Dynamic Robust Bootstrap Algorithm for Linear Model Selection Using Least Trimmed Squares

The Ordinary Least Squares (OLS) method is often used to estimate the parameters of a linear model. Under certain assumptions, the OLS estimates are the best linear unbiased estimates. One of the important assumptions of the linear model is that the error terms are normally distributed. Unfortunately, many researchers are not aware that the performance of the OLS can be very poor when the data set that one often makes a normal assumption, has a heavy-tailed distribution which may arise as a result of the presence of outliers. One way to deal with this problem is to use robust statistics which is less affected by the presence of outliers. Another possibility is to apply a bootstrap technique which does not rely on the normality assumption. In this thesis the usage of bootstrap technique is emphasized. It was a computer intensive method that can replace theoretical formulation with extensive use of computer. Unfortunately, many statistics practitioners are not aware of the fact that most of the classical bootstrap techniques are based on the OLS estimates which is sensitive to outliers. The problems are further complicated when the percentage of outliers in the bootstrap samples are greater than the percentage of outliers in the original sample. To rectify this problem, we propose a Dynamic Robust Bootstrap-LTS based (DRBLTS) algorithm where the percentage of outliers in each bootstrap sample is detected. We modified the classical bootstrapping algorithm by developing a mechanism based on the robust LTS method to detect the correct number of outliers in the each bootstrap sample. Kallel et al. ( 2002 ) proposed utilizing the bootstrap technique for model selection. They used the classical bootstrap method to estimate the bootstrap location and the scale parameters based on calculating the Mean of Squared Residual (MSR). It is now evident that the classical mean and classical standard deviation are easily affected by the presence of outliers. In this respect, we propose to incorporate our proposed DRBLTS in the bootstrap model selection technique. We also proposed to use an alternative robust location and scale estimates which are less affected by outliers instead of using the classical mean and classical standard deviation. The performances of the newly proposed methods are investigated extensively by real data sets and simulations study. The effect of outliers is investigated at various percentage, i.e , 0%, 5%, 10%, 15% and 20%. The results show that the DRBLTS is more efficient than other estimators discussed in this thesis. The results on the model selection again signify that our proposed robust bootstrap model selection method is more robust than the classical bootstrap model selection.

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