Abstract
Supposing that the disturbance terms in the standard linear statistical model are independent and follow a common Laplacian, Gaussian, or rectangular distribution, then the principle of maximum likelihood suggests that we should choose estimates of the slope parameters to minimise the \(L_t\)-norm of the residuals with \(t=1,\ t=2\) or \(t=\infty \) respectively. In this context, we outline the small sample and asymptotic theory relating to these maximum likelihood estimators and the related Likelihood Ratio, Lagrange Multiplier and Wald tests of linear restrictions on the parameters of the model. We also demonstrate that a simple modification of the standard linear programming implementation of the \(l_1\) -norm or \(L_{\infty }\)-norm fitting problem yields (pseudo-unbiased) estimators that are symmetrically distributed about the true parameter values when the disturbances are symmetrically distributed about zero.
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Farebrother, R.W. (2013). Statistical Theory. In: L1-Norm and L∞-Norm Estimation. SpringerBriefs in Statistics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36300-9_5
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DOI: https://doi.org/10.1007/978-3-642-36300-9_5
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