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Statistics for High-Dimensional Data: Methods, Theory and Applications

We investigate a robust penalized logistic regression algorithm based on a minimum distance criterion. Influential outliers are often associated with the explosion of parameter vector estimates, but in the context of standard logistic regression, the bias due to outliers always causes the parameter vector to implode, that is, shrink toward the zero vector. Thus, using LASSO-like penalties to perform variable selection in the presence of outliers can result in missed detections of relevant covariates. We show that by choosing a minimum distance criterion together with an elastic net penalty, we can simultaneously find a parsimonious model and avoid estimation implosion even in the presence of many outliers in the important small n large p situation. Minimizing the penalized minimum distance criterion is a challenging problem due to its nonconvexity. To meet the challenge, we develop a simple and efficient MM (majorization–minimization) algorithm that can be adapted gracefully to the small n large p context...

https://amstat.tandfonline.com/doi/pdf/10.1080/10618600.2012.737296

Robust Parametric Classification and Variable Selection by a Minimum Distance Criterion

We investigate a robust penalized logistic regression algorithm based on a minimum distance criterion. Influential outliers are often associated with the explosion of parameter vector estimates, but in the context of standard logistic regression, the bias due to outliers always causes the parameter vector to implode, that is shrink towards the zero vector. Thus, using LASSO-like penalties to perform variable selection in the presence of outliers can result in missed detections of relevant covariates. We show that by choosing a minimum distance criterion together with an Elastic Net penalty, we can simultaneously find a parsimonious model and avoid estimation implosion even in the presence of many outliers in the important small $n$ large $p$ situation. Implementation using an MM algorithm is described and performance evaluated.

Provided is a maximum likelihood estimation method for machine learning. The method comprises the following steps of: performing trimming for processing outlier values and heavy tailed noise on a loss function for a sample set; performing I_1 normalization in consideration of a penalty determined by performing the trimming; and estimating the maximum likelihood of a model associated with the sample set based on the l_1 normalization result.

L1 apparatus and method for m-estimation with trimmed l1 penalty

Parametric Classification and Variable Selection by the Minimum Integrated Squared Error Criterion

Parametric classification and variable selection by the minimum integrated squared error criterion

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Parametric classification and variable selection by the minimum integrated squared error criterion