M-estimator with asymmetric influence function for estimating the Burr type III parameters with outliers

• Main Authors: Chih-wen Lee, 李志文
• Other Authors: Fu-kwun Wang
• Format: Others
• Language: en_US
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/13447452466176508003

博士 === 國立臺灣科技大學 === 工業管理系 === 99 === The Burr type III distribution allows for a wider region for the skewness and kurtosis plane, which covers several distributions including: the log-logistic, and the Weibull and Burr type XII distributions. However, outliers may occur in the data set. The robust regression method such as M-estimator with symmetric influence function has been successfully used to diminish the effect of outliers on statistical inference. However, when the data distribution is asymmetric, these methods yield biased estimators. We want to develop a mathematical model which can diminish the effect of outliers on statistical inference. The traditional method suggests that a simple way to handle outliers is to detect them and remove them from the data set. There are many methods for detecting outliers. Deleting an outlier, although better than doing nothing, still poses a number of problems. When is deletion justified? Deletion requires a subjective decision. When is an observation “outlying enough” to be deleted? The user of the data may think that “an observation is an observation” and hence feel uneasy about deleting them. Since there is generally some uncertainty as to whether an observation is really atypical, there is a risk of deleting “good” observations, which results in underestimating data variability. Most published papers presented the outlier estimation based on the CDF or PDF functions. However, the skew logistic distribution has only quantile function. If we wanted to compute the values of the CDF or PDF for these distributions, we would have to be made of numerical techniques to obtain specific approximate value. We present an M-estimator with asymmetric influence function (AM-estimator) based on the quantile function of the Burr type III distribution to estimate the parameters for complete data with outliers. The simulation results show that the M-estimator with asymmetric influence function generally outperforms the maximum likelihood and traditional M-estimator methods in terms of bias and root mean square errors. One real example is used to demonstrate the performance of our proposed method.