Abstract
Outliers in a statistical analysis strongly affect the performance of the ordinary least squares, such outliers need to be detected and extreme outliers deleted. Thisp is aimed at proposing a Redescending M-estimator which is more efficient and robust compared to other existing methods. The results show that the proposed method is effective in detection and deletion of extreme outliers compared to the other existing ones.
Highlights
The standard multiple regression model in matrix notation is given as Y = X + ε (1)OLS estimates are obtained by minimizing the sum of squared error (SSE) given asSSE=∑ni=1 i2 = = (Y − Xβ) (Y − Xβ) (2)Some of the assumptions of Ordinary Least Squares are: E ( ) = 0, E ( ) = 21n, X is a non–stochastic matrix and N(0, 21n).In the context of outlier detection, many researchers developed various methods
The Simulated results for the proposed estimator and that of OLS, Huber, Hampel, Bisquare (Biweight) and Alarm estimators are discussed as follows: 4.1 Discussion of Simulated results for data without outlier
Based on data generated from 30% outliers in x- direction in a multiple regression, shown in table 9, the result indicates that the Alarm estimator is the most efficient having the smallest Mean Square Errors (MSE) among others while Hampel estimator is the second
Summary
OLS estimates are obtained by minimizing the sum of squared error (SSE) given as. Some of the assumptions of Ordinary Least Squares are: E ( ) = 0, E ( ) = 21n, X is a non–stochastic matrix and N(0, 21n). Nguyena and Welch (2010) studied outlier detection and proposed a new trimmed square approximation. Armin (2008) proposed a method for detection of outliers using Dixon’s test statistics. Zhang et al (2015) proposed an enhanced Monte Carlo outlier detection method by establishing cross-prediction models based on normal samples and analysing the distribution of prediction errors for dubious samples. Robust regression is use for improving the results of the least square estimates in the presence of outliers. Some methods of robust regression include those of: Huber (1964) who discovered M-estimators which are the generalization of the Maximum Likelihood Estimators (MLE). Rousseeuw (1983) proposed the Least Trimmed Squares (LTS) estimators. Some Redescending M-estimators for detection and deletion of outliers are given in: Andrew et al (1972), Beaton and Tukey (1974), Hampel et al (1986) and Alamgir et al (2013). To compare the proposed Redescending M-estimator with some existing Mestimators and Redescending M-estimators in terms of efficiency and robustness
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