Several large volatility matrix estimation procedures have been recently developed for factor-based It processes whose integrated volatility matrix consists of low-rank and sparse matrices. Their performance depends on the accuracy of input volatility matrix estimators. When estimating co-volatilities based on high-frequency data, one of the crucial challenges is non-synchronization for illiquid assets, which makes their co-volatility estimators inaccurate. In this paper, we study how to estimate the large integrated volatility matrix without using co-volatilities of illiquid assets. Specifically, we pretend that the co-volatilities for illiquid assets are missing, and estimate the low-rank matrix using a matrix completion scheme with a structured missing pattern. To further regularize the sparse volatility matrix, we employ the principal orthogonal complement thresholding method (POET). We also investigate the asymptotic

The effect of errors in variables in nonparametric regression estimation is examined. To account for errors in covariates, deconvolution is involved in the construction of a new class of kernel estimators. It is shown that optimal local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution. In fact, there are two types of rates of convergence according to whether the error is ordinary smooth or super smooth. It is also shown that these results hold uniformly over a class of joint distributions of the response and the covariate, which is rich enough for many practical applications. Furthermore, to achieve optimality, we show that the convergence rates of all possible estimators have a lower bound possessed by the kernel estimators.

We propose a novel technique to boost the power of testing a highdimensional vector <i>H</i>:<i><b></b></i>=<b>0</b> against sparse alternatives where the null hypothesis is violated by only a few components. Existing tests based on quadratic forms such as the Wald statistic often suffer from low powers due to the accumulation of errors in estimating highdimensional parameters. More powerful tests for sparse alternatives such as thresholding and extreme value tests, on the other hand, require either stringent conditions or bootstrap to derive the null distribution and often suffer from size distortions due to the slow convergence. Based on a screening technique, we introduce a power enhancement component, which is zero under the null hypothesis with high probability, but diverges quickly under sparse alternatives. The proposed test statistic combines the power enhancement component with an asymptotically pivotal statistic, and

We consider the fundamental problem of solving quadratic systems of m equations in n variables. We propose a novel method, which starting with an initial guess computed by means of a spectral method, proceeds by minimizing a nonconvex functional as in the Wirtinger flow approach. There are several key distinguishing features, most notably, a distinct objective functional and novel update rules, which operate in an adaptive fashion and drop terms bearing too much influence on the search direction. These careful selection rules provide a tighter initial guess, better descent directions, and thus enhanced practical performance. On the theoretical side, we prove that for certain unstructured models of quadratic systems, our algorithms return the correct solution in linear time, i.e. in time proportional to reading the data and as soon as the ratio m/n between the number of equations and unknowns exceeds a fixed numerical constant. We extend the theory to deal with noisy systems and prove that our algorithms achieve a statistical accuracy, which is nearly un-improvable. We complement our theoretical study with numerical examples showing that solving random quadratic systems is both computationally and statistically not much harder than solving linear systems of the same size---hence the title of this paper. For instance, we demonstrate empirically that the computational cost of our algorithm is about four times that of solving a least-squares problem of the same size.

Motivated by the sampling problems and heterogeneity issues common in high-dimensional big datasets, we consider a class of discordant additive index models. We propose method of moments based procedures for estimating the indices of such discordant additive index models in both low and high-dimensional settings. Our estimators are based on factorizing certain moment tensors and are also applicable in the overcomplete setting, where the number of indices is more than the dimensionality of the datasets. Furthermore, we provide rates of convergence of our estimator in both high and low-dimensional setting. Establishing such results requires deriving tensor operator norm concentration inequalities that might be of independent interest. Finally, we provide simulation results supporting our theory. Our contributions extend the applicability of tensor methods for novel models in addition to making progress on understanding theoretical properties of such tensor methods.