Measurement Error Effect on the Power of Control Chart for Doubly Truncated Normal Distribution under Standardization Procedure

Researchers in various quality control procedures consider the possibility of measurement error effect on the power of control charts as an important issue. In this paper the effect of measurement errors on the power curve of standardization procedure will be studied for doubly truncated normal distribution. A method of obtaining the expression of the power of control chart for doubly truncated normal distribution is being proposed. The effect of truncation will be shown accordingly. To study the sensitivity of the monitoring procedure, average run length ( ARL ) is also considered.


Introduction
The truncated distributions are quite important and widely used by researchers. Kendall and Stuart (1979) have discussed an interesting example of a practical application of various types of truncated distributions: Suppose the underlying variate x simply cannot be observed in part or parts of its range. For example, if x is the distance from the centre of a vertical circular target for fixed radius R on a shooting target, we can only observe x for shots actually hitting the target. If we have no knowledge of how many shots were fired at the target (say n ) we simply have to accept m values of x observed on the target as coming from a distribution ranging from 0 to R . We then say that x is truncated on the right at R . Similarly, if we define Y in this example as the distance of a shot from the vertical line through the centre of the target, Y may range from R  to R and its distribution is doubly truncated. Similarly, we may have a variate truncated on the left i.e. if observations below a certain value are not recorded.
The normal distribution in statistical quality control (process control) arises because observations of a measurable quality characteristic vary under large number of small random disturbances. And it varies similarly on both sides of the average. X and R charts are the typical statistical control charts for variables where normal distribution is employed to control the location and dispersion of the characteristic under consideration. However, there are some practical situations where truncated form of a normal distribution may provide better approximation to the actual distribution. Since Cohen's initial contribution in the 1940s, several studies have looked at the various aspects of the truncated normal distributions. See, for example, Cohen (1949Cohen ( , 1950Cohen ( , 1991, Schneider (1986), Johnson (2001), Johnson and Thomopoulos, (2004), Horrace (2015). A number of authors used truncated normal distribution in statistical quality control. Rai (1966) constructed CUSUM for truncated normal distribution. Schneider (1985) studied the performance of the standard sampling plans when the variable understudy is distributed as truncated normal distribution. He showed how variable sampling plans can be designed when the truncation points and the normal population standard deviation  are known. The case of unknown  was also discussed by him. Kakoty and Chakraborty (1990) studied continuous inspection sampling plan based CUSUM chart for controlling the mean of a doubly truncated normal distribution. They used Fredholm integral equation to obtain Type C-OC curve. Recently, Cox (2009) applied truncated normal distribution to derive integral equations for the average run length (ARL), a key measure of the performance of a control chart.
In this paper the effect of errors on the power curve of standardization procedure will be studied for DTND and the effect of truncation will be shown accordingly along with the calculation of average run length ( ARL ).

Measurement Error
Measurement errors, which often exist in practice, may considerably affect the performance of control charts (Ryan, 2011). The sources of error may be due to inherent variability in the process and the error due to measurement instrument. The efficiency and the ability of the control chart to detect the shift of the process level will be affected if the measurement error is large relative to the process variability (Chakraborty and

Doubly Truncated Normal Distribution
A random variable X has doubly truncated normal distribution if its probability density function is given by are probability density function and cumulative distribution function of standardized normal variate respectively. If A is replaced by   or B by  , the distribution is singly truncated from above or below respectively.
We refer an interested reader to Schneider (1986)

Assumptions and Notations
In the development of the power of the control chart and ARL for equation (1), the following assumptions are made and notations are used: i.
The measurement of items is considered to ascertain the magnitude of the variable characteristics in the lot. ii. The process has doubly truncated normal distribution with mean t  and variance 2 p  ; iii. The applied measurement process (which is independent of the manufacturing process) has a variance 2 m  . Thus, the overall variability is given by iv. Measurements of the items are taken to classify the produced units into defective and non-defective ones; v.
The process is in a state of statistical control at the time of determining the control limits and the same measuring instrument is used for later measurements; vi. When the process parameter shifts, the data still comes restricted from DTND, however, with mean t  and variance ) ( Thus, considering the above assumptions, exact Shewhart control limits will be

Power of control chart for standardized doubly truncated normal distribution
Instead of plotting measurement values in the control chart, we can standardize the variables as given below and plot accordingly (Khurshid and Chakraborty, 2014). This standardization procedure not only stabilizes the variables, but also stabilizes the resulting control chart. In this case the control limits as well as central lines are invariant with sample size . n Thus, equation (4) can be expressed in terms of standardized normal variable Z (when sample size is large and varies): Now, following Kanazuka (1986), Chakraborty and Khurshid (2013 b) and using equation (5), when the process parameter changes from t  to t  , the power of the control chart for equation (1) is

Average Run Length ( ARL ) for DTND under measurement error
To study the sensitivity of the monitoring procedure, one can also study ARL which is the average number of points that must be plotted before a point indicates an out of control condition when operating is statistical control (Khurshid and Chakraborty, 2014).
For any Shewhart control chart, the where P is the probability that a single point exceeds the control limits. Thus instead of drawing conclusion based on X P , one can interpret the results of the power of control chart in terms of ARL just by reversing equation (6) i.e., in this case Pak.j.stat.oper.res. Vol.XI No.3 2015 pp387-396 392 The values of ARL are shown in Table 7.

Concluding Remarks
The effects of truncation as well as measurement errors on the power of detecting the changes in the process parameters by  3 control limits with the control chart for DTND are shown in Tables 1-7. It has been observed from Table 1

B
For fixed d , Table 4 shows the values of d P which increases as the ratio to t  to t  decreases.
From the Tables 1 and 5, it has been observed that as we increase the range of truncations, there is a decrease in the values of d P when the ratios of t  to t  are same.