On Performance of Two-Parameter Gompertz-Based X Control Charts

In this paper, two methods of control chart were proposed to monitor the process based on the two-parameter Gompertz distribution. .e proposed methods are the Gompertz Shewhart approach and Gompertz skewness correction method. A simulation study was conducted to compare the performance of the proposed chart with that of the skewness correction approach for various sample sizes. Furthermore, real-life data on thickness of paint on refrigerators which are nonnormal data that have attributes of a Gompertz distribution were used to illustrate the proposed control chart. .e coverage probability (CP), control limit interval (CLI), and average run length (ARL) were used to measure the performance of the two methods. It was found that the Gompertz exact method where the control limits are calculated through the percentiles of the underline distribution has the highest coverage probability, while the Gompertz Shewhart approach and Gompertz skewness correction method have the least CLI and ARL. Hence, the two-parameter Gompertz-based methods would detect out-of-control faster for Gompertz-based X charts.


Introduction
Shewhart X control chart is one of the most widely used statistical process control techniques developed to monitor the process average [1]. Shewhart control charts are often based on the assumption that the sample observations are independently and identically distributed and the process observations follow a normal distribution. However, skewed data often violate the normality assumption and cause an increase in error probability when control charts are used to monitor the process. Skew probability distributions always occur in the monitoring process of real-life data during the production process, but gamma distribution is often selected to examine the performances of control charts [2]. Furthermore, whenever skew distributions exist, mean and variance may not be appropriate summary statistics to measure the process variation [3]. When the distribution of data is known, the use of the exact method proposed to provide accurate control limits is more likely to detect whether a process is in control or not [4,5].
Some exact control limits based on the form of underlying distribution are investigated in the literature. Construction of control charts using the theory of confidence intervals, when the random variable follows inverse Gaussian distribution, is considered by Edgeman [6]. Edgeman [6] justified his assumption by relying on the central limit theorem for nonnormal processes whenever the sample size used for control charting is less than 10. However, quality characteristics are always better modelled by using probability distribution with nonnegative support rather than a normal distribution. Kantam and Sriram [7] developed control charts to be used when the process characteristic follows a gamma distribution. Kantam et al. [8] developed control charts for log-logistic distribution, Kan and Yazici [9] developed individual control charts for Burr distributed and Weibull distributed data, Subba Rao and Kantam [10] introduced control charts for double exponential distribution, Yazici and Kan [11] developed asymmetric control limits for small samples, Srinivasa Rao and Sarath Babu [12] proposed control charts for linear failure rate distribution, and Srinivasa Rao and Kantam [13] developed control charts for half-logistic distribution percentiles. Srinivasa Rao et al. [14] developed control charts for a variable quality characteristic that is assumed to follow size-biased Lomax distribution based on the evaluated percentiles of sample statistics like mean, median, midrange, range, and standard deviation, Srinivasa Rao et al. [15] developed control charts for a variable quality characteristic that is assumed to follow new Weibull-Pareto distribution based on the evaluated percentiles of sample statistics like mean, median, midrange, range, and standard deviation, Wang et al. [16] proposed control charts for monitoring the lower Weibull percentiles under complete data and Type-II censoring, and Rao [17] considered an exponentiated halflogistic distribution to develop an attribute control chart for time-truncated life tests with a known or unknown shape parameter and references therein.
In this paper, the X Shewhart-type control chart based on Gompertz distribution is proposed for monitoring a nonnormal process. A simulation study was conducted to compare the performance of the proposed chart with that of the skewness correction approach for various sample sizes. Furthermore, real-life data on thickness of paint on refrigerators which are nonnormal data that have attributes of a Gompertz distribution were used to measure the performance of the proposed control chart.

Two-Parameter Gompertz Distribution
Gompertz distribution is an exponentially increasing, continuous probability distribution. It is basically a truncated extreme value distribution [18]. Gompertz distribution is a lifetime distribution and is often applied to describe the distribution of adult life spans by actuaries and demographers. It is considered for the analysis of survival in some sciences such as biology, gerontology, computer science, and marketing science [19].
e Gompertz distribution is an extension of the exponential distribution. According to Alizadeh et al. [20], the cumulative distribution function (CDF) and probability density function (PDF) of the Gompertz generalized family of distributions are given, respectively, by where θ and c are additional shape parameters which are introduced to vary tail weights and G(x) and g(x) are, respectively, the CDF and PDF of the exponential distribution which is the parent distribution. e CDF and PDF of the exponential distribution with the parameter λ are given by the following equations: erefore, the PDF of the Gompertz exponential distribution is derived by inserting the expression in equations (3) and (4) into equation (2). is gives e CDF of the Gompertz exponential distribution is derived by inserting equation (3) into equation (1). us, we have the expression c, λ, and θ in (5) and (6) may be combined into two independent parameters, say, t and z defined as follows: z � θ/c and t � λc.
en, f(x) in equation (5) becomes e CDF, F(x) in equation (6), becomes e expressions in equations (7) and (8) are, respectively, the PDF and CDF of Gompertz distribution with parameters t and z.

Gompertz X Control Chart Using the Shewhart Approach.
Consider the Shewhart X chart which contains the center line (CL) that represents the average value of quality characteristics corresponding to the in-control state. ere are two horizontal lines, namely, the lower control limit (LCL) and upper control limit (UCL). ese control limits are selected so that if the process is in control, nearly all sample points will fall within them. If w is a statistic that measures the quality characteristic and if the mean and the variance of w are given as μ Gw and σ 2 Gw , respectively, then the general model for the Shewhart control chart is given as where L is the distance of the control limits from the center line.
In the construction of control charts, it is common to set L � 3. Vysochanskij and Petunin [21] refined Chebyshev's inequality by including the factor of 4/9 and made it possible to set 3-sigma limits for any unimodal distribution. e Vysochanskij-Petunin inequality allows the inference that, for any unimodal distribution, at least 95% of the data will be captured by limits placed at 3-sigma. erefore, for Gompertz-based X control charts, the control limits can be obtained as follows: When the process data are assumed to follow Gompertz distribution, we used the expression in equation (7) to derive the mean and variance of the Gompertz distribution (μ G and σ 2 G ). Using Maple software, the mean of Gompertz exponential distribution is derived in an integral form (see details in Appendix). e derived mean is given as Similarly, the variance of the Gompertz exponential distribution is derived and presented in Appendix. e derived variance is where Ei(a, z) is an exponential integral and hypergeom (.) is a generalized hypergeometric function.

Gompertz X Control Chart Using the Skewness Correction
Approach. e skewness correction (SC) method is used for constructing the X control charts for skewed distributions. Its asymmetric control limits are obtained by taking into consideration the degree of skewness estimated from subgroups, and with no assumptions on the distributions. According to Chan and Cui [4], if the parameters of a process are known, the control limits of the X control chart are given by where C * 4 is the control chart constant for the SC method. It should be noted that C * 4 � 0 if the underlying distribution is symmetric and the control limits of the SC method for X control chart will reduce to the traditional control limits of the Shewhart X control chart. However, if the distribution is asymmetrical, Chan and Cui [4] gave the constant C * 4 in equation (13) to be where k 3 (X) is the skewness of the subgroup mean X.
Let x 1 , x 2 , . . . , x n be samples from Gompertz distribution with mean μ Gw and standard deviation σ Gw . e control limits and the center line for a skewness correction method for the X chart are To obtain the constant C * 4 , Bowley's formula is chosen for finding the coefficient of skewness which is given by where Q i (i � 1, 2, 3) is the ith quartile of the Gompertz distribution.
Hence, the constant where k 3G (X) is the skewness of the subgroup mean X.

Using Exact Control Limits
. Let x 1 , x 2 , . . . , x n be a random sample subgroup of industrial process data of size n supposed to have been drawn from the two-parameter Gompertz exponential distribution with a targeted population average; under repeated sampling, the statistic X gives whether the process average is around the targeted mean or not. Statistically speaking, we have to find the "most probable" limits within which X falls. It is well known that 3σ limits of normal distribution include 99.73% of probability. Hence, we have to search for two limits of the sampling distribution of sample mean in Gompertz exponential distribution such that the probability content of those limits is 0.9973 (see Srinivasa Rao et al. [14] and Srinivasa et al. [15]). Symbolically, we have to find L and U such that where x i is the mean of sample size n. Taking the equitailed concept, L and U are, respectively, 0.00135 and 0.99865 percentiles of the sampling distribution of x i .
where x is the grand mean and x i is the ith subgroup mean. us, A * 2p and A * * 2p are the percentile constants of the x chart. Hence, the control limits and the center line for an exact X control chart are Journal of Probability and Statistics 3

Performance Evaluation.
In this work, three indicators were used to measure the performance of the derived control charts discussed above. ey are the coverage probability (CP), control limit interval (CLI), and average run length (ARL). e coverage probability is the probability of the number of points within the control limits. It is used to compare the stability of the simulated process under different methods. e CLI and ARL were

Simulation Study
e steps for the simulation study are presented as follows.

Results
Using the two steps presented in Section 3, data were generated from a class of distribution of Gompertz with parameters t � 2.2 and z � 4.7 using Monte Carlo simulation. e coefficient of skewness of the generated data is 0.9226. e control limits for the two proposed methods and the exact method were computed to determine the stability of the simulated process. e coverage probability of the X charts based on classical Shewhart was adopted. e results obtained were compared with those of the skewness correction method. e control limits of the skewness correction method and the Gompertz Shewhart approach, Gompertz skewness correction method, and Gompertz exact method using equations (10), (15), and (20), respectively, are computed and presented in Table 1.   Journal of Probability and Statistics To measure the performance of the charts, the coverage probability was computed for the four methods. e obtained results are presented in Table 2.

Data on ickness of Paint on Refrigerators.
e data on the thickness of paint on refrigerators presented in Table 3 were obtained from Priya and Kantam [22]. e data are for 20 subgroups of size n � 5 from a process that is known to be in control.
To test for the normality of the data in Table 3, the density and the Q-Q plots of the mean of the data are determined and presented in Figure 1. Furthermore, the Jarque-Bera normality test was obtained to be 10.556 with a p value of 0.005.
From Figure 1, it is clear that the data are not normally distributed. e p value of the Jarque-Bera normality also confirmed the nonnormality of the data. Furthermore, the skewness coefficient was computed to be 1.2195. Hence, the data are skewed data. Based on the fact that the data are skewed data, the Gompertz distribution was used to model the data, and it was discovered that the CDF of the data is approximately Gompertz (see Figure 2). us, the data are approximately a Gompertz random variable. Hence, we can use the Gompertz-based X charts to monitor the process that produces the data. e mean of the thickness of paint of one hundred refrigerators in twenty subgroups for sample size 5 was computed and used to compute the control limits derived in Section 2. e skewness correction, Gompertz Shewhart, Gompertz skewness correction, and Gompertz exact methods are used for the construction of the X charts. e control charts for the four methods are presented in  e performance indices, namely, coverage probability (CP), control limit interval (CLI), and average run length (ARL), for the four methods are presented in Table 4.

Discussion of Results
e results of the control limits in Table 1 for the proposed methods showed improved limits compared to the skewness correction limits. e results of the coverage probabilities in Table 2 revealed that as the sample size increases, the coverage probabilities decrease for Gompertz Shewhart and Gompertz skewness correction methods. However, the coverage probabilities for skewness correction and Gompertz exact methods are more stable than those of Gompertz Shewhart and Gompertz skewness correction methods. e obtained results of the real-life data in Table 4 showed that the CLI and ARL of Gompertz Shewhart and Gompertz skewness correction methods are smaller but those of the Gompertz exact method are the smallest.

Conclusion
is study proposed control limits for the Gompertz X control chart using different approaches. Coverage probability was used to compare the stability of the methods. e results show that as the sample size increases, the coverage probabilities for the Gompertz Shewhart approach and Gompertz skewness correction approach decrease. e coverage probabilities for skewness correction and Gompertz exact methods are more stable than those of the Gompertz Shewhart approach and Gompertz skewness correction method. e results of the real-life data using the CLI and ARL showed that the Gompertz Shewhart approach and Gompertz skewness correction method would be able to detect out-of-control faster than the exact method for Gompertz-based X charts. e CLI and ARL of the skewness correction method are too small and thereby would raise false alarm when there are none. Hence, it is recommended that the Gompertz Shewhart approach and Gompertz skewness correction method can be used to monitor skewed process data that have the attribute of a two-parameter Gompertz distribution.