L 1 law of large numbers for functions of i.i.d. random variables that are translated by a consistent estimator]]>

摘要

We develop a newL1law of large numbers where theith summand is given by a functionh(?)evaluated atXi?θn, and whereθn?θn(X1,X2,…,Xn)is an estimator converging in probability to some parameterθ∈R. Under broad technical conditions, the convergence is shown to hold uniformly in the set of estimators interpolating betweenθand another consistent estimatorθn?. Our main contribution is the treatment of the case where|h|blows up at 0, which is not covered by standard uniform laws of large numbers.