Slepian and Sudakov-Fernique type inequalities, which compare expectations of maxima of Gaussian random vectors under certain restrictions on the covariance matrices, play an important role in the probability theory, especially in empirical process and extreme value theories. Here we give...

We derive a Gaussian approximation result for the maximum of a sum of high dimensional random vectors. Specifically, we establish conditions under which the distribution of the maximum is approximated by that of the maximum of a sum of the Gaussian random vectors with the same covariance...

This paper considers the problem of testing many moment inequalities where the number of moment inequalities, denoted by p, is possibly much larger than the sample size n. There are a variety of economic applications where the problem of testing many moment inequalities appears; a notable...

In this work we consider series estimators for the conditional mean in light of three new ingredients: (i) sharp LLNs for matrices derived from the non-commutative Khinchin inequalities, (ii) bounds on the Lebesgue factor that controls the ratio between the L∞ and L2-norms, and (iii)...

We develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating whole empirical processes in the supremum norm. We prove an abstract approximation theorem that is applicable to a...

We derive a central limit theorem for the maximum of a sum of high dimensional random vectors. More precisely, we establish conditions under which the distribution of the maximum is approximated by the maximum of a sum of the Gaussian random vectors with the same covariance matrices as the...

We develop a new direct approach to approximating suprema of general empirical processes by a sequence of suprema of Gaussian processes, without taking the route of approximating empirical processes themselves in the sup-norm. We prove an abstract approximation theorem that is applicable to a...

Modern construction of uniform confidence bands for non-parametric densities (and other functions) often relies on the classical Smirnov-Bickel-Rosenblatt (SBR) condition; see, for example, Giné and Nickl (2010). This condition requires the existence of a limit distribution of an extreme value...

<p>Instrumental variables are often associated with low estimator precision. This paper explores efficiency gains which might be achievable using moment conditions which are nonlinear in the disturbances and are based on flexible parametric families for error distributions. We show that these...</p>

Using many valid instrumental variables has the potential to improve efficiency but makes the usual inference procedures inaccurate. We give corrected standard errors, an extension of Bekker (1994) to nonnormal disturbances, that adjust for many instruments. We find that this adujstment is...