High-dimensional statistical tests often ignore correlations to gain simplicity and stability leading to null distributions that depend on functionals of correlation matrices such as their Frobenius norm and other r norms. Motivated by the computation of critical values of such tests, we investigate the difficulty of estimation the functionals of sparse correlation matrices. Specifically, we show that simple plug-in procedures based on thresholded estimators of correlation matrices are sparsity-adaptive and minimax optimal over a large class of correlation matrices. Akin to previous results on functional estimation, the minimax rates exhibit an elbow phenomenon. Our results are further illustrated in simulated data as well as an empirical study of data arising in financial econometrics.

Kernel density estimates are frequently used, based on a second order kernel. Thus, the bias inherent to the estimates has an order of O(<i>h</i>²_<i>n</i>). In this note, a method of correcting the bias in the kernel density estimates is provided, which reduces the bias to a smaller order. Effectively, this method produces a higher order kernel based on a second order kernel. For a kernel function <i>K</i>, the functions W_k(x)=^k-1₁₌₀(^k_l+1)x^lK^(l)(x)/l! and [1/<i>K</i>^{(<i>k</i> 1)}(<i>x</i>)/<i>x</i> d <i>x</i>]<i>K</i>^{(<i>k</i> 1)}(<i>x</i>)/<i>x</i> are kernels of order <i>k</i>, under some mild conditions.

We develop a specification test for the transition density of a discretely-sampled continuous-time diffusion process, based on a comparison of a nonparametric estimate of the transition density or distribution function to their corresponding parametric counterparts assumed by the null hypothesis. We provide a direct comparison of the two densities for an arbitrary specification of the null parametric model using three different discrepancy measures between the null and alternative transition density and distribution functions. We establish the asymptotic null distributions of proposed test statistics and compute their power functions. The finite sample properties are critically investigated via simulation studies and are compared with other tests. Our approaches are illustrated by applications to implied volatility data and Treasury bill data.

An aggregated method of nonparametric estimators based on time-domain and state-domain estimators is proposed and studied. To attenuate the <i>curse of dimensionality</i>, we propose a factor modeling strategy. We first investigate the asymptotic behavior of nonparametric estimators of the volatility matrix in the time domain and in the state domain. Asymptotic normality is separately established for nonparametric estimators in the time domain and state domain. These two estimators are asymptotically independent. Hence, they can be combined, through a dynamic weighting scheme, to improve the efficiency of volatility matrix estimation. The optimal dynamic weights are derived, and it is shown that the aggregated estimator uniformly dominates volatility matrix estimators using time-domain or state-domain smoothing alone. A simulation study, based on an essentially affine model for the term structure, is

This paper is concerned with the problem of top- K ranking from pairwise comparisons. Given a collection of K items and a few pairwise comparisons across them, one wishes to identify the set of K items that receive the highest ranks. To tackle this problem, we adopt the logistic parametric model---the Bradley-Terry-Luce model, where each item is assigned a latent preference score, and where the outcome of each pairwise comparison depends solely on the relative scores of the two items involved. Recent works have made significant progress towards characterizing the performance (eg the mean square error for estimating the scores) of several classical methods, including the spectral method and the maximum likelihood estimator (MLE). However, where they stand regarding top- K ranking remains unsettled.