Applications of Methods of Integrals and Function Theory to Statistics

Abstract

Let ${Y_i} = {x'_i}{\beta _0} + {e_i},1 \leqslant i \leqslant n,n \geqslant 1$ be a linear regression model. Denote by ${\hat \beta _n}$ the M-estimate of β ₀, using a convex function ρ. In [1], a basic theorem (Theorem A below) concerning the weak consistency of ${\hat \beta _n}$ is established. This theorem raises further problems concerning the consistency of ${\hat \beta _n}$ . In this note, some of these questions are considered for the special cases of LAD and LS estimates, and the methods of integrals and real function theory are used.