Asymptotic behavior of robust estimators of regression and scale parameters with fixed carriers
For a linear regression model, \(y_ i=x_ i\beta +\epsilon_ i\), where \(x_ i=(1,x_{i2},...,x_{ip})\) are known, the estimates (\({\hat \beta}\),\({\hat \sigma}\)) of the vector of parameters \(\beta\) and the scale of the errors \(\epsilon_ i\), respectively, are considered. The estimates were constructed by minimizing the Huber-Dutter loss function \[ \sum \sigma \rho \{(y_ i-x_ i\beta)/\sigma \}+A_ n\sigma \] where \(\rho\geq 0\) is convex, \(\rho (0)=0\), \(t^{-1}\rho (t)\to k\) as \(| t| \to \infty\) for some k, and \(\{A_ n\}\) is some suitably chosen sequence of constants. The asymptotic normality of (\({\hat \beta}\),\({\hat \sigma}\)) under rather general conditions is proved.
scientific article; zbMATH DE number 3965233