Friedrich Liese, Igor Vajda
On $\sqrt n$--Consistency and Asymptotic Normality of Consistent Estimators in Models with Independent Observations
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. [57], [3]-[51]([2003])
62J02 General nonlinear regression
62F12 Asymptotic properties of estimators
Abstract: The paper presents relatively simple verifiable conditions
for $\sqrt{n}$-consistency and asymptotic normality of
M-estimators of vector parameters in a wide class of statistical
models. The conditions are established for the $M$-estimators with
absolutely continuous $\rho$-function of locally bounded
variation, and for the class of models including e.g. the linear
and the nonlinear regression, the generalized linear models and
the proportional hazards models as special cases. The conditions
are verified on $L_{1 }$ and $L_{2}$ estimators embedded into a
continuum of their alternative versions, as well as on one new
class of M-estimators of parameters of exponential families which
are shown to be robust in the sense of bounded gross-error
sensitivity. Comparisons with known conditions for special models
indicate that the present general conditions are not too
restrictive in special situations and that sometimes they are even
weaker than the previously published special conditions.

Keywords: General nonlinear regression;Asymptotic properties of estimators