**1. Introduction**

Statistical process control is widely applied to monitor the quality of a production process, where no matter of how thoroughly it is maintained, a natural variability always exists. Control charts help the practitioners to identify assignable causes so that the state of statistical control can be accomplished. Generally speaking, when a cast-off shift in the process takes place, a control chart should detect it as quickly as possible and produce an out-of-control signal.

Shewhart-type control charts were introduced in the early work of Shewhart [1], and since then, several modifications have been established and studied in detail. For a thorough study on statistical process control, the interested reader is referred to the classical textbooks of Montgomery [2] or Qiu [3]. Most of the monitoring schemes are distribution-based procedures, even though this assumption is not always realized in practice. To overcome this obstruction without disrupting the primary formation of the traditional control charts, several nonparametric (or distribution-free) monitoring schemes have been proposed in the literature. The plotted statistics being utilized for constructing such type of control charts are related to well-known nonparametric testing procedures. Among others, a variety of distribution-free control charts appeared already in the literature are based on order statistics; see, e.g., Chakraborti et al. [4], Balakrishnan et al. [5], or Triantafyllou [6, 7]. For an up-to-date account on nonparametric statistical process control, the reader is referred to the review chapter of Koutras and Triantafyllou [8], the recent monograph of Chakraborti and Graham [9], or Qiu [10, 11].

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 under several out-of-control scenarios.

• A quantile *Y <sup>j</sup>*:*<sup>n</sup>* of the test sample which is compared with the control limits

*Simulation-Based Comparative Analysis of Nonparametric Control Charts with Runs-Type Rules*

• The number of observations from the test sample that lie between the control

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

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

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

**Condition 1.** The statistic *Y <sup>j</sup>*:*<sup>n</sup>* of the test sample should lie between the obser-

**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

**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*.

In the present section, we describe the step-by-step procedure which has been followed in order to determine the basic performance characteristics of monitoring schemes mentioned previously. Two well-known runs-type rules are implemented in order to improve the performance of the control charts being considered. More precisely, if we denote by *LCL* and *UCL* the lower and the upper control limit of the

• The 2-of-2 rule. Under this scenario, an out-of-control signal is produced from the control chart, whenever two consecutive plotted points fall all of them

use of an in-control rule, which embraces the following three conditions:

vations *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>*.

*Xa*:*<sup>m</sup>* ≤*Y <sup>j</sup>*:*<sup>n</sup>* ≤*Xb*:*<sup>m</sup>* and *R*≥*r*, (1)

*Xa*:*<sup>m</sup>* ≤*Y <sup>j</sup>*:*<sup>n</sup>* ≤ *Yk*:*<sup>n</sup>* ≤*Xb*:*<sup>m</sup>* and *R*≥*r*, (2)

ð Þ *Xa*, *Xb*

limits

where *r* is a positive integer.

*DOI: http://dx.doi.org/10.5772/intechopen.91040*

where *r* is a positive integer.

sample, namely, *Yc*:*<sup>n</sup>* ≤*Xi*:*<sup>m</sup>* ≤ *Yd*:*<sup>n</sup>*.

**199**

**3. The simulation procedure and some results**

underlying monitoring scheme, we apply the following runs rules

conditions hold true
