Summary We consider estimation for a multivariate location family. Between all confidence regions with volume less than a fixed value L , we find the equivariant confidence region with the biggest coverage probability. This region maximizes the infimum of the coverage probability over all...

Some laws of the iterated logarithm for empirical processes rescaled in the "time" parameter are presented. These laws of the iterated logarithm are applied to obtain strong limit theorems for M-estimators. In particular, a law of the iterated logarithm for M-estimators with unusual rates of...

The law of the iterated logarithm for canonical or completely degenerate U-statistics with square integrable kernel h is proved, for h taking values in 1, 7 and, in general, in a type 2 separable Banach space. The LIL is also obtained for U-processes indexed by canonical Vapnik-Cervonenkis...

We give some easy methods to check sufficient conditions for the weak convergence of stochastic processes indexed by smooth functions. The main condition is a moment condition on the increments of the process. We apply this to empirical processes and U-processes in the independent identically...

We study the weak convergence for the row sums of a triangular array of empirical processes under bracketing conditions involving majorizing measures. As an application, we consider the weak convergence of stochastic processes of the form where {Xj}j=1[infinity] is a sequence of i.i.d.r.v.s with...

Exponential inequalities, the law of the iterated logarithm and the bootstrap central limit theorem for U-processes indexed by Vapnik-Cervonenkis classes of functions are derived. These results are then applied to the asymptotics and the bootstrap of U-statistics with estimated parameters, in...

We discuss the large and moderate deviations of a type of empirical processes whose finite-dimensional distributions do not satisfy the Cramer condition. Nonstandard speeds and rate functions appear.

An approximate M-estimator is defined as a value that minimizes certain random function up to a [var epsilon]n, where {[var epsilon]n} is a sequence of real numbers converging to zero. We determine the rate of [var epsilon]n so that the approximate M-estimator is asymptotically normal with rate...

It is shown that if {Xt: t [set membership, variant] T} is a Gaussian process such that (T, [varrho]) is a separable metric space, where [varrho](t, s) = Cov(Xt,Xs), then, with probability 1, no sample path of X can achieve its supremum at two distinct points of T. Conversely if Pr* {supt[set...

We discuss the asymptotic linearization of multivariate M-estimators, when the limit distribution is stable. We consider two different types of kernels: VC and bracketing. When applied to the case of normal limits, our work improves the known results to obtain the limit distribution of...