Asymptotic Normality under Nonstandard Conditions

Abstract

The standard approach for deriving the asymptotic distribution of M-estimators outlined in the previous chapter relies on the assumption that the objective function Q n is twice continuously differentiable w.r.t. both the parameter of interest β and the nuisance parameter τ. (Or, if the estimator \({\hat \beta _n}\) is derived as an approximate solution of a set of estimating equations F n = 0, it is maintained that F n is continuously differentiable.) In a number of applications this smoothness assumption is too stringent. E.g., if the objective function corresponds to the least absolute deviation estimator or Huber’s M-estimator this assumption is violated. Also in a semiparametric context, where τ represents an infinite dimensional nuisance parameter that varies in a metric space T which is not a subset of Euclidean space, the notion of differentiability w.r.t. τ may not be available, although Q n may be smooth as a function of β for every given value of τ. Such situations can often be handled by a refinement of the argument underlying Lemma 8.1. The basic idea is again to show that the (normalized) estimator \({\hat \beta _n}\) is asymptotically equivalent to a linear transformation of the score vector evaluated at the true parameter, and then to invoke a CLT for the score vector. Of course, in the absence of the smoothness assumptions of Chapter 8, establishing such a linear transformation is now more delicate (and, if Q n is not differentiable at all, special care has to be given to defining the notion of a score vector properly). The linearization is frequently attempted by showing that the objective function can — in a certain sense — be replaced by its asymptotic counterpart \({\bar Q_n}\) and by exploiting the usually greater degree of smoothness of the latter function in the linearization argument. The fact that \({\bar Q_n}\) is frequently a smooth function, even when Q n is not, originates from the fact that \({\bar Q_n}\) is frequently equal to EQ n or lim n →∞ EQ n and taking expectations is a smoothing operation. This approach was pioneered by Daniels (1961) and Huber (1967). For a modern exposition see Pollard (1985). The following discussion will be informal.