focus of Siegmund and Venkatraman [25] was the development of an asymptotic approximation formula for the average run length in the initial-state framework for both in- and out-of-control Gaussian processes. Practical aspects of their control chart along with a few comparisons were given. Choosing a ‘reference’ IC ARL 400 and an initial-state situation, we compared their generalized-likelihood-ratio (GLR) and combined Shewhart–CUSUM charts to our proposed TS chart. The TS chart outperformed GLR and combined Shewhart–CUSUM charts (cf. [25, Table 2]). Thus, we only focus on the CUSUM FIR chart for comparison purposes.

Although not the focus of our paper, the following theoretical aspect related to the usual CUSUM chart deserves mentioning. According to classic results of Lorden [17] and Moustakides [21], the optimally tuned CUSUM chart exhibits certain minimax-type optimality properties. They proved for the optimally tuned CUSUM chart that the essential supremum of its run length distribution over all steady-state sizes and all data streams that stay in-control before a shift of size δ occurs does not exceed the essential supremum associated with any other (non-randomized) chart for sufficiently large IC ARL values. This result is separate from the context of our paper as we use ARLs as the performance measure. Despite mathematical rigor, optimality results of Lorden [17] and Moustakides [21] have somewhat limited practical applicability. Apart from the fact the proportion of data streams surviving over increasingly large time horizons without issuing a false alarm goes to 0 (and this is exactly where the CUSUM chart performs best), the comparison framework adopted does not match practice when the chart restarts every time an out-of-control signal is issued.

Since IC run length distributions of TS and CUSUM FIR charts are not geometric and not identical, an adequate performance measure needs to incorporate both shift detection time as measured by ARL and a false alarm rate as measured by the number of chart restarts due to false alarms for a selected IC ARL. We put forth such a measure. A performance comparison is made between our TS chart and CUSUM FIR chart using the average run length to detection following the beginning of a sustained shift in the process mean and the false alarm rate that occurs before some prescribed beginning of the sustained shift in the process mean. We hold σ constant and the assumed IC ARL at the same level for both methods. The $X_t$'s are modeled as i.i.d. standard normal for an assumed stable process. Parenthetically, our approach is applicable for i.i.d. random variables of known distribution with given mean and variance. Selected initial and steady state (sizes 1, 3 and 5) processes are considered. For a variety of scenarios, a decision rule for choosing either the TS or CUSUM FIR chart is established that takes into account both the false alarm (restart) rate and average run length to the detection of a selected shift in the mean. We conclude the paper by comparing false alarm rates and detection performance of the TS and CUSUM FIR two-sided charts for data examples Lucas and Crosier [18] presented.

The rest of this article is structured as follows. In Section 2, we discuss the taut string estimator and develop our new TS chart. In Section 3, we present a simulation study that includes an IC simulation to determine the control limits as well as an OC simulation to estimate the OC ARLs for both the TS and the CUSUM FIR charts, propose an ‘economic’ decision rule based on cost analysis and apply this rule to identify scenarios where the TS chart is preferred over the CUSUM FIR chart. A numeric example is also given. In Section 4, we summarize and discuss the results of our paper. The algorithm for estimating the IC ARLs of our TS chart is given in Appendix 1 while Appendix 2 discusses the asymptotics of the TS estimator. Supplemental Sections I through IV contain additional figures, tables and auxiliary proofs.

2. Methodology: the taut string and TS control chart

We briefly discuss the total variation approach to nonparametric regression and its connection to the taut string estimator proposed by Davies and Kovac [5]. For a more detailed treatise on nonparametric inference, we refer the reader to [7, Chapter 2], [12, Chapter 4] or [28, Chapter 5]. Further, we introduce our new TS chart built upon the taut string estimate of the underlying process conditional mean and study the asymptotic behavior of the associated TS signal process. It should be emphasized that we solely consider univariate processes in this paper. Despite recent developments in total variation denoising for multivariate processes, to the best of our knowledge, only approximate computational heuristics for multivariate taut strings currently exist (cf. [8] and references therein). Once appropriate methodologies become available in the literature, we envision a possibility to extend our TS chart to multivariate process data. See also discussion at the end of Section 4. We refer to a recent contribution by Liu et al. [16] for an alternative approach that does not involve taut string smoothing.

Having started with a discrete-time univariate process $(X_t{)}_{t\in T}$ indexed by an equidistant set $T=\{t_1,t_2,\dots ,t_n,\dots \}$, at each time step $t_n\in T$, we rescale the truncated index set $\{t_1,t_2,\dots ,t_n\}$ to $I_n=\{\frac{1}{n},\frac{2}{n},\dots ,1\}$ and treat the truncated process $(X_{t_k}{)}_{1\le k\le n}$ as if it was sampled from a continuous-time random process $(Y_t{)}_{t\in [0,1]}$ on an equidistant lattice $I_n$. This enables us to transform the time index set onto a compact interval and invoke the nonparametric techniques described in Section 2.1.

2.1. Total variation regression and taut strings

For a probability space $(\mathrm{\Omega },\mathcal{F},\mathbf{P})$, let $(Y_t{)}_{t\in [0,1]}$ be a univariate stochastic process such that $\mu (t)=\mathbf{E}[Y_t]$ and $\sigma (t)=(\mathbf{V}\mathbf{a}\mathbf{r}[Y_t]{)}^{1/2}$ exist for any $t\in [0,1]$. According to Tukey, the process can be decomposed as

$$Y_t=\mu (t)+\sigma (t){\epsilon }_t\text{for every}t\in [0,1]\mathbf{P}\text{-a.s.},$$ (4)

where $({\epsilon }_t{)}_{t\in [0,1]}$ is an error process with zero mean and unit variance for $t\in [0,1]$. In the present paper, we assume ${\epsilon }_t$'s are i.i.d. with $\mathbf{E}[{\epsilon }_t]=0$, $\mathbf{V}\mathbf{a}\mathbf{r}[{\epsilon }_t]=1$ and $\sigma (t)\equiv {\sigma }_0>0$ constant. For $n\in \mathbb{N}$, we define a finite index set $I_n:=\{\frac{1}{n},\frac{2}{n},\dots ,1\}$ and consider the discrete random process $(Y_t{)}_{t\in I_n}$. Our thrust is to obtain a nonparametric estimate ${\hat{\mu }}_n(t)$ of the function $\mu (t)$ based on a process realization $(y_n(\frac{1}{n}),y_n(\frac{2}{n}),\dots ,y_n(1))$ of $(Y_t{)}_{t\in I_n}$.

In contrast to parametric regression, standard approaches like ordinary least-squares or maximum-likelihood techniques fail in the nonparametric case due to the lack of compactness. A way to overcome this deficiency is to use a Tychonoff regularization, which can furnish compactness in the $L^2$-topology. In our case, an appropriate Tychonoff regularization is given by the total variation functional

$$\mathrm{TV}(f)=sup\{\sum _{k=1}^m|f({\xi }_k)-f({\xi }_{k-1})||0={\xi }_0<\cdots <{\xi }_m=1\text{ for }m\in \mathbb{N}\},$$ (5)

where $f(t)$ plays the role of a candidate for ${\hat{\mu }}_n(t)$. This type of regularization is particularly helpful in the context of statistical process control as it allows for the estimation of discontinuous trends $\mu (t)$. As proved by Grasmair [11], the solution to the standard least-squares problem penalized by the TV-functional is equivalent to the taut string estimator earlier proposed by Davies and Kovac [5] and outlined below.

For a realization $(y_n(\frac{1}{n}),y_n(\frac{2}{n}),\dots ,y_n(1))$ of $(Y_t{)}_{t\in I_n}$, consider the cumulative process

$$y_n^{\circ }(0)=0,y_n^{\circ }\left({\displaystyle \frac{k}{n}}\right)={\displaystyle \frac{1}{n}}\sum _{j=1}^k{y}_n\left({\displaystyle \frac{j}{n}}\right)\mathrm{for}k=1,\dots ,n.$$ (6)

For ${\lambda }_n>0$, in the context of the taut string methodology, the lower and upper bounds $l_n(t)$ and $u_n(t)$ for $y_n^{\circ }(t)$ are defined as

$$l_n(t)=\{\begin{array}{ll}0,& t=0,\\ {\displaystyle y_n^{\circ }\left(\frac{k}{n}\right)-{\lambda }_n,}& t={\displaystyle \frac{k}{n}},\end{array}{u}_n(t)=\{\begin{array}{ll}0,& t=0,\\ y_n^{\circ }\left({\displaystyle \frac{k}{n}}\right)+{\lambda }_n,& t={\displaystyle \frac{k}{n}}\end{array}$$ (7)

for $k=1,\dots ,n$ and extended onto $[0,1]$ by a piecewise linear interpolation. Next, a function tube of radius ${\lambda }_n$ around $y_n^{\circ }(t)$ can be expressed as

$$\begin{array}{rl}B(y_n^{\circ }(t),{\lambda }_n)=\{s:[0,1]\to \mathbb{R}& |s(t)\text{ absolutely continuous, }s(0)=0,\\ & l_n(t)\le s(t)\le u_n(t)\text{ for }0\le t\le 1\}.\end{array}$$ (8)

In this paper, the tube radius ${\lambda }_n$ is selected as

$$\begin{array}{rl}{\lambda }_n& ={\sigma }_{0}max\{0.8\sqrt{2\log (\log (max\{n,3\}))},\mathrm{med}(\underset{t\in [0,1]}{max}|W(t)|)\}{n}^{-1/2}\\ & \approx {\sigma }_{0}max\{0.8\sqrt{2\log (\log (max\{n,3\}))},1.149\}{n}^{-1/2},\end{array}$$ (9)

where $(W(t){)}_{t\in [0,1]}$ denotes a standard univariate Wiener process and $\mathrm{med}(\cdot )$ stands for the median. If ${\sigma }_0$ is not known, an estimate ${\hat{\sigma }}_0$ is used, e.g. the one in Equation (10). While the second argument of the max-function in Equation (9) corresponds to the choice recommended by Davies and Kovac [5, p. 22], the first argument is required to invoke the Law of the Iterated Logarithm in the proof of our Theorem A.1 and later used to insure an eventual no-break condition for the taut strings associated with in-control data streams.

For $s(t)\in B(y_n^{\circ }(t),{\lambda }_n)$, the length functional or curve length reads as

$$\mathcal{L}(s(t))={\int }_0^1\sqrt{1+(s^{(1)}(t){)}^2}\mathrm{d}t.$$

From [5], the functional $\mathcal{L}(\cdot )$ is known to attain its unique minimum over the set $B(y_n^{\circ }(t),{\lambda }_n)$ at some function $s_n^{\ast }(t)$, which is exactly the taut string running through the tube $B(y_n^{\circ }(t),{\lambda }_n)$ ‘clamped’ at $s_n^{\ast }(0)=0$ and $s_n^{\ast }(1)=y_n^{\circ }(1)$, i.e.

$$s_n^{\ast }(t)=\underset{s(t)}{\arg \min }\{\mathcal{L}(s(t))|s(t)\in B(y_n^{\circ }(t),{\lambda }_n)\text{ such that }s(0)=0,s(1)=y_n^{\circ }(1)\}.$$

Moreover, $s_n^{\ast }(t)$ is a piecewise linear function uniquely determined by its values at the grid points $\{0,\frac{1}{n},\frac{2}{n},\dots ,1\}$ and continuously depends on these values (cf. the proof of Theorem A.1). In particular, $s_n^{\ast }(t)$ is absolutely continuous. The taut string estimator ${\hat{\mu }}_n(t)$ for $\mu (t)$ is then given as the local slope of $s_n^{\ast }(t)$ at $t\in [0,1]$, more precisely, the cádlág (right continuous with left limits) realization of the (weak) derivative $s_n^{\ast (1)}(t)$. With ${\hat{\mu }}_n(t)$ being uniquely determined by its values on $I_n$, we do not need to distinguish between ${\hat{\mu }}_n(t)$ and $({\hat{\mu }}_n{|}_{I_n})(t)$.

Figure 2 illustrates the taut string estimator applied to a Lake Erie dataset of n = 600 monthly levels measured over 1921–1970 Qlik $\circledR$ [24]. For illustration purposes, the data were transformed by subtracting the sample mean and ${\sigma }_0$ was estimated by

$${\hat{\sigma }}_0=(1.48/\sqrt{2})\cdot \mathrm{med}\{|y_n\left(\frac{2}{n}\right)-y_n\left(\frac{1}{n}\right)|,\dots ,|y_n(1)-y_n\left(\frac{n-1}{n}\right)|\}.$$ (10)

We now extend the basic notions and terminologies previously presented and needed for applications in the control chart setting. Based on a single (truncated) stream of observed process data $x_1,\dots ,x_{n_0}$ of length $n_0\in \mathbb{N}$ realized at equidistant times $t_1<\cdots <t_{n_0}$, a total of $n_0$ taut strings denoted by $s_1^{\ast }(t),\dots ,s_{n_0}^{\ast }(t)$ are constructed with $s_k^{\ast }(t)$ being the taut string computed from the first k observations. For the kth taut string, the taut string estimator ${\hat{\mu }}_k(t)$ (given as the local slope of $s_k^{\ast }(t)$) estimates the ( time-scaled) conditional mean of the data process. For the Taut String chart, the kth taut string produces one signal, ${\mathrm{TS}}_k$. Finally, when the phrase ‘taut string estimator’ is used, we are not referring to an estimate of the taut string, rather its slope. A better terminology would be ‘taut-string-based estimator’.

Figure 2.

Plots of $y_n(t)$, ${\hat{\mu }}_n(t)$, $y_n^{\circ }(t)$, $B(y_n^{\circ },{\lambda }_n)$ and $s_n^{\ast }(t)$.

As Equation (5) suggests, a natural norm for measuring taut string estimation quality is ‘the’ total variation norm

$$\Vert {\hat{\mu }}_n(t)-\mu (t){\Vert }_{{\mathrm{BV}}_0([0,1])}:=|{\hat{\mu }}_n(0)-\mu (0)|+\mathrm{TV}({\hat{\mu }}_n(t)-\mu (t)).$$ (11)

Being stronger than the uniform or integral norms typically employed in nonparametric statistics, the total variation norm is more sensitive and, therefore, particularly suitable for statistical process control. With the total variation norm defined in Equation (11), we introduce our new TS control chart in Section 2.2. In our analysis, we discovered the TS chart run length distribution to be rather heavy-tailed to the right; therefore, the control limit determination procedure needed an asymptotic approximation of the TS signal process. An expanded discussion of this approximation is given in supplemental Section IV(i). In Appendix 2, necessary asymptotic consistency theory for the taut string estimator is presented.

2.2. The TS control chart signal and control limit

We now introduce the signal statistic of our TS chart and describe a stopping rule for the associated signal process. Let $x_1,x_2,\dots ,x_n,\dots$ be a realization of a univariate process $(X_{t_n}{)}_{t_n\in T}$ with $T=\{t_1,t_2,\dots ,t_n,\dots \}$ such that

$$X_{t_n}=\mu (t_n)+\sigma (t_n){\epsilon }_n\text{ for }{t}_n\in T\text{ with }\mu (t_n)=\mathbf{E}[X_{t_n}],\sigma (t_n)=\sqrt{\mathbf{V}\mathbf{a}\mathbf{r}[X_{t_n}]},$$

where ${\epsilon }_n$'s are i.i.d. standard normal random variables. The IC standards ${\mu }_0\in \mathbb{R}$ and ${\sigma }_0>0$ are assumed known and $\sigma (t_n)\equiv {\sigma }_0$.

Selecting $\alpha =\frac{3}{5}$, the TS signal statistic at the nth period $t_n$ is defined as

$$\begin{array}{rl}{\mathrm{TS}}_n& =n^{\alpha }{\Vert \frac{{\hat{\mu }}_n(t)-{\mu }_0}{{\sigma }_0}\Vert }_{{\mathrm{BV}}_0([0,1])}\\ & \equiv \frac{n^{3/5}}{{\sigma }_0}(|{\hat{\mu }}_n\left(\frac{1}{n}\right)-{\mu }_0|+\sum _{k=2}^n|{\hat{\mu }}_n\left(\frac{k}{n}\right)-{\hat{\mu }}_n\left(\frac{k-1}{n}\right)|),\end{array}$$ (12)

where ${\hat{\mu }}_n(t)$ denotes the taut string estimator applied to the dataset $(x_1,\dots ,x_n)$ treated as a realization of the stochastic process $(X_{t_k}{)}_{1\le k\le n}$ as if it was sampled on $I_n=\{\frac{1}{n},\frac{2}{n},\dots ,1\}$ which makes the theory in Section 2.1 applicable. The tube radius ${\lambda }_n$ is chosen according to Equation (9). The rationale behind the choice $\alpha =\frac{3}{5}$ in Equation (12) is that it appears, based on preliminary simulations with other α, to produce a reasonable trade-off between chart performance and tail thickness of the IC run length distribution. Since the TS signal is affine equivariant, the standards ${\mu }_0=0$, ${\sigma }_0=1$ were assumed for the in-control simulation.

We decided to employ the total variation norm in Equation (12) to define the TS signal process for two main reasons. On the one hand, this norm appears to generate the finest topology with respect to which the asymptotic convergence theory from Appendix 2 remains valid. On the other hand, this choice is particularly natural in the light of equivalence between the taut string problem and total variation regression (cf. [11,22]). Weaker norms can be used to define alternative signal processes, e.g.

$${\mathrm{WL}}_n^1:=\frac{n^{\alpha }}{{\sigma }_0}\sum _{k=1}^n|{\hat{\mu }}_n\left(\frac{k}{n}\right)-{\mu }_0|$$

using the weighted ${\ell }_1$-norm. With appropriate adaptations, implementation of the methodology developed in this paper can potentially analyze this and other signal types. This is beyond the scope of our present contribution though.

Because TS signals are nonnegative, the TS chart is stopped, say at the nth time period, as soon as ${\mathrm{TS}}_n>L$ where $L>0$ is a selected control limit. This yields the stopping time $\tau =t_n$ and the associated run length

$$\mathrm{RL}=min\{n^{\mathrm{\prime }}\in \mathbb{N}|{\mathrm{TS}}_{n^{\mathrm{\prime }}}>L\}=n.$$ (13)

The IC ARL and IC run length standard deviation for the stopping time $\tau (L)\equiv t_N(L)$ become

$$\text{IC ARL}(L)={\mathbf{E}}_{\mathrm{\infty }}[N(L)]\mathrm{and}{\sigma }_{\mathrm{RL}}(L)=({\mathbf{V}\mathbf{a}\mathbf{r}}_{\mathrm{\infty }}[N(L)]{)}^{1/2}$$

with the probability ${\mathbf{P}}_{\mathrm{\infty }}$ induced by $X_{t_n}\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}.}{\sim }\mathcal{N}({\mu }_0,{\sigma }_0^2)$ for $n\in \mathbb{N}$.

Identifying appropriate control limits is a fundamental aspect of TS chart construction. For a desired IC ARL value ϑ, e.g. $\vartheta =370$, the control limit $L_{\vartheta }$ is such that

$$\mathbf{E}[N(L_{\vartheta })]=\vartheta \mathrm{with}\tau (L)\equiv t_{N(L)}.$$ (14)

No explicit equation for the distribution of the IC run length $N(L)$ is known. Hence, a Monte Carlo simulation was essential to estimate the IC ARL ${\mathbf{E}}_{\mathrm{\infty }}[N(L)]$. As simulation size grows, both CPU time and memory requirements become prohibitively large due to the run length distribution being heavy-tailed to the right.

To overcome this barrier, necessary TS signal asymptotic theory expressed in Theorems A.1 and A.2 in the appendix and supplemental Theorems IV.1 and IV.2. was developed. This theory provides support for the simulation method used to find TS control limits for IC ARLs of interest. In particular, the asymptotic value of the TS chart signal ${\mathrm{TS}}_n$ in the IC situation, where $n\ge n_0$ for large $n_0$, was derived to be

$${\mathrm{TS}}_n={\sigma }_0{n}^{-2/5}\left|\sum _{k=1}^n{\epsilon }_n\right|\stackrel{d}{=}|{\mathrm{TS}}_{n_0}+{\sigma }_0{n}^{-2/5}\sum _{k=n_0}^n{\epsilon }_k|\mathrm{with}{\epsilon }_k\stackrel{\mathrm{i}.\mathrm{i}.\mathrm{d}.}{\sim }\mathcal{N}(0,1).$$ (15)

Note that Equation (15) is true if and only if none of the taut strings $s_{n_0}^{\ast }(t),\dots ,s_n^{\ast }(t)$ computed from the first $n_0,n_0+1,\dots ,n$ points from a particular data stream break, i.e. if the local slope ${\hat{\mu }}_k(t)$ does not change within a particular taut string $s_k^{\ast }(t)$. This is equivalent to saying that each of the taut string estimators ${\hat{\mu }}_k(t)$ for $k=n_0,n_0+1,\dots ,n$ is a constant function. The conditional run length expectation associated with the right-hand side of Equation (15) is given in Equation (16) and derived in supplemental Theorem IV.1.

For our Monte Carlo simulation, a set of $N_{\mathrm{MC}}=1,000,000$ i.i.d. standard normal data streams of size $n_0=\mathrm{10,000}$ were simulated. A grid of 2000 control limits L ranging from 1.40 to 3.00 was considered. For a given L, from the set of $N_{\mathrm{MC}}$ streams, two disjoint subsets were identified: those streams having actual run lengths less than or equal to $n_0=\mathrm{10,000}$ ( $D_{n_0,L}$) and those streams having actual run lengths greater than $n_0$ ( $D_{n_0,L}^c$). For the selected control limit, the IC ARL was estimated as the weighted average of the average run length for all streams in $D_{n_0,L}$ and average of the expected run length values of all streams in $D_{n_0,L}^c$.

Consider the selected control limits L = 1.7706, 2.1233 and 2.3261; the three sets $D_{n_0,L}^c$ contained 25, 472 and 1381 streams from the $N_{\mathrm{MC}}=\mathrm{1,000,000}$ randomly generated streams. The question becomes how run length can be estimated for each stream in $D_{n_0,L}^c$. On the strength of Theorem A.1, the probability that for $n\ge n_0$ all taut strings $s_n^{\ast }(t)$ do no break converges to 1 as the truncation time point $n_0\to \mathrm{\infty }$. For such streams, asymptotic equation (15) is valid for determining run length. To empirically illustrate the value of $n_0=\mathrm{10,000}$ was sufficiently large to invoke Theorem A.1, the following procedure was performed. All streams from $D_{n_0,L}^c$ such that the associated taut string $s_k^{\ast }(t)$ does not break for all $k\in \{\lfloor 0.75n_0\rfloor ,\dots ,n_0\}$ were counted. Using the same three control limits as above, none of the data streams in $D_{n_0,L}^c$ happened to violate this condition.

Hence, in line with Theorem A.1, for each data stream in $D_{n_0,L}^c$, instead of simulating the TS signal beyond $n_0$, the TS signal process was replaced with the discrete half-Gaussian random process in Equation (15). Probability theory was used to compute the average run length for each of these data streams. Letting $m_0$ play the role of some discretization constant explained in the appendix, we set $m_0=1$ as this choice produces the most conservative approximation of estimated run length while other values of $m_0$ such as 10, 100, 500 or 1000 lead to essentially the same results. The conditional run length expectation of streams in $D_{n_0,L}^c$ can be expressed as

$$\text{Cond. RL expectation}=n_0+m_0{V}_1(\frac{n_0}{m_0},m_0^{1/2-\alpha }{\mathrm{TS}}_{n_0};m_0^{1/2-\alpha }L)$$ (16)

with the ‘value function’ $V_1(\cdot ,\cdot ;\cdot )$ from Equation (IV.6) in supplemental Section IV(ii) Theoretical justification of Equation (16) is given in supplemental Theorem IV.1. The IC run length standard deviation ${\sigma }_{\mathrm{IC}\mathrm{RL}}$ was estimated in a similar fashion to that of IC ARL. Both estimation procedures are detailed in Appendix 1.

A summary of IC run length standard deviations ( $s_{\mathrm{IC}\mathrm{RL}}$), $\widehat{\mathrm{IC}\mathrm{ARL}}$ standard deviations ( $s_{\widehat{\mathrm{IC}\mathrm{ARL}}}=s_{\mathrm{IC}\mathrm{RL}}/\sqrt{N_{\mathrm{MC}}}$) and estimated control limits for selected nominal IC ARLs are given in Table 1. TS charts are a function of a tuning number α. In this paper, we restrict attention to $\alpha =0.60$. Since standard deviations of the $\widehat{\mathrm{IC}\mathrm{ARL}}$s become noticeably larger for larger IC ARLs, modified control limits for an IC ARL are given. These modified limits correspond to limits from simulation for the selected $\mathrm{IC}\mathrm{ARL}+3s_{\mathrm{IC}\mathrm{RL}}/\sqrt{N_{\mathrm{MC}}}$. As can be seen in Table 1, although standard deviations of $\widehat{\mathrm{IC}\mathrm{ARL}}$ values become large as the nominal IC ARL increases, the modified control limits change little. The TS chart control limit curve is illustrated in Figure 3 for IC ARLs. Table II(2) in supplemental Section II contains a more extensive set of TS chart control limits. Table II(3) in supplemental Section II contains CUSUM and CUSUM FIR control limits used in our simulations and example.

Table 1.

Summary characteristics of TS chart for a stable process.

	Nominal IC ARL
Taut string (TS) chart	100	370	750
Run length std. dev. ( $s_{\mathrm{IC}\mathrm{RL}}$)	447.99	1928.89	4078.65
Std. dev. of $\widehat{\mathrm{IC}\mathrm{ARL}}$ ( $s_{\widehat{\mathrm{IC}\mathrm{ARL}}})=s_{\mathrm{IC}\mathrm{RL}}/\sqrt{N}$	0.4480	1.9289	4.0786
$\mathrm{IC}\mathrm{ARL}+3s_{\widehat{\mathrm{IC}\mathrm{ARL}}}$	101.3440	375.7867	762.2359
Estimated control limit for $\mathrm{IC}\mathrm{ARL}+3s_{\widehat{\mathrm{IC}\mathrm{ARL}}}$	1.7706	2.1233	2.3261
Estimated control limit for $\mathrm{IC}\mathrm{ARL}$	1.7673	2.1193	2.3214

Figure 3.

TS chart ( $\alpha =0.6$) conservative control limits for IC ARLs 100, 370 & 750.

3. Application: simulation and comparisons

Several factors affect chart performance – type of shift, time of shift, manner of shift, how the chart is tuned, shift size and tolerance level for incorrectly flagging a shift. We investigated and compared the impact of these on the two-sided CUSUM FIR and TS charts' ability to correctly detect a shift type referred to as a sustained change in the process mean from some standard. Throughout the simulation, data from a stable process stream were assumed to be standard normal. If the simulation initiated the time of shift at time 1 or before, the simulation was identified as an initial-state simulation of a steady-state process of size 0. Simulations of steady-state processes with size $n_{\mathrm{lead}}$ assume the shift begins on or before the first period after some $n_{\mathrm{lead}}$ time periods of a stable process. Letting $t_{\mathrm{start}}$ be the last point in time of a stable process and $t_{\mathrm{end}}$ the time of first occurrence of an out-of-control signal after the $n_{\mathrm{lead}}$ time periods of a stable process, the run length for a steady-state process was recorded as $(t_{\mathrm{end}}-t_{\mathrm{start}})/(\mathrm{\Delta }t)$ with uniform spacing $\mathrm{\Delta }t$. Any signal of a shift in the mean or upset in the system at or before time $t_{\mathrm{start}}$ is counted as a false alarm or restart and the chart re-started. As previously stated, steady-state sizes 0, 1, 3 and 5 were considered.

Shifts in the mean can occur in many fashions. For our simulation and analysis, different sustained shifts away from the standard process mean 0 with constant variance 1 were selected. When the process mean increased from 0 to 0.5, we let $\delta =0.5$ denote shift size. To evaluate and compare chart detection performances, we identified seven sustained shifts ( $\delta =0.2,0.50,1.00,1.50,2.00,2.50$ and 3.00) and seven k values for the CUSUM FIR chart, k = 0.10, 0.25, 0.50, 0.75, 1.00, 1.25 and 1.50. Simulations of the two-sided TS chart and CUSUM FIR chart were generated for each of the situations, steady-state sizes 0, 1, 3, 5, IC ARLs 100, 370 and 750. Except for the false alarm or restart issue described earlier, simulations were performed in the same fashion as those that produced the control limits with the data streams generated from up to $n_0\mathcal{N}(\delta ,1)$ random variates for each of $N=\mathrm{1,000,000}$ streams. No run lengths in excess of $n_0=\mathrm{10,000}$ occurred for all shifted scenarios. Hence, there was no need to implement the asymptotic theory required for IC ARL calculations. So, ARL and $s_{\mathrm{ARL}}$ statistics were calculated with the usual approach. A complete set of run length standard deviations $s_{\mathrm{RL}}$ and ARL standard deviations $s_{\mathrm{ARL}}$ from our TS and two-sided CUSUM FIR charts based on initial and steady-state simulations for the set of δ's are given in the supplement.

Assuming an IC ARL of 370 and steady-state size 0, Table 2 contains CUSUM FIR and TS chart ARLs as a function of $(\delta ,k)$ and average number of restarts (false alarms) as a function of k. Table 3 contains ARL and average number of restart differences (TS – CUSUM FIR). Based on the decision rule described later in the paper, see inequality (18), the TS chart is preferred over the CUSUM FIR for all (δ, k) considered.

Table 2.

ARLs & average restart numbers, IC ARL 370, steady-state size 0.

ARL
	CUSUM FIR							TS
$\delta \mathrm{\setminus }k$	0.10	0.25	0.50	0.75	1.00	1.25	1.50	$\alpha =0.60$
0.20	69.876	96.778	153.701	199.551	235.149	262.670	282.977	37.455
0.50	19.989	18.416	27.330	43.512	63.779	86.831	110.126	11.221
1.00	9.053	6.585	6.197	7.596	10.528	15.175	21.431	4.321
1.50	5.943	4.081	3.301	3.272	3.708	4.625	6.101	2.544
2.00	4.479	3.020	2.311	2.098	2.126	2.335	2.719	1.797
2.50	3.630	2.443	1.817	1.589	1.541	1.587	1.706	1.414
3.00	3.084	2.094	1.507	1.310	1.259	1.264	1.306	1.201
${\mathrm{Avg}}_{\delta }$	16.579	19.060	28.024	36.990	45.441	53.498	60.910	8.565
Average restart number
	CUSUM FIR							TS
k	0.10	0.25	0.50	0.75	1.00	1.25	1.50	$\alpha =0.60$
–	0.000	0.000	0.000	0.000	0.000	0.000	0.000	0.000

Table 3.

Differences in ARL & average restart number, IC ARL 370, steady-state size 0.

ARL difference
	TS minus CUSUM FIR
$\delta \mathrm{\setminus }k$	0.10	0.25	0.50	0.75	1.00	1.25	1.50
0.20	−32.421	−59.323	−116.246	−162.096	−197.695	−225.215	−245.522
0.50	−8.768	−7.195	−16.109	−32.291	−52.558	−75.611	−98.905
1.00	−4.732	−2.264	−1.876	−3.275	−6.207	−10.854	−17.110
1.50	−3.399	−1.537	−0.757	−0.728	−1.164	−2.080	−3.557
2.00	−2.682	−1.223	−0.514	−0.301	−0.329	−0.537	−0.922
2.50	−2.216	−1.029	−0.403	−0.174	−0.126	−0.173	−0.291
3.00	−1.882	−0.893	−0.305	−0.109	−0.058	−0.062	−0.105
${\mathrm{Avg}}_{\delta }$	−8.014	−10.495	−19.459	−28.425	−36.877	−44.933	−52.345
Average restart number difference
	TS minus CUSUM FIR
k	0.10	0.25	0.50	0.75	1.00	1.25	1.50
–	0.000	0.000	0.000	0.000	0.000	0.000	0.000

Notes: Inequality (18) is satisfied for all $(\delta ,k)$. Always choose TS chart here.

Continuing, assuming IC ARL 370 and a process of steady-state size 1, Table 4 contains CUSUM FIR and TS chart ARLs as a function of $(\delta ,k)$ and average number of restarts (false alarms) as a function of k. Table 5 contains CUSUM FIR and TS chart ARLs and average number of restart differences (TS − CUSUM FIR). Table 6 summarizes the decision rule described later, see inequality (17), under an IC 370 and steady-state size 1. The complete set of tables for IC 100, 370, 750, and steady-state sizes (0, 1, 3 and 5) along with standard deviations appear in supplemental Sections III(i), III(ii) and III(iii).

Table 4.

ARLs and average restart numbers, IC ARL 370, steady-state size 1.

ARL
	CUSUM FIR							TS
$\delta \mathrm{\setminus }k$	0.10	0.25	0.50	0.75	1.00	1.25	1.50	$\alpha =0.60$
0.20	70.491	97.903	154.880	200.991	236.229	262.942	282.833	37.445
0.50	20.223	19.224	28.955	45.574	66.118	89.060	111.969	11.518
1.00	9.165	6.908	7.004	8.849	12.105	16.934	23.281	4.690
1.50	6.015	4.280	3.766	4.014	4.656	5.729	7.282	2.851
2.00	4.533	3.164	2.633	2.598	2.752	3.042	3.481	2.050
2.50	3.673	2.550	2.069	1.973	1.986	2.062	2.200	1.613
3.00	3.113	2.167	1.736	1.622	1.586	1.590	1.629	1.340
${\mathrm{Avg}}_{\delta }$	16.745	19.456	28.721	37.946	46.490	54.480	61.811	8.787
Average restart number
	CUSUM FIR							TS
k	0.10	0.25	0.50	0.75	1.00	1.25	1.50	$\alpha =0.60$
–	0.000	$0.1\cdot {10}^{-4}$	0.004	0.015	0.023	0.024	0.021	0.034

Table 5.

Differences in ARL & average restart numbers, IC ARL 370, steady-state size 1.

ARL difference
	TS minus CUSUM FIR
$\delta \mathrm{\setminus }k$	0.10	0.25	0.50	0.75	1.00	1.25	1.50
0.20	−33.046	−60.457	−117.435	−163.546	−198.784	−225.497	−245.388
0.50	−8.705	−7.706	−17.437	−34.055	−54.600	−77.541	−100.451
1.00	−4.475	−2.218	−2.314	−4.160	−7.415	−12.244	−18.591
1.50	−3.163	−1.428	−0.915	−1.162	−1.804	−2.878	−4.430
2.00	−2.483	−1.113	−0.583	−0.548	−0.701	−0.992	−1.431
2.50	−2.060	−0.937	−0.457	−0.360	−0.373	−0.449	−0.587
3.00	−1.773	−0.827	−0.396	−0.282	−0.246	−0.250	−0.289
${\mathrm{Avg}}_{\delta }$	−7.958	−10.670	−19.934	−29.159	−37.703	−45.693	−53.024
Average restart number difference
	TS minus CUSUM FIR
k	0.10	0.25	0.50	0.75	1.00	1.25	1.50
–	0.034	0.034	0.030	0.019	0.010	0.009	0.012

Table 6.

Decision rule for IC ARL 370 and steady-state size 1: choose TS over CUSUM FIR if $\frac{d_{\mathrm{avg}}}{1+\rho }$ (see Table 7) satisfies the inequality in a particular cell.

$\delta \mathrm{\setminus }k$	0.10	0.25	0.50	0.75	1.00	1.25	1.50
0.20	<976.5	<1786.8	<3863.9	<8591.9	<18579.7	<23669.2	<19243.1
0.50	<262.1	<232.1	<585.1	<1834.1	<5400.6	<9196.1	<8186.0
1.00	<133.6	<66.2	<76.9	<222.6	<734.5	<1350.6	<1472.5
1.50	<93.8	<42.4	<30.3	<61.0	<175.1	<307.0	<362.2
2.00	<73.7	<33.0	<19.2	<29.1	<66.9	<111.7	<114.9
2.50	<61.2	<27.9	<15.1	<19.0	<34.9	<48.7	<46.5
3.00	<52.7	<24.6	<13.1	<14.9	<23.8	<27.2	<23.2

Statistical software by Gan [9,10] and Knoth [15] (viz. xcusum.crit() and xcusum.arl() routines from the spc R-package) provided assistance with identifying optimal control limits $(h,-h)$ for given $k=\delta /2$ and selected IC ARLs for CUSUM FIR charts. The programs were also helpful as we checked ARLs from our initial and steady-state simulations for known $(h,k)$ pairs from optimally and not optimally designed CUSUM FIR charts for the δ's of interest.

As described earlier, $N_{\mathrm{MC}}=\mathrm{1,000,000}$ data streams, each $n_0=\mathrm{10,000}$ time periods long, were created to determine TS chart control limits for IC ARLs of interest. The separate data streams were split into 50 independent runs, and the total variation of the taut string estimator for these data ( $n_0$ values for each of the streams) was precomputed. The individual runs finished in about 12 h with total CPU or core hours summing to almost 600. The simulations were executed in Matlab $\circledR$ on Dell $\circledR$ PowerEdge^TM 410 machines with Intel $\circledR$ Xeon $\circledR$ E5520 processors having 24 GB of memory each. Simulations for the out-of-control situations were carried out on Dell $\circledR$ Precision 7920 Tower^TM with 32 Intel $\circledR$ Xeon $\circledR$ Gold 6130 processor cores with a total of 128 GB of shared DDR4 RAM. The total core hours amounted to 1800. Both examples from Section 3.3 were run in real time. A Matlab $\circledR$ code by Dümbgen (2002) was used to produce the taut string estimator. For real-time data, the Matlab $\circledR$ program produces the TS signals and associated control chart instantly.

3.1. Decision rule for choosing TS or CUSUM FIR chart

We make the following definitions and assumptions regarding costs associated with a false alarm and average run length to detection following a sustained shift in the process mean. Based on these definitions and assumptions, a decision rule is derived for choosing either the TS chart or CUSUM FIR chart that minimizes costs or maximizes gain.

First, consider opportunity costs or lost profits associated with false alarms. We assume opportunity costs equal to lost profits from a cease in production due to an inappropriate process shutdown are the only costs to be considered from a false alarm. The lost profit from a false alarm is assumed to be a constant per period of shutdown independent of whether the TS or CUSUM FIR chart is applied. Further, for a given shift δ, the average number of time periods lost from a single false alarm is assumed constant and defined as $d_{\mathrm{avg}}$, independent of whether the TS or CUSUM FIR chart is applied. Let λ be the average number of items produced per time period, p the profit per item produced from an IC process, c the cost to replace a produced item from a shifted process, $n_{\mathrm{TS}}$ the average number of false alarms before the first correct detection alarm for a selected IC ARL from the TS chart and $n_{\mathrm{FIR}}(k)$ the average number of false alarms before the first correct detection alarm for a selected IC ARL from the CUSUM FIR chart. The average number of false alarms from the CUSUM FIR chart is a function of k because control limits for the CUSUM FIR chart are based on k, whereas this is not true for the TS chart. Thus,

$$(n_{\mathrm{FIR}}(k)d_{\mathrm{avg}}-n_{\mathrm{TS}}{d}_{\mathrm{avg}})\lambda p$$

is the difference (CUSUM FIR minus TS) in average lost profit from false alarms before correctly detecting a process shift. If this difference is negative, the CUSUM FIR chart is preferred to the TS chart based only on lost profits associated with false alarms. Likewise, if this difference is positive, the TS chart is preferred to the CUSUM FIR chart based only on lost profits associated with false alarms.

Second, consider saved replacement costs and lost profits based on the difference in average run lengths, CUSUM FIR minus TS, required to detect a shift δ in the process mean. Assuming the process mean shifted immediately after some time $t^{\mathrm{\prime }}$, we define the average number of time periods to the first detection of this shift as $r_{\mathrm{TS}}(\delta )$ and $r_{\mathrm{FIR}}(\delta ,k)$ for the TS and CUSUM FIR charts and a selected IC (ARL). Further, we assume a worst-case scenario where all items produced after time $t^{\mathrm{\prime }}$ until the shift is detected are scrapped, i.e. no sorting out good items that are within specifications from those outside specifications. The reader should understand control limits are different from specification limits. The argument k is used for the CUSUM FIR chart average run length because control limits for the CUSUM FIR chart are dependent on k. Thus,

$$(r_{\mathrm{FIR}}(\delta ,k)-r_{\mathrm{TS}}(\delta ))\lambda p$$

is the difference (CUSUM FIR minus TS) in average profit lost from a CUSUM FIR chart compared to the use of the TS chart due to a shift in the process mean. Similarly,

$$(r_{\mathrm{FIR}}(\delta ,k)-r_{\mathrm{TS}}(\delta ))\lambda c$$

is the difference (CUSUM FIR minus TS) in average replacement cost due to the difference in ARL to detect a shift in the process mean from a CUSUM FIR chart compared to the use of the TS chart. Note, if the average run length to first detection of a δ process mean shift is greater for the CUSUM FIR than the TS chart for a selected IC ARL, i.e. $r_{\mathrm{FIR}}(\delta ,k)-r_{\mathrm{TS}}(\delta )>0$, both expressions above are greater than 0. Using this assumption and ignoring lost profit due to false alarms, the TS chart would be the choice that minimizes the average profit loss plus average replacement costs.

For some selected IC ARL and process mean shift δ, immediately following some time $t^{\mathrm{\prime }}$, the following expression combines the differences (CUSUM FIR minus TS) in lost profit due to false alarms with lost profit and replacement costs due to excess average run length following a process mean shift δ,

$$(n_{\mathrm{FIR}}(k)d_{\mathrm{avg}}-n_{\mathrm{TS}}{d}_{\mathrm{avg}})\lambda p+(r_{\mathrm{FIR}}(\delta ,k)-r_{\mathrm{TS}}(\delta ))\lambda p+(r_{\mathrm{FIR}}(\delta ,k)-r_{\mathrm{TS}}(\delta ))\lambda c.$$

If this sum exceeds 0, the TS chart should be chosen. Likewise, if this sum is below 0, the CUSUM FIR chart should be chosen. Letting $\rho =c/p$ and simplifying, if

$$(r_{\mathrm{FIR}}(\delta ,k)-r_{\mathrm{TS}}(\delta ))>(n_{\mathrm{TS}}-n_{\mathrm{FIR}}(k))\cdot d_{\mathrm{avg}}/(1+\rho ),$$ (17)

the TS chart should be chosen. Otherwise, choose the CUSUM FIR.

Assuming a steady-state process of size 0, inequality (17) becomes

$$(r_{\mathrm{FIR}}(\delta ,k)-r_{\mathrm{TS}}(\delta ))>0$$ (18)

because $n_{\mathrm{TS}}$ and $n_{\mathrm{FIR}}(k)$ are 0, i.e. there are no false alarms. With no false alarms, the TS chart is chosen if inequality (18) is true, otherwise choose CUSUM FIR.

3.2. Decision rule application

Assuming steady-state process of size 0, the decision rule for choosing TS or CUSUM FIR chart reduces to the validity of inequality (18) for selected IC ARL and $(\delta ,k)$ pair. Assuming an IC ARL 370, ARL differences in Table 3 should be compared to 0. Assuming IC ARL levels 100 and 750, ARL differences in Tables III(2) and III(32) in the supplement should be compared to 0. If inequality (18) is satisfied, choose TS, otherwise choose the CUSUM FIR. These comparisons identify the TS chart is preferred to the CUSUM FIR chart for each of the three IC ARL levels, all $(\delta ,k)$ considered and steady-state size 0. Assuming a process with steady-state size 1 and some IC ARL level, the decision rule for choosing TS or CUSUM FIR chart reduces to whether inequality (17) is valid for selected $(\delta ,k)$ and $(\rho ,d_{\mathrm{avg}})$ pairs. Selecting an IC ARL 370 level, evaluation of the validity of inequality (17) becomes a comparison of the content of Table 6 to the values in Table 7. This comparison implies for processes of steady-state size 1; the TS chart is preferred to the CUSUM FIR chart for all $(\delta ,k)$ and $(\rho ,d_{\mathrm{avg}})$ considered. Assuming an IC ARL 100 or 750 and processes with steady-state size 1, comparing the content of Tables III(6) and III(36) in the supplement to the values in Table 7 implies the TS chart is preferred to the CUSUM FIR chart for all $(\delta ,k)$ and $(\rho ,d_{\mathrm{avg}})$ considered. Turning now to processes with steady-state sizes 3 and 5, comparing the content of Tables III(10) and III(14) in the supplement to values in Table 7 determines when the TS chart is preferred to the CUSUM FIR chart for IC ARL 100. Comparing content in Tables III(25) and III(29) in the supplement to values in Table 7 determines when the TS chart is preferred to the CUSUM FIR chart for IC ARL 370. Comparing the contents of Tables III(40) and III(44) in the supplement to values in Table 7 determines when the TS chart is preferred to the CUSUM FIR chart for IC ARL 750. Table 8 summarizes all TS/CUSUM FIR chart performance comparisons considered. Details follow next.

Table 7.

Tabulated values for $d_{\mathrm{avg}}/(1+\rho )$.

$\rho \mathrm{\setminus }{d}_{\mathrm{avg}}$	0.5	1.0	1.5	2.0	2.5	3.0	3.5	4.0	4.5	5.0
0.1	0.4545	0.9091	1.3636	1.8182	2.2727	2.7273	3.1818	3.6364	4.0909	4.5455
0.5	0.3333	0.6667	1.0000	1.3333	1.6667	2.0000	2.3333	2.6667	3.0000	3.3333
2.0	0.1667	0.3333	0.5000	0.6667	0.8333	1.0000	1.1667	1.3333	1.5000	1.6667
4.0	0.1000	0.2000	0.3000	0.4000	0.5000	0.6000	0.7000	0.8000	0.9000	1.0000
5.0	0.0833	0.1667	0.2500	0.3333	0.4167	0.5000	0.5833	0.6667	0.7500	0.8333
6.0	0.0714	0.1429	0.2143	0.2857	0.3571	0.4286	0.5000	0.5714	0.6429	0.7143
8.0	0.0556	0.1111	0.1667	0.2222	0.2778	0.3333	0.3889	0.4444	0.5000	0.5556
9.0	0.0500	0.1000	0.1500	0.2000	0.2500	0.3000	0.3500	0.4000	0.4500	0.5000
10.0	0.0455	0.0909	0.1364	0.1818	0.2273	0.2727	0.3182	0.3636	0.4091	0.4545

Table 8.

The number of cases out of $49\times 90=4410$ $(\delta ,k)$ & $(\rho ,d_{\mathrm{avg}})$ pairs where the TS is preferred to CUSUM FIR chart based on decision rule inequalities (17) and (18).

		Steady-state size
		0	1	3	5
IC ARL	100	4410/4410	4410/4410	4193/4410	3833/4410
	370	4410/4410	4410/4410	4392/4410	4223/4410
	750	4410/4410	4410/4410	4410/4410	4341/4410

Consider first a comparison of IC ARL 100 TS and CUSUM FIR chart performances for processes with steady-state size 3. For 26 of the 49 $(\delta ,k)$ combinations, the TS chart was better than the CUSUM FIR chart for all 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7. These 26 $(\delta ,k)$ combinations correspond to ( $\delta \le 1.0$, all k) and ( $\delta =1.5$, $k=0.10,0.75,1.0,1.25,1.5$) combinations. The worst performance of the TS chart compared to the CUSUM FIR chart corresponds to the $(\delta =3.0,k=1.25,1.5)$ pairs where the TS chart is preferred to the CUSUM FIR chart for 66 of the 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7 for each k = 1.25 and 1.50. The second comparison of IC ARL 100 TS and CUSUM FIR chart performances is for processes with steady-state size 5. For 18 of the 49 $(\delta ,k)$ combinations, the TS chart was better than the CUSUM FIR chart for the 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7. These 18 $(\delta ,k)$ combinations correspond to ( $\delta \le 0.5$, all k) and $(\delta =1.0$, $k\ge 0.75$) combinations. The worst performance of the TS chart compared to the CUSUM FIR chart corresponds to the $(\delta =3.0,k=1.25)$ and $(\delta =3.0,k=1.5)$ combinations. The TS chart is preferred to the CUSUM FIR chart for 37 of the 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7 for each of the two $(\delta ,k)$ combinations. The third comparison is for IC ARL 370 and processes with steady-state size 3. For 44 of the 49 $(\delta ,k)$ combinations, the TS chart was better than the CUSUM FIR chart for the 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7. These 44 $(\delta ,k)$ combinations correspond to ( $\delta \le 2.5$, all k) and $(\delta =3.0,k\ge 0.5)$ combinations. The worst performance of the TS chart relative to the CUSUM FIR chart corresponds to the $(\delta =3.0,k=1.5)$ pair where the TS chart is preferred to the CUSUM FIR chart for 82 of the 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7. The fourth comparison is for IC ARL 370 and processes with steady-state size 5. For 28 of the 49 $(\delta ,k)$ combinations, the TS chart was better than the CUSUM FIR chart for all 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7. These 28 $(\delta ,k)$ combinations correspond to ( $\delta \le 1.0$, all k), $(\delta =1.5,k\ge 0.5)$ and $(\delta =2.0,k=0.10)$ combinations. The worst performance of the TS chart relative to the CUSUM FIR chart corresponds to the $(\delta =3.0,k=1.5)$ pair where the TS chart is preferred to the CUSUM FIR chart for 63 of the 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7. The fifth comparison is for IC ARL 750 and processes with steady-state size 3. For 49 of the 49 $(\delta ,k)$ combinations, the TS chart was better than the CUSUM FIR chart for the 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7. The sixth comparison is for IC ARL 750 and processes with steady-state size 5. For 37 of the 49 $(\delta ,k)$ combinations, the TS chart was better than the CUSUM FIR chart for the 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7. These 37 $(\delta ,k)$ combinations correspond to ( $\delta \le 2.0$, all k), $(\delta =2.5,k=0.10)$ and $(\delta =3.0,k=0.10)$ combinations. The worst performance of the TS chart relative to the CUSUM FIR chart corresponds to the $(\delta =3.0,k=1.5)$ pair where the TS chart is preferred to the CUSUM FIR chart for 75 of the 90 $(\rho ,d_{\mathrm{avg}})$ pairs in Table 7. Finally, Table 8 gives an overall performance summary of TS vs. CUSUM FIR charts.

If Table 7 does not contain the $(\rho ,d_{\mathrm{avg}})$ combination of interest, an analyst can easily compute $d_{\mathrm{avg}}/(1+\rho )$. Based on an IC ARL of interest (100, 370, 750) and using the tables presented in this paper, the analyst can decide whether to apply the TS or CUSUM FIR for processes with steady-state sizes (0, 1, 3 and 5).

3.3. Example

Two sets of data were proposed and analyzed by Lucas and Crosier [18] with the CUSUM and CUSUM FIR chart. Assuming an IC ARL 370, k = 0.5, and control limits ±4.86 for the CUSUM FIR and ±4.77 for the CUSUM charts, Figure 4 compares the application of the proposed two-sided TS chart, CUSUM and CUSUM FIR charts for the IC data presented by Lucas and Crosier [18]. Control limits for CUSUM and CUSUM FIR are found in supplemental Section II, Table II(3). Table 9 presents results from the application of the two-sided TS, CUSUM FIR and CUSUM charts to the shifted data presented by Lucas and Crosier [18]. TS and CUSUM FIR charts detected the shift at the same time index 3, i.e. the detection run lengths are equal. The CUSUM chart detected this same shift at time index 6. Assuming IC ARL levels 100 and 750, the two-sided TS chart detected the shifted process one period earlier than the two-sided CUSUM FIR chart. Comparing two-sided TS, CUSUM and CUSUM FIR chart performance for IC data from Lucas and Crosier [18] is only relevant in the sense of comparing the number of false alarms (restarts) within the 19 time periods. Figure 4 shows two false alarms for the TS chart, two false alarms for the CUSUM FIR chart and one false alarm for the CUSUM chart. Although of no real interest, false alarms for the TS chart are at time indices 16 and 18, for the CUSUM FIR chart at 16 and 19 and for the CUSUM chart at 16. These false alarms occur at somewhat different time periods than presented by Lucas and Crosier [18] because when an alarm occurred, we restarted the chart and Lucas and Crosier [18] did not. This restart feature does not apply to what is presented in Table 9 because the alarms were not false.

Figure 4.

Signal processes and restart time indices for TS, CUSUM FIR and CUSUM charts, IC data.

Table 9.

TS, CUSUM FIR and CUSUM signal processes, shifted data from the beginning.

			CUSUM FIR		CUSUM
i	$x_i$	${\mathrm{TS}}_i$	$L_i$	$U_i$	$L_i$	$U_i$
1	0.8	0.80	−1.09	2.69	0	0.30
2	1.9	2.05	0	4.09	0	1.70
3	1.4	2.64	0	4.99	0	2.60
4	2.0	–	–	–	0	4.10
5	1.1	–	–	–	0	4.70
6	0.7	–	–	–	0	4.90
7	2.6	–	–	–	–	–

4. Conclusion and discussion

Simulations of initial state processes with the application of an optimally designed CUSUM FIR and TS charts accurately match the scenario where an analyst knows a change in the mean has occurred at or before process monitoring begins. The observed ARL differences ( $\text{CUSUM FIR}$ – $\mathrm{TS}$) obtained from these simulations produced overwhelming evidence in support of identifying our TS chart as uniformly superior to the CUSUM FIR chart using IC ARL 100, 370 and 750 for all $(\delta ,k)$ considered. Similarly, for a steady-state process of size 1, the decision rule showed the TS chart to be superior to the CUSUM FIR chart for all $(\rho ,d_{\mathrm{avg}})$ and $(\delta ,k)$ combinations considered for IC ARL 100, 370 and 750.

For a given IC ARL and steady-state size 3 or 5, the range of threshold values $d_{\mathrm{avg}}/(1+\rho )$ guaranteeing superiority of the TS chart, i.e. satisfying inequality (17), shrinks for larger δ and k. However, the worst performance of the TS chart compared to the CUSUM FIR chart was for IC ARL 100, steady-state size 5, $\delta =3.0$ and k = 1.5. In this setting, the TS chart was superior to CUSUM FIR for 37 of the 90 $(\rho ,d_{\mathrm{avg}})$ pairs. Further, for IC ARL level 100, 370 and 750, as steady-state size increases from 3 to 5, the proportion of the 4410 scenarios considered where the TS chart is advantageous decreases slightly. For steady-state sizes 3 or 5, the TS chart is judged superior to the CUSUM FIR chart for an increasing proportion of the 4410 possibilities considered as IC ARL level increases from 100 to 370 to 750. In fact, for IC ARL 750, steady-state size 3, the TS chart is judged superior to the CUSUM FIR chart for 90 of the 90 $(\rho ,d_{\mathrm{avg}})$ combinations for all $(\delta ,k)$ pairs considered. See Table 8. For a selected $d_{\mathrm{avg}}$ in Table 7, as the profit per item (p) becomes a smaller portion of cost per item (c), say $\rho =10$, most if not all $(\delta ,k)$ pairs dictate selecting the TS chart. Likewise, if the profit per item is a larger proportion of cost per item, say $\rho =0.10$, the likelihood slightly increases that more $(\delta ,k)$ pairs will indicate the choice of CUSUM FIR.

We applied and compared the detection performance of the TS chart to the CUSUM and CUSUM FIR charts using two datasets presented by Lucas and Crosier [18]. The TS chart was the clear choice for quick detection of a shift in the mean for IC 100 and 750. For IC 370, the TS and CUSUM FIR charts produced equivalent detection run lengths for the initial state data with a shift in the mean. Application of IC ARL 370 TS and CUSUM FIR charts to the set of IC data produced an equal number of false alarms. Lucas and Crosier [18] originally presented the analysis of the IC data to demonstrate in an IC scenario, CUSUM and CUSUM FIR performed essentially the same after the early time periods pass. Asymptotic theory for TS signals in the IC situation was developed. Using this theory and exploiting the Fokker–Planck equation associated with the underlying Markovian structure, we developed a mathematically sound method for determining the TS chart control limit for an IC ARL of interest. In addition to being asymptotically consistent, the technique is conservative, assuring actual TS chart performance is better than reported.

Additional work can be pursued such as the one-sided TS chart, determining TS chart performance for different α in Equation (12) and developing a decision rule to compare the typical CUSUM and TS chart that includes both false alarm rate and average run length to the detection of a process shift in the mean initiated at some time $t^{\mathrm{\prime }}$ for a multitude of $t^{\mathrm{\prime }}$'s on a time horizon at least as big as the IC ARL level. A thorough comparison of modified EWMA charts [14] proposed to our TS chart for process steady-state sizes 1, 3 and 5 that account for false alarms would be worth exploring. Another important and promising direction is the multivariate extension of our methodology. While no algorithms for computing multivariate taut strings currently exist, some heuristic approximations are readily available [8]. With ${\hat{\mathit{\mu }}}_n(t)$ denoting the multivariate (approximate) taut string estimate, the multivariate version of the signal statistic from Equation (12) reads as

$${\mathrm{TS}}_n=n^{\alpha }\Vert {\mathbf{\Sigma }}_0^{-1/2}({\hat{\mathit{\mu }}}_n(t)-{\mathit{\mu }}_0){\Vert }_{{\mathrm{BV}}_0([0,1])}\text{for some}\alpha \in [0,1],$$

where ${\mathit{\mu }}_0$ and ${\mathbf{\Sigma }}_0$ denote the multivariate standards. Additional extensive work would be needed to establish asymptotic consistency akin to Appendix 2 as well as the average exit time computation for the limiting Markovian diffusion.

Appendices.

Appendix 1. Algorithm for computing $\widehat{\mathrm{IC}\mathrm{ARL}}$ and $s_{\mathrm{IC}\mathrm{RL}}^2$

Input:	Control limit L>0
Output:	Estimates $\widehat{\mathrm{IC}\mathrm{ARL}}$ and $s_{\mathrm{IC}\mathrm{RL}}^2$ of IC ARL and ${\sigma }_{\mathrm{IC}\mathrm{RL}}^2$
Tuning Constants:	$\alpha =\frac{3}{5}$, $N_{\mathrm{MC}}=\mathrm{1,000,000}$, $n_0=\mathrm{10,000}$, $m_0=1$

1. Generate $N_{\mathrm{MC}}$ data streams of $n_0$ i.i.d. standard normal univariates $x_1,\dots$, $x_{n_0}$.

2. For the kth data stream and $j=1,\dots ,n_0$, construct the taut string estimator ${\hat{\mu }}_j(t)$ for the dataset $(x_1,\dots ,x_j)$ and compute ${\mathrm{TS}}_j$ according to Equation (12).

3. Assign all streams with ${\mathrm{TS}}_j\ge L$ for some $j<n_0$ to $D_{n_0,L}$ and the remaining streams to $D_{n_0,L}^c$.

4. For the kth data stream in $D_{n_0,L}$, $k=1,\dots ,N_1$, record

   $${\mathrm{RL}}_k=j^{\ast }\mathrm{and}{\mathrm{RL}}_k^2=(j^{\ast }{)}^2\text{ with }{j}^{\ast }=min\{j\in \mathbb{N}|{\mathrm{TS}}_j>L\}.$$
5. Let ‘Cnd. Exp.’ abbreviate ‘conditional expectation’ (given the TS chart has not stopped at $n_0$ or before and the signal value at $n_0$ is ${\mathrm{TS}}_{n_0}$). For the kth data stream in $D_{n_0,L}^c$, $k=1,\dots ,N_2$,

   $$\text{Cnd. Exp. }{\mathrm{RL}}_k^p=n_0^p+m_0^p{V}_p(\frac{n_0}{m_0},m_0^{1/2-\alpha }{\mathrm{TS}}_{n_0};m_0^{1/2-\alpha }L)\mathrm{for}p=1,2.$$

   Functions $V_1(\cdot ,\cdot ;\cdot )$, $V_2(\cdot ,\cdot ;\cdot )$ are found in Equation (IV.6) in supplemental Section IV.(i).
6. Noting $N=N_1+N_2$, where $N_1$ and $N_2$ are the sizes of $D_{n_0,L}$ and $D_{n_0,L}^c$,

   $$\begin{array}{rl}\widehat{\text{IC ARL}}& =\frac{1}{N_{\mathrm{MC}}}\sum _{k=1}^{N_1}{\mathrm{RL}}_k+\frac{1}{N_{\mathrm{MC}}}\sum _{k=1}^{N_2}\text{ Cnd. Exp. }{\mathrm{RL}}_k,\\ s_{\mathrm{IC}\mathrm{RL}}^2& =\frac{1}{N_{\mathrm{MC}}-1}(\sum _{k=1}^{N_1}{\mathrm{RL}}_k^2+\sum _{k=1}^{N_2}\text{ Cnd. Exp. }{\mathrm{RL}}_k^2)-\frac{N_{\mathrm{MC}}\cdot (\widehat{\text{IC ARL}}{)}^2}{N_{\mathrm{MC}}-1}.\end{array}$$

Appendix 2. Asymptotics of the taut string estimator

A seminal property that distinguishes the taut string estimator and makes it appropriate for designing the TS chart is that the taut string estimator is asymptotically consistent in the total variation norm defined in Equation (11) in the in-control situation, whereas in the out-of-control situation, without necessarily being consistent, the estimate obtained with the taut string estimator is asymptotically distinguishable from the standards given mean ${\mu }_0$. Hence, the TS estimator ${\hat{\mu }}_n(t)$ can effectively be used to decide if the conditional mean $\mu (t)$ departs from a given standard ${\mu }_0$. This is proved in Theorems A.1 and A.2.

Asymptotic consistency of the taut string estimator in the uniform topology has earlier been studied by Davies and Kovac [5] under $C^2$-smoothness assumptions on $\mu (t)$, etc. Among other things, Kim et al. [13] established convergence in a dual Sobolev topology. We also refer the reader to a recent result [4, Section 7.3.1] studying an alternative version of the taut string estimator. Neither approach is applicable to describe the TS signal asymptotic behavior which motivated the theory presented below.

Theorem A.1 proved in supplemental Section IV(iii.i) is essential for deriving the run length tail behavior in the in-control situation. Namely, combining Theorem A.1 with the results of supplemental Sections IV(i) and IV(ii), the TS chart run length tail distribution can be approximated by the exit time distribution of a non-autonomous Itô process from Equation (IV.4) in supplemental Section IV(ii). As before, we assume ${\epsilon }_t$'s are i.i.d. random variables with $\mathbf{E}[{\epsilon }_t]=0$, $\mathbf{V}\mathbf{a}\mathbf{r}[{\epsilon }_t]=1$.

Theorem A.1

Let the functions $\mu (t)$ and $\sigma (t)$ in Equation (4) satisfy $\mu (t)\equiv {\mu }_0$ and $\sigma (t)\equiv {\sigma }_0$ for every $t\in [0,1]$ and some ${\mu }_0\in \mathbb{R}$, ${\sigma }_0>0$. Further, let ${\hat{\mu }}_n(t)$ denote the taut string estimator based on the tube radius

$${\lambda }_n={\sigma }_0(\sqrt{2}/2+\delta )\sqrt{2\log (\log (n))}{n}^{-1/2}forn\gg 1$$ (A1)

for an arbitrary $\delta >0$ (e.g. $\delta =(0.8-\sqrt{2}/2)>0$, cf. Equation (9)). Then:

1. $\mathbf{P}(\underset{n\to \mathrm{\infty }}{lim inf}\{{\hat{\mu }}_n(t)\equiv {\mu }_0+{\sigma }_0{n}^{-1}\sum _{k=1}^n{\epsilon }_{\frac{k}{n}}\})=1$,

2. $\underset{n_0\to \mathrm{\infty }}{lim}\mathbf{P}(\{{\hat{\mu }}_n(t)\equiv {\mu }_0+{\sigma }_0{n}^{-1}\sum _{k=1}^n{\epsilon }_{\frac{k}{n}}\mathrm{for}n\ge n_0\})=1$,

3. $(n^{1/2}/{\sigma }_0)\Vert {\hat{\mu }}_n-{\mu }_0{\Vert }_{{\mathrm{BV}}_0([0,1])}\stackrel{d}{\to }|\mathcal{N}(0,1)|$ as $n\to \mathrm{\infty }$.

Turning to the out-of-control situation, Theorem A.2 proved in supplementary Section IV(iii.ii) guarantees the TS estimator does not converge to the standards given mean ${\mu }_0$. Precisely, the further the conditional mean $\mu (t)$ departs from ${\mu }_0$, the larger the TS signals grow and the shorter the run lengths of the TS chart become boosting detection power of our chart. This theoretical observation is consistent with simulation results in Section 3.

Theorem A.2

Let the function $\mu :[0,1]\to \mathbb{R}$ in Equation (4) be piecewise Hölder-continuous. With ${\mathrm{BV}}_0^{\ast }([0,1])$ denoting the topological dual of ${\mathrm{BV}}_0([0,1])$, we have

$$\Vert {\hat{\mu }}_n-{\mu }_0{\Vert }_{{\mathrm{BV}}_0([0,1])}\ge \Vert \mu -{\mu }_0{\Vert }_{{\mathrm{BV}}_0^{\ast }([0,1])}+o_{\mathbf{P}}(1)asn\to \mathrm{\infty },$$

i.e. ${\hat{\mu }}_n(t)$ stays away from ${\mu }_0$ in ${\mathrm{BV}}_0([0,1])$ unless $\mu (t)\equiv {\mu }_0$.

Funding Statement

This work has been funded by a research grant from Zukunftskolleg (University of Konstanz, Germany). The second author has also been partially supported by the Farmer School of Business (Miami University) and thanks Professor Roland Schnaubelt (Karlsruhe Institute of Technology, Germany) for the financial support and hospitality during his stay in Karlsruhe.

Data availability statement

All data presented in the article are available in the Supplement.

Disclosure statement

No potential conflict of interest was reported by the author(s).