Robust Method for Logistic Profiles Monitoring in Phase I
In this paper, a new robust method based on weighted maximum likelihood estimation (WMLE) is proposed to estimate the regression parameters in logistic profiles in Phase I. This approach reduces the outlier’s effects on the statistical performance of T2 control chart in terms of probability of Type...
| Main Authors: | , , |
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| Format: | Article |
| Language: | fas |
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University of Isfahan
2018-04-01
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| Series: | مدیریت تولید و عملیات |
| Subjects: | |
| Online Access: | http://jpom.ui.ac.ir/article_22953_7e5d58be2c0e8e708648787788649183.pdf |
| Summary: | In this paper, a new robust method based on weighted maximum likelihood estimation (WMLE) is proposed to estimate the regression parameters in logistic profiles in Phase I. This approach reduces the outlier’s effects on the statistical performance of T2 control chart in terms of probability of Type I error. A numerical example is used to evaluate the performance of the proposed method. The results show the better performance of the proposed estimator compared to the maximum likelihood estimation method in terms of power in T2 control chart. Introduction: Yeh et al. (2009) proposed five based T2 statistics to monitor the binary logistic regression profile in Phase I. Different approaches are proposed to monitor logistic regression profiles in Phase II. So far, few researches have been done on monitoring the profile with the presence of contaminated data. In this area, Ebadi and Shahriari (2014) proposed robust estimation approach to monitor the simple linear profile based on two classic and robust methods (M-estimator) with two functions including Huber weighted and double square functions. The aim of this paper is to monitor the logistic regression profiles with the presence of outliers in Phase I based on weighted maximum likelihood robust estimator and T2 control chart. The main questions of this papers are as follows: a) Evaluating the effect of outliers on the mean and standard deviation of the proposed and classic estimators of the logistic regression profile parameters and probability of Type I error in common T2 control chart, b) Comparing performance of the proposed and classic estimators on the T2 control chart power for different shifts in logistic regression profile parameters under outliers in Phase I. Materials and Methods: Sometimes, there are outliers in the gathered data which lead to incorrect estimation of the profile parameters. Hence, to decrease or remove the effect of outlier(s), robust estimation methods are applied. In this paper, a robust approach called weighted maximum likelihood estimator (WMLE) is applied to estimate the parameters of the logistic regression profiles as follows (Maronna et al., 2006): (1) where, is the probability of response variable in each level of logistic regression profile using the estimated parameters. A robust estimate for obtaining parameters is achieved by minimizing the above function. However, in order to give less weight to outliers, we can consider the following relationship and minimize it. (2) where is the weight of the ith observation which is calculated as Equation (3) (3) in which W is a non-ascending function and computed based on Carroll and Pederson (1993) as follows: (4) Results and Discussion The Type I error probability of T2 control chart considering the outlier using MLE and WMLE methods is summarized in Table 1. Table 1- Type I error probability of T2 control chart considering MLE and WMLE methods WMLE MLE Estimation Method 0.0718 0.1242 Type I error probability In this section, r percentage of the total data is contaminated with an increase in the variance of the errors. For r equal to 0.07 and 0.15, the variance error is changed from 1 to 4 and Type I error probability of the T2 control chart with both classic and proposed estimators are calculated and reported in Table 2. Table 2- Type I error probability of the T2 control chart with both classic and proposed estimators under different r Type I error probability Estimation methods r=0.07 Type I error probability Estimation methods r=0.15 0.1801 MLE 0.2779 MLE 0.1021 WMLE 0.1541 WMLE Based on Table 2, Type I error probability of the T2 control chart under the classic method is more than the robust one and this result shows the better performance of the proposed method rather than the classic one. Conclusion In this paper, a robust approach was developed to estimate the logistic regression profiles with the presence of outliers in Phase I. The performance of the proposed robust estimator was compared with the classic method (MLE) based on Type I error probability and power of T2 control chart in Phase I. Results showed that the WMLE method outperforms the MLE in estimating the logistic regression profile parameter under outliers. References Ebadi, M., & Shahriari, H., (2014), "Robust Estimation of Parameters in Simple Linear Profiles Using M-Estimators", Communications in Statistics - Theory and Methods, 43(20), 4308-4323. Maronna, AR., Martin, R.D., & Yohai, V.J., (2006), Robust Statistics Theory and Methods, John Wiley, New York. Yeh A.B., Huwang L., & Li Y.M., (2009), "Profile Monitoring for a Binary Response", IIE Transactions, 41(13), 931-941. |
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| ISSN: | 2251-6409 2423-6950 |