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Problems of nonparametric ltering arises frequently in engineering and nancial economics. Nonparametric lters often involve some ltering parameters to choose. These parameters can be chosen to optimize the performance locally at each time point or globally over a time interval. In this article, the ltering parameters are chosen via minimizing the prediction error for a large class of lters. Under a general martingale setting, with mild conditions on the time series structure and virtually no assumption on lters, we show that the adaptive lter with ltering parameter chosen by historical data performs nearly as well as the one with the ideal lter in the class, in terms of ltering errors. The theoretical result is also veri ed via intensive simulations. Our approach is also useful for choosing the orders of parametric models such as AR or GARCH processes. It can also be applied to volatility estimation in nancial economics. We illustrate the proposed methods by estimating the volatility of the returns of the S&P500 index and the yields of the three-month Treasury bills.

The varying coefficient model is an important class of nonparametric statistical model, which allows us to examine how the effects of covariates vary with exposure variables. When the number of covariates is large, the issue of variable selection arises. In this article, we propose and investigate marginal nonparametric screening methods to screen variables in sparse ultra-high-dimensional varying coefficient models. The proposed nonparametric independence screening (NIS) selects variables by ranking a measure of the nonparametric marginal contributions of each covariate given the exposure variable. The sure independent screening property is established under some mild technical conditions when the dimensionality is of nonpolynomial order, and the dimensionality reduction of NIS is quantified. To enhance the practical utility and finite sample performance, two data-driven iterative NIS (INIS) methods are

Regularity properties such as the incoherence condition, the restricted isometry property, compatibility, restricted eigenvalue and iq sensitivity of covariate matrices play a pivotal role in high-dimensional regression and compressed sensing. Yet, like computing the spark of a matrix, we first show that it is NP-hard to check the conditions involving all submatrices of a given size.

In this paper, we propose a penalised pseudo-partial likelihood method for variable selection with multivariate failure time data with a growing number of regression coefficients. Under certain regularity conditions, we show the consistency and asymptotic normality of the penalised likelihood estimators. We further demonstrate that, for certain penalty functions with proper choices of regularisation parameters, the resulting estimator can correctly identify the true model, as if it were known in advance. Based on a simple approximation of the penalty function, the proposed method can be easily carried out with the NewtonRaphson algorithm. We conduct extensive Monte Carlo simulation studies to assess the finite sample performance of the proposed procedures. We illustrate the proposed method by analysing a dataset from the Framingham Heart Study.