This paper studies a weighted local linear regression smoother for longitudinal/clustered data, which takes a form similar to the classical weighted least squares estimate. As a hybrid of the methods of Chen and Jin (2005) and Wang (2003), the proposed local linear smoother maintains the advantages of both methods in computational and theoretical simplicity, variance minimization and bias reduction. Moreover, the proposed smoother is optimal in the sense that it attains linear minimax efficiency when the within-cluster correlation is correctly specified. In the special case that the joint density of covariates in a cluster exists and is continuous, any working within-cluster correlation would lead to linear minimax efficiency for the proposed method.

We consider diffusivity of random walks with transition probabilities depending on the number of consecutive traversals of the last traversed edge, the so called senile reinforced random walk (SeRW). In one dimension, the walk is known to be sub-diffusive with identity reinforcement function. We perturb the model by introducing a small probability \delta of escaping the last traversed edge at each step. The perturbed SeRW model is diffusive for any \delta , with enhanced diffusivity (\delta ) in the small \delta regime. We further study stochastically perturbed SeRW models by having the last edge escape probability of the form \delta with \delta 's being independent random variables. Enhanced diffusivity in such models are logarithmically close to the so called residual diffusivity (positive in the zero \delta limit), with diffusivity between \delta and \delta . Finally, we generalize our results to higher dimensions where the unperturbed model is already diffusive. The enhanced diffusivity can be as much as \delta .

The problem of estimating the density function and the regression function involving errors-in-variables in time series is considered. Under appropriate conditions, it is shown that the rates obtained in Fan (1991), Fan and Truong (1990) are also achievable in the context of dependent observations. Consequently, the results presented here extend our previous results for cross-sectional data to the longitudinal ones. oAbbreviated title. Measurement Errors in Time Series AMS 1980 subject classification. Primary 62G20. Secondary 62G05, 62J99.

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This article proposes a consistent and efficient estimator of the high-frequency covariance (quadratic covariation) of two arbitrary assets, observed asynchronously with market microstructure noise. This estimator is built on the marriage of the quasimaximum likelihood estimator of the quadratic variation and the proposed generalized synchronization scheme and thus is not influenced by the Epps effect. Moreover, the estimation procedure is free of tuning parameters or bandwidths and is readily implementable. Monte Carlo simulations show the advantage of this estimator by comparing it with a variety of estimators with specific synchronization methods. The empirical studies of six foreign exchange future contracts illustrate the time-varying correlations of the currencies during the 2008 global financial crisis, demonstrating the similarities and differences in their roles as key currencies in the global market.