**2. The setup of well-known nonparametric control charts based on order statistics**

Let us assume that a reference random sample of size *m*, say *X*1,*X*2, … ,*Xm*, is drawn independently from an unknown continuous distribution *F*, namely, when the process is in-control. The control limits of the distribution-free monitoring scheme are determined by exploiting specific order statistics of the reference sample. In the sequel, test samples are drawn independently of each other (and also of the reference sample) from a continuous distribution *G*, and the decision whether the process is still in-control or not rests on suitably chosen test statistics. The framework for constructing nonparametric control charts based on order statistics calls for the following step-by-step procedure.

**Step 1.** Draw a reference sample of size *m*, namely, *X*1,*X*2, … ,*Xm*, from the process when it is known to be in-control.

**Step 2.** Form an interval by choosing appropriately a pair of order statistics from the reference sample (say, e.g., ð Þ *Xa*,*Xb* , where 1≤ *a*< *b*≤ *m*).

**Step 3.** Draw independently future (test) samples of size *n*, namely,

*Y*1, *Y*2, … , *Yn*, from the underlying process.

**Step 4.** Pick out *l* order statistics (0 <*l* ≤*n*) from each test sample.

**Step 5.** Determine the number of observations of each test sample, say *R* that lie between the limits of the interval ð Þ *Xa*, *Xb* .

**Step 6.** Configure the signaling rule by utilizing both the statistics *R* and the *l* ordered test sample observations as monitoring statistics.

The implementation of the above mechanism does not require the assumption of any specific probability distribution for the underlying process (measurements). The reference sample (usually of large size) is drawn from the underlying in-control process, while test (Phase II) samples are picked out from the future process in order to decide whether the process remains in-control or it has shifted to an out-ofcontrol state. The proposed monitoring scheme is likely to possess the robustness feature of standard nonparametric procedures and is, consequently, less likely to be affected by outliers or the presence of skewed or heavy-tailed distributions for the underlying populations.

It is straightforward that the proposed framework requires the construction of more than one control charts, which monitor simultaneously the underlying process. In fact, the design parameter *l* is connected to the number of the control charts which are needed to be built for trading on the aforementioned mechanism. Indeed, for each one of the *l* order statistics from the test sample, a separate two-sided control chart should be constructed.

The family of distribution-free monitoring schemes presented earlier includes as special cases some nonparametric control charts, which have been already established in the literature. For example, the monitoring scheme established by Balakrishnan et al. [5] calls for the following plotted statistics:

*Simulation-Based Comparative Analysis of Nonparametric Control Charts with Runs-Type Rules DOI: http://dx.doi.org/10.5772/intechopen.91040*


It goes without saying that the control chart introduced by Balakrishnan et al. [5] (*Chart* 1, hereafter) belongs to the family of monitoring schemes described previously. In fact, the *BTK* chart could be seen as a special case of the aforementioned class of distribution-free control schemes with *l* ¼ 1. According to *Chart* 1, the process is declared to be in-control, if the following conditions hold true

$$X\_{a:m} \le Y\_{j:n} \le X\_{b:m} \quad \text{and} \quad R \ge r,\tag{1}$$

where *r* is a positive integer.

In the present chapter, we study well-known distribution-free *Shewhart*-type monitoring schemes based on order statistics. The general setup of the control charts being in consideration is presented in Section 2, while their performance characteristics are investigated based on the algorithm described in Section 3. In order to enhance the ability of the proposed monitoring schemes for detecting possible shifts of the process distribution, some well-known runs-type rules are considered. In Section 4, we carry out extensive simulation-based numerical comparisons that reveal that the underlying control charts outperform the existing ones

*Quality Control - Intelligent Manufacturing, Robust Design and Charts*

**2. The setup of well-known nonparametric control charts based on**

Let us assume that a reference random sample of size *m*, say *X*1,*X*2, … ,*Xm*, is drawn independently from an unknown continuous distribution *F*, namely, when the process is in-control. The control limits of the distribution-free monitoring scheme are determined by exploiting specific order statistics of the reference sample. In the sequel, test samples are drawn independently of each other (and also of the reference sample) from a continuous distribution *G*, and the decision whether the process is still in-control or not rests on suitably chosen test statistics. The framework for constructing nonparametric control charts based on order statistics

**Step 1.** Draw a reference sample of size *m*, namely, *X*1,*X*2, … ,*Xm*, from the

**Step 3.** Draw independently future (test) samples of size *n*, namely,

**Step 4.** Pick out *l* order statistics (0 <*l* ≤*n*) from each test sample.

**Step 2.** Form an interval by choosing appropriately a pair of order statistics from

**Step 5.** Determine the number of observations of each test sample, say *R* that lie

The implementation of the above mechanism does not require the assumption of any specific probability distribution for the underlying process (measurements). The reference sample (usually of large size) is drawn from the underlying in-control process, while test (Phase II) samples are picked out from the future process in order to decide whether the process remains in-control or it has shifted to an out-ofcontrol state. The proposed monitoring scheme is likely to possess the robustness feature of standard nonparametric procedures and is, consequently, less likely to be affected by outliers or the presence of skewed or heavy-tailed distributions for the

It is straightforward that the proposed framework requires the construction of more than one control charts, which monitor simultaneously the underlying process. In fact, the design parameter *l* is connected to the number of the control charts which are needed to be built for trading on the aforementioned mechanism. Indeed, for each one of the *l* order statistics from the test sample, a separate two-sided

The family of distribution-free monitoring schemes presented earlier includes as

special cases some nonparametric control charts, which have been already established in the literature. For example, the monitoring scheme established by

Balakrishnan et al. [5] calls for the following plotted statistics:

**Step 6.** Configure the signaling rule by utilizing both the statistics *R* and the *l*

under several out-of-control scenarios.

calls for the following step-by-step procedure.

the reference sample (say, e.g., ð Þ *Xa*,*Xb* , where 1≤ *a*< *b*≤ *m*).

ordered test sample observations as monitoring statistics.

process when it is known to be in-control.

*Y*1, *Y*2, … , *Yn*, from the underlying process.

between the limits of the interval ð Þ *Xa*, *Xb* .

underlying populations.

**198**

control chart should be constructed.

**order statistics**

In addition, the monitoring scheme introduced by Triantafyllou [6] (*Chart* 2, hereafter) takes into account the location of two order statistics of the test sample drawn from the process along with the number of its observations between the control limits. In other words, the aforementioned control chart could be viewed as a member of the general class of nonparametric monitoring schemes with *l* ¼ 2. According to *Chart* 2, the process is declared to be in-control, if the following conditions hold true

$$X\_{a:m} \le Y\_{j:n} \le Y\_{k:n} \le X\_{b:m} \quad \text{and} \quad R \ge r,\tag{2}$$

where *r* is a positive integer.

In a slightly different framework, Triantafyllou [7] proposed a distribution-free control chart based on order statistics (*Chart* 3, hereafter) by taking advantage of the position of single ordered observations from both test and reference sample. More precisely, *Chart* 3 asks for an order statistic of each test sample (say *Y <sup>j</sup>*:*<sup>n</sup>*) to be enveloped by two prespecified observations *Xa*:*<sup>m</sup>* and *Xb*:*<sup>m</sup>* of the reference sample, while at the same time an ordered observation of the reference sample (say *Xi*:*<sup>m</sup>*) be enclosed by two predetermined values of the test sample ð Þ *Yc*:*<sup>n</sup>*, *Yd*:*<sup>n</sup>* . *Chart* 3 makes use of an in-control rule, which embraces the following three conditions:

**Condition 1.** The statistic *Y <sup>j</sup>*:*<sup>n</sup>* of the test sample should lie between the observations *Xa*:*<sup>m</sup>* and *Xb*:*<sup>m</sup>* of the reference sample, namely, *Xa*:*<sup>m</sup>* ≤ *Y <sup>j</sup>*:*<sup>n</sup>* ≤*Xb*:*<sup>m</sup>*.

**Condition 2.** The interval ð Þ *Yc*:*<sup>n</sup>*, *Yd*:*<sup>n</sup>* formulated by two appropriately chosen order statistics of the test sample should enclose the value *Xi*:*<sup>m</sup>* of the reference sample, namely, *Yc*:*<sup>n</sup>* ≤*Xi*:*<sup>m</sup>* ≤ *Yd*:*<sup>n</sup>*.

**Condition 3.** The number of observations of the *Y*-sample that are placed enclosed by the observations *Xa*:*<sup>m</sup>* and *Xb*:*<sup>m</sup>* should be equal to or more than *r*, namely, *R*≥*r*.
