Smooth backfitting in generalized additive models

Generalized additive models have been popular among statisticians and data analysts in multivariate nonparametric regression with non-Gaussian responses including binary and count data. In this paper, a new likelihood approach for fitting generalized additive models is proposed. It aims to maximize a smoothed likelihood. The additive functions are estimated by solving a system of nonlinear integral equations. An iterative algorithm based on smooth backfitting is developed from the Newton--Kantorovich theorem. Asymptotic properties of the estimator and convergence of the algorithm are discussed. It is shown that our proposal based on local linear fit achieves the same bias and variance as the oracle estimator that uses knowledge of the other components. Numerical comparison with the recently proposed two-stage estimator [Ann. Statist. 32 (2004) 2412--2443] is also made.Comment: Published in at http://dx.doi.org/10.1214/009053607000000596 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Tyler's Covariance Matrix Estimator in Elliptical Models with Convex Structure

We address structured covariance estimation in elliptical distributions by assuming that the covariance is a priori known to belong to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the General Method of Moments (GMM) optimization applied to robust Tyler's scatter M-estimator subject to these convex constraints. Unfortunately, GMM turns out to be non-convex due to the objective. Instead, we propose a new COCA estimator - a convex relaxation which can be efficiently solved. We prove that the relaxation is tight in the unconstrained case for a finite number of samples, and in the constrained case asymptotically. We then illustrate the advantages of COCA in synthetic simulations with structured compound Gaussian distributions. In these examples, COCA outperforms competing methods such as Tyler's estimator and its projection onto the structure set.Comment: arXiv admin note: text overlap with arXiv:1311.059

A Generalized Framework on Beamformer Design and CSI Acquisition for Single-Carrier Massive MIMO Systems in Millimeter Wave Channels

In this paper, we establish a general framework on the reduced dimensional channel state information (CSI) estimation and pre-beamformer design for frequency-selective massive multiple-input multiple-output MIMO systems employing single-carrier (SC) modulation in time division duplex (TDD) mode by exploiting the joint angle-delay domain channel sparsity in millimeter (mm) wave frequencies. First, based on a generic subspace projection taking the joint angle-delay power profile and user-grouping into account, the reduced rank minimum mean square error (RR-MMSE) instantaneous CSI estimator is derived for spatially correlated wideband MIMO channels. Second, the statistical pre-beamformer design is considered for frequency-selective SC massive MIMO channels. We examine the dimension reduction problem and subspace (beamspace) construction on which the RR-MMSE estimation can be realized as accurately as possible. Finally, a spatio-temporal domain correlator type reduced rank channel estimator, as an approximation of the RR-MMSE estimate, is obtained by carrying out least square (LS) estimation in a proper reduced dimensional beamspace. It is observed that the proposed techniques show remarkable robustness to the pilot interference (or contamination) with a significant reduction in pilot overhead

Covariance Estimation in Elliptical Models with Convex Structure

We address structured covariance estimation in Elliptical distribution. We assume it is a priori known that the covariance belongs to a given convex set, e.g., the set of Toeplitz or banded matrices. We consider the General Method of Moments (GMM) optimization subject to these convex constraints. Unfortunately, GMM is still non-convex due to objective. Instead, we propose COCA - a convex relaxation which can be efficiently solved. We prove that the relaxation is tight in the unconstrained case for a finite number of samples, and in the constrained case asymptotically. We then illustrate the advantages of COCA in synthetic simulations with structured Compound Gaussian distributions. In these examples, COCA outperforms competing methods as Tyler's estimate and its projection onto a convex set

High-dimensional estimation with geometric constraints

Consider measuring an n-dimensional vector x through the inner product with several measurement vectors, a_1, a_2, ..., a_m. It is common in both signal processing and statistics to assume the linear response model y_i = + e_i, where e_i is a noise term. However, in practice the precise relationship between the signal x and the observations y_i may not follow the linear model, and in some cases it may not even be known. To address this challenge, in this paper we propose a general model where it is only assumed that each observation y_i may depend on a_i only through . We do not assume that the dependence is known. This is a form of the semiparametric single index model, and it includes the linear model as well as many forms of the generalized linear model as special cases. We further assume that the signal x has some structure, and we formulate this as a general assumption that x belongs to some known (but arbitrary) feasible set K. We carefully detail the benefit of using the signal structure to improve estimation. The theory is based on the mean width of K, a geometric parameter which can be used to understand its effective dimension in estimation problems. We determine a simple, efficient two-step procedure for estimating the signal based on this model -- a linear estimation followed by metric projection onto K. We give general conditions under which the estimator is minimax optimal up to a constant. This leads to the intriguing conclusion that in the high noise regime, an unknown non-linearity in the observations does not significantly reduce one's ability to determine the signal, even when the non-linearity may be non-invertible. Our results may be specialized to understand the effect of non-linearities in compressed sensing.Comment: This version incorporates minor revisions suggested by referee