For multiple index models, it has recently been shown that the sliced inverse regression (SIR) is consistent for estimating the sufficient dimension reduction (SDR) subspace if and only if the dimension p and sample size n satisfies that ρ=limp/n=0. Thus, when p is of the same or a higher order of n, additional assumptions such as sparsity have to be imposed in order to ensure consistency for SIR. By constructing artificial response variables made up from top eigenvectors of the estimated conditional covariance matrix, \widehat{var(𝔼[x|y])}, we introduce a simple Lasso regression method to obtain an estimate of the SDR subspace. The resulting algorithm, Lasso-SIR, is shown to be consistent and achieve the optimal convergence rate under certain sparsity conditions when p is of order o(n^2λ^2) where λ is the generalized signal noise ratio. We also demonstrate the superior performance of Lasso-SIR compared with existing approaches via extensive numerical studies and several real data examples.

In deconvolution in R^d, d≥1, with mixing density p(∈\mathscr{P}) and kernel h, the mixture density f_p(∈\mathscr{F}_p) is estimated with MDE f_{\hat{p}n}, having upper L1-error rate, an, in probability or in risk; \hat{p}_n∈\mathscr{P}. In one application, \mathscr{P} consists of L1-separable densities in R with differences changing sign at most J times and h(x−y) Totally Positive. When h is known and p is \tilde{q}-smooth, vanishing outside a compact in R^d, plug-in upper bounds are provided for the L2-error rate of \hat{p}_n and its [s]-th mixed partial derivative \hat{p}^{(s)}_n, via ∥f_{\hat{p}_n}−f_p∥_1, with rates (log a^{−1}_n)^{−N_1} and a^{N_2}_n, respectively, for h super-smooth and smooth; \tilde{q}∈R^+, [s] ≤ \tilde{q}, d≥1, N_1>0, N_2>0. For a_n ∼ (logn)^ζ⋅(n^{−δ}), the former rate is optimal for any δ>0 and the latter misses the optimal by the factor (logn)^ξ when δ=.5; ζ>0, ξ>0. N_1 and N_2 appear in optimal rates and lower error and risk bounds in the deconvolution literature.

This paper deals with the estimation of a high-dimensional covariance with a con- ditional sparsity structure and fast-diverging eigenvalues. By assuming sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the Principal Orthogonal complEment Thresholding (POET) method to explore such an approximate factor structure with sparsity. The POET estimator includes the sample covariance matrix, the factor-based covariance matrix (Fan, Fan, and Lv, 2008), the thresholding estimator (Bickel and Levina, 2008) and the adaptive thresholding estimator (Cai and Liu, 2011) as specic examples. We provide mathe- matical insights when the factor analysis is approximately the same as the principal component analysis for high-dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the impact of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the un- observed factors and their factor loadings are derived. The asymptotic results are also veried by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.

A decisive question in nonparametric smoothing techniques is the choice of the bandwidth or smoothing parameter. The present paper addresses this question when using local polynomial approximations for estimating the regression function and its derivatives. A fully-automatic bandwidth selection procedure has been proposed by Fan and Gijbels (1995a), and the empirical performance of it was tested in detail via a variety of examples. Those experiences supported the methodology towards a great extend. In this paper we establish asymptotic results for the proposed variable bandwidth selector. We provide the rate of convergence of the bandwidth estimate, and obtain the asymptotic distribution of its error relative to the theoretical optimal variable bandwidth. These asymptotic properties give extra support to the proposed bandwidth selection procedure. It is also demonstrated how the proposed selection

Double-blind randomized controlled trials are traditionally seen as the gold standard for causal inferences as the difference-in-means estimator is an unbiased estimator of the average treatment effect in the experiment. The fact that this estimator is unbiased over all possible randomizations does not, however, mean that any given estimate is close to the true treatment effect. Similarly, while predetermined covariates will be balanced between treatment and control groups on average, large imbalances may be observed in a given experiment and the researcher may therefore want to condition on such covariates using linear regression. This article studies the theoretical properties of both the difference-in-means and OLS estimators conditional on observed differences in covariates. By deriving the statistical properties of the conditional estimators, we can establish guidance for how to deal with covariate imbalances.