Outlier detection and parameter estimation in multivariate multiple regression (MMR)

Abstract

Outlier detection in Y-direction for multivariate multiple regression data is interesting since there are correlations between the dependent variables which is one cause of difficulty in detecting multivariate outliers, furthermore, the presence of the outliers may change the values of the estimators arbitrarily. Having an alternative method that can detect those outliers is necessary so that reliable results can be obtained. The multivariate outlier detection methods have been developed by many researchers. But in this study, Mahalanobis Distance method, Minimum Covariance Determinant method and Minimum Volume Ellipsoid method were considered and compared to the proposed method which tried to solve outlier detection problem when the data containing the correlated dependent variables and having very large sample size. The proposed method was based on the squared distances of the residuals to find the robust estimates of location and covariance matrix for calculating the robust distances of Y. The behavior of the proposed method was evaluated through Monte Carlo simulation studies. It was demonstrated that the proposed method could be an alternative method used to detect those outliers for the cases of low, medium and high correlations/variances of the dependent variables. Simulations with contaminated datasets indicated that the proposed method could be applied efficiently in the case of data having large sample sizes. That is, the principal advantage of the proposed algorithm is to solve the complicated problem of resampling algorithm which occurs when the sample size is large. When data contain outliers, the ordinary least-squares estimator is no longer appropriate. For obtaining the parameter estimates of data with outliers, we analyze Multivariate Weighted Least Squares (MWLS) estimator. The estimates of the regression coefficients using the proposed method were compared to those of using MCD and MVE method. For comparing the properties of the estimation procedures, we focus on the values of Bias and Mean Squared Error (MSE) of the estimated coefficients. For most of the values of Bias and MSE in the case of large sample size, the proposed method gave lower values of Bias and MSE than the others with any percentages of Y-outliers.