In this paper, higher-order expansions for distributions and densities of powered extremes of standard normal random sequences are established under an optimal choice of normalized constants. Our findings refine the related results in Hall (1980). Furthermore, it is shown that the rate of convergence of distributions/densities of normalized extremes depends in principle on the power index.

We consider a multivariate time series model which represents a high dimensional vector process as a sum of three terms: a linear regression of some observed regressors, a linear combination of some latent and serially correlated factors, and a vector white noise. We investigate the inference without imposing stationary conditions on the target multivariate time series, the regressors and the underlying factors. Furthermore we deal with the endogeneity that there exist correlations between the observed regressors and the unobserved factors. We also consider the model with nonlinear regression term which can be approximated by a linear regression function with a large number of regressors. The convergence rates for the estimators of regression coefficients, the number of factors, factor loading space and factors are established under the settings when the dimension of time series and the number of regressors may both tend to infinity together with the sample size. The proposed method is illustrated with both simulated and real data examples.

We show that, when the double bootstrap is used to improve performance of bootstrap meth- ods for bias correction, techniques based on using a single double-bootstrap sample for each single-bootstrap sample can produce third-order accuracy for much less computational expense than is required by conventional double-bootstrap methods. However, this improved level of per- formance is not available for the single double-bootstrap methods that have been suggested to construct confidence intervals or distribution estimators.

This paper considers the maximum generalized empirical likelihood (GEL) estimation and inference on parameters identified by high dimensional moment restrictions with weakly dependent data when the dimensions of the moment restrictions and the parameters diverge along with the sample size. The consistency with rates and the asymptotic normality of the GEL estimator are obtained by properly restricting the growth rates of the dimensions of the parameters and the moment restrictions, as well as the degree of data dependence. It is shown that even in the high dimensional time series setting, the GEL ratio can still behave like a chi-square random variable asymptotically. A consistent test for the over- identification is proposed. A penalized GEL method is also provided for estimation under sparsity setting.

We study a marginal empirical likelihood approach in scenarios when the number of variables grows exponentially with the sample size. The marginal empirical likelihood ratios as functions of the parameters of interest are sys- tematically examined, and we find that the marginal empirical likelihood ratio evaluated at zero can be used to differentiate whether an explanatory vari- able is contributing to a response variable or not. Based on this finding, we propose a unified feature screening procedure for linear models and the gen- eralized linear models. Different from most existing feature screening ap- proaches that rely on the magnitudes of some marginal estimators to identify true signals, the proposed screening approach is capable of further incorpo- rating the level of uncertainties of such estimators. Such a merit inherits the self-studentization property of the empirical likelihood approach, and extends the insights of existing feature screening methods. Moreover, we show that our screening approach is less restrictive to distributional assumptions, and can be conveniently adapted to be applied in a broad range of scenarios such as models specified using general moment conditions. Our theoretical results and extensive numerical examples by simulations and data analysis demon- strate the merits of the marginal empirical likelihood approach.

The transition density of a diffusion process does not admit an explicit expression in general, which prevents the full maximum likelihood estimation (MLE) based on discretely observed sample paths. Aït-Sahalia [J. Finance 54 (1999) 1361–1395; Econometrica 70 (2002) 223–262] proposed asymptotic expansions to the transition densities of diffusion processes, which lead to an approximate maximum likelihood estimation (AMLE) for parameters. Built on Aït-Sahalia’s [Econometrica 70 (2002) 223–262; Ann. Statist. 36 (2008) 906–937] proposal and analysis on the AMLE, we establish the consistency and convergence rate of the AMLE, which reveal the roles played by the number of terms used in the asymptotic density expansions and the sampling interval between successive observations. We find conditions under which the AMLE has the same asymptotic distribution as that of the full MLE. A first order approximation to the Fisher information matrix is proposed.