We treat all the bivariate lack-of-memory (BLM) distributions in a unified approach and develop some new general properties of the BLM distributions, including joint moment generating function, product moments, and dependence structure. Necessary and sufficient conditions for the survival functions of BLM distributions to be totally positive of order two are given. Some previous results about specific BLM distributions are improved. In particular, we show that both the Marshall–Olkin survival copula and survival function are totally positive of all orders, regardless of parameters. Besides, we point out that Slepian’s inequality also holds true for BLM distributions.

We propose to model multivariate volatility processes on the basis of the newly defined conditionally uncorrelated components (CUCs). This model represents a parsimonious representation for matrixvalued processes. It is flexible in the sense that each CUC may be fitted separately with any appropriate univariate volatility model. Computationally it splits one high dimensional optimization problem into several lower dimensional subproblems. Consistency for the estimated CUCs has been established. A bootstrap method is proposed for testing the existence of CUCs. The methodology proposed is illustrated with both simulated and real data sets.

Largescale multiple testing with correlated test statistics arises frequently in much scientific research. Incorporating correlation information in approximating the false discovery proportion (FDP) has attracted increasing attention in recent years. When the covariance matrix of test statistics is known, Fan and his colleagues provided an accurate approximation of the FDP under arbitrary dependence structure and some sparsity assumption. However, the covariance matrix is often unknown in many applications and such dependence information must be estimated before approximating the FDP. The estimation accuracy can greatly affect the FDP approximation. In the current paper, we study theoretically the effect of unknown dependence on the testing procedure and establish a general framework such that the FDP can be well approximated. The effects of unknown dependence on approximating the FDP are in the

Abstract Markowitz (1952, 1959) laid down the ground-breaking work on mean-variance analysis without gross exposure constraints. Under this framework, the theoretical optimal allocation vector can be different from the estimated one due to intrinsic difficulty of estimating a large covariance matrix and return vector. This can result in adverse performance in portfolio selected based on empirical data due to noise accumulation on estimation errors (Jagannathan and Ma, 2003; Fan, Fan and Lv, 2008). We address this problem by introducing the gross-exposure constrained mean-variance portfolio selection. We show that with gross-exposure constraint the theoretical optimal portfolios have similar performance as empirically selected ones based on estimated covariance matrices and there is no noise accumulation effect from estimation of covariance matrices. This gives theoretical justification to the empirical results

We extend the idea of crossvalidation to choose the smoothing parameters of the double-kernel local linear regression for estimating a conditional density. Our selection rule optimises the estimated conditional density function by minimising the integrated squared error. We also discuss three other bandwidth selection rules, an ad hoc method used by Fan et al. (1996), a bootstrap method of Hall et al. (1999) for bandwidth selection in the estimation of conditional distribution functions, modified by Bashtannyk & Hyndman (2001) to cover conditional density functions, and finally a simple approach proposed by Hyndman & Yao (2002). The performance of the new approach is compared with these three methods by simulation studies, and our method performs outstandingly well. The method is illustrated by an application to estimating the transition density and the Value-at-Risk of treasury-bill data.