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Abstract:
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We introduce a newly proposed envelope huber regression. It is known that huber regression is robust in the scenario where there are outliers and error follows heavy tail distribution, while maintaining acceptable efficiency loss compared to OLS when error follows normal distribution. In this paper we generalize huber regression using envelope model. Assuming the true signal lies in a reducing subspace of the covariance matrix of predictors, it is known that envelope model could produce estimator with smaller standard error compared to OLS. We borrow this intuition, propose the formulation of envelope huber regression, and introduce a non-likelihood based estimation procedure through Generalized Method of Moments (GMM). The asymptotic normality of our estimator is established. Moreover, we show that envelope huber regression is more efficient than usual huber regression, namely, it could produce estimator with smaller standard error compared to usual huber regression. This helps us achieve better performance in hypothesis testing and statistical inference, especially in the case where error distribution is heavy-tailed other than just normal.
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