As will be shown in detail in

Section 4, the actual applicability of the SRC is often hampered due to its virtually unacceptable performance in certain scenarios. More specifically, the SRC suffers from a lack of agility, i.e., given a distributional shift actually occurred the SRC may require an undue amount of time to signal (i.e., it exhibits a large detection delay); as will be shown, this is especially pronounced if a change occurs soon after monitoring commenced. In such case, the amount of data gathered by the SRC may be grossly insufficient, thus resulting in a prolonged time to signal. It should be noted that, as McDonald correctly points out (see [

18], pg. 628–629), the above mentioned lack of agility (compared to an optimal parametric approach) and the poor detection of changes occurring after a relatively small number of observations is to some degree inherent to all nonparametric procedures.

The idea behind the adaptive control limit SRC (AC-SRC) proposed by this author [

16,

17] is to mitigate the SRC’s drawbacks while maintaining its ease-of-use, robustness, and the ability to obtain generally valid control limits ahead of time. This is facilitated by the AC-SRC being inspired by and incorporating large parts of a distribution-free bootstrap based CUSUM proposed by Chatterjee and Qiu [

19]. Said authors in 2009 proposed an elegant procedure where the conventional fixed control limit is swapped for a sequence of control limits obtained from the conditional distribution of the test statistic (i.e., the CUSUM) given the last time it was zero. Chatterjee and Qiu estimate these conditional distributions by means of bootstrapping; note that among other things this implies the need of a large amount of representative training data as well as a high computational burden. However, transferring the key idea of the approach by Chatterjee and Qiu to the SRC results in the AC-SRC described and analyzed in the following.

Akin to the SRC described in

Section 2.2 let

${R}_{n}$ and

${C}_{{\mathrm{AC-SRC}}_{n}}$ denote the sequential rank of

${x}_{n}$ and the respective SRC as provided by Equations (

4) and (

5), respectively. Furthermore, let

${Y}_{{\mathrm{AC-SRC}}_{j}}$ be a random variable following the conditional distribution

where

${T}_{{\mathrm{AC-SRC}}_{n}}$, also referred to as

sprint length, denotes the time elapsed since

${C}_{{\mathrm{AC-SRC}}_{n}}$ was last zero, i.e.,

Central to the method by Chatterjee and Qiu is the fact that the conditional distributions in Equation (

6) depend only on

j and

F but not on

n [

19]. Then for any positive integer

${j}_{\mathrm{max}}\phantom{\rule{3.33333pt}{0ex}}\le \phantom{\rule{3.33333pt}{0ex}}n$ the (unconditional) distribution of

${C}_{{\mathrm{AC-SRC}}_{n}}$ can be approximated by means of the conditional distributions in Equation (

6) as

with

I being the common indicator function and

${Y}^{*}\sim \left[{C}_{{\mathrm{AC-SRC}}_{n}}|{T}_{{\mathrm{AC-SRC}}_{n}}>{j}_{\mathrm{max}}\right].$ Since the AC-SRC is based on sequential ranks which, given the process is in control, are independent and discrete uniform on

$\left\{\frac{1}{n+1},\frac{2}{n+1},\cdots ,\frac{n}{n+1}\right\}$ (see

Section 2.2) the sequence of control limits

$\left\{{h}_{j}\right\}$ can be determined (ahead of time) without the need for training data by means of Monte Carlo simulations as outlined in Algorithm 1.

The AC-SRC then signals if

${T}_{{\mathrm{AC-SRC}}_{n}}=j$ and

${C}_{{\mathrm{AC-SRC}}_{n}}>{h}_{j}$ for

$1\le j\le {j}_{\mathrm{max}}$ or if

${T}_{{\mathrm{AC-SRC}}_{n}}>{j}_{\mathrm{max}}$ and

${C}_{{\mathrm{AC-SRC}}_{n}}>{h}_{{j}_{\mathrm{max}}}.$ Note that following recommendations by Chatterjee and Qiu the

${h}_{j}$ are only calculated up to a reasonably small

${j}_{\mathrm{max}}$ after which, if the test statistic does not bounce back to zero, they are kept fixed at

${h}_{{j}_{\mathrm{max}}}$. Furthermore,

${k}_{\mathrm{AC-SRC}}$ is linked to

${j}_{\mathrm{max}}$ such that a desired sprint length

$tE{T}_{n}$, which is set to be proportional to

${j}_{\mathrm{max}}$ (see

Section 3.1), e.g., as

$tE{T}_{n}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\lfloor \frac{3{j}_{\mathrm{max}}}{4}\rfloor $, is approximately attained by the average sprint length. That is,

${k}_{\mathrm{AC-SRC}}:\phantom{\rule{3.33333pt}{0ex}}{\overline{T}}_{{\mathrm{AC-SRC}}_{n}}\approx tE{T}_{n}=\lfloor \frac{3{j}_{\mathrm{max}}}{4}\rfloor $.

#### 3.1. Remarks on and Suggestions for the Selection of AC-SRC Parameters

The aim of this Section is twofold: first, to complete the description of the proposed procedure by justifying some seemingly completely arbitrary design choices of

Section 3 , and second, to provide guidance to practitioners in order to facilitate the applicability of our method.

Average in-control and out-of-control run lengths, which are commonly referred to as

$AR{L}_{0}$ and

$AR{L}_{1}$ in the SPC literature, play a crucial role in the design and use of control charts. The

$AR{L}_{0}$ characterizes the chart’s propensity to false alarms in terms of the average number of samples they are separated by, whereas

$AR{L}_{1}$ describes what we referred to as the control chart’s agility, i.e., the average delay between the occurrence of an actual change and its detection. Clearly, then, there exists an inherent trade off between the objectives of low false alarm rates (large

$AR{L}_{0}$) and small detection delays (small

$AR{L}_{1}$). In this paper, we assume that the common approach of choosing an acceptable

$AR{L}_{0}$ followed by attempts to minimize

$AR{L}_{1}$ is pursued. The suitability of an

$AR{L}_{0}$ highly depends on the particular problem at hand and is influenced, among other things, by crucial aspects such as weighting the aforementioned conflicting objectives to ensure compliance with requirements as well as detailed knowledge of the specific application. Accordingly, we find further discussions pertaining

$AR{L}_{0}$ to be beyond the scope of this paper and, again, refer the interested reader to selected representatives of the established SPC literature [

1,

2,

3].

Recall that, given a fixed and pre-determined

${j}_{\mathrm{max}}$, Algorithm 1 starts out by calibrating

${k}_{\mathrm{AC-SRC}}$ such that the average sprint length

${\overline{T}}_{{\mathrm{AC-SRC}}_{n}}$ equals the desired sprint length

$tE{T}_{n}$ within a reasonable margin of error. Following Chatterjee and Qiu [

19] ,we fix the desired sprint length

$tE{T}_{n}$ as a in theory arbitrary ratio of

${j}_{\mathrm{max}}$; throughout this work

$tE{T}_{n}=\lfloor \frac{3{j}_{\mathrm{max}}}{4}\rfloor $ is used. Note that, although the rationale for linking

${k}_{\mathrm{AC-SRC}}$ and

${j}_{\mathrm{max}}$ is compelling, doing so is not required.

The behavior of the AC-SRC’s test statistic

${C}_{{\mathrm{AC-SRC}}_{n}}$ is crucially influenced by the specific choice of

${k}_{\mathrm{AC-SRC}}$ in that the propensity of

${C}_{{\mathrm{AC-SRC}}_{n}}$ bouncing back to zero decreases for smaller

${k}_{\mathrm{AC-SRC}}$ (and vice versa for larger values of the reference constant). In other words, the average sprint length

${\overline{T}}_{{\mathrm{AC-SRC}}_{n}}$ increases with a reduction of

${k}_{\mathrm{AC-SRC}}$, whereas increasing the reference value results in smaller sprint lengths. Furthermore, the sensible constraint of choosing

$tE{T}_{n}\ll {j}_{\mathrm{max}}$ restricts the computational burden of Algorithm 1 and reasonably ensures its algorithmic stability. In fact, in the absence of constraints on

${j}_{\mathrm{max}}$ and

${k}_{\mathrm{AC-SRC}}$, `inappropriate’ combinations such as, e.g., (very) large

${k}_{\mathrm{AC-SRC}}$ and

${j}_{\mathrm{max}}$ could easily result in the inability to evaluate Equation (

6) which, in turn, is required in Part II of Algorithm 1 . While we find the aforementioned to establish sufficient and convincing justification for the choice of calibrating

${k}_{\mathrm{AC-SRC}}$ such that

$\phantom{\rule{3.33333pt}{0ex}}{\overline{T}}_{{\mathrm{AC-SRC}}_{n}}\approx tE{T}_{n}=\lfloor \frac{3{j}_{\mathrm{max}}}{4}\rfloor $ holds, it is arbitrary in that other reasonable but not necessarily superior design choices are readily discernible (see [

19]).

As expected, and consistent with the considerations expressed by Chatterjee and Qiu [

19] pertaining to their bootstrap-based method, we observed diminishing returns with increasing the length

${j}_{\mathrm{max}}$ of the sequence of adaptive control limits

${\left\{{h}_{j}\right\}}_{j=1}^{{j}_{\mathrm{max}}}$.

While we are unable to provide specific guidelines pertaining to the selection of AC-SRC’s pertinent tuning parameters and further research in this area is required, we advocate the use of rather short sequences ${\left\{{h}_{j}\right\}}_{j=1}^{{j}_{\mathrm{max}}}$ with $6\le {j}_{\mathrm{max}}\ll 30$. Based on our current understanding and evidence, we recommend to set up the AC-SRC as discussed above and to adjust it to the requirements of the specific scenario by means of choosing either smaller or larger ${j}_{\mathrm{max}}$.

As will be corroborated by simulation results in

Section 4.2, a reasonably consistent degree of fine-tuning is attainable with smaller

${j}_{\mathrm{max}}$ allowing for good agility, whereas using slightly larger values for

${j}_{\mathrm{max}}$ yields improved robustness at the expense of an increased detection delay.