On Asymptotic Behavior of Certain Estimates of Shift and Scale Parameters

Assume that we have $n$ observations of the form $x_i=\theta+\sigma\varepsilon_i$, where the errors ei are independent and identically distributed with distribution function $F(x)$.“Polynomial” Pitman estimates are derived for the parameters $\theta$ and $\sigma$, and their asymptotic behavior is investigated. The results are applied to studying the Pitman estimates of $\theta$ and $\sigma$.