Asymptotic equivalence and adaptive estimation for robust nonparametric regression

Asymptotic equivalence theory developed in the literature so far are only for bounded loss functions. This limits the potential applications of the theory because many commonly used loss functions in statistical inference are unbounded. In this paper we develop asymptotic equivalence results for robust nonparametric regression with unbounded loss functions. The results imply that all the Gaussian nonparametric regression procedures can be robustified in a unified way. A key step in our equivalence argument is to bin the data and then take the median of each bin. The asymptotic equivalence results have significant practical implications. To illustrate the general principles of the equivalence argument we consider two important nonparametric inference problems: robust estimation of the regression function and the estimation of a quadratic functional. In both cases easily implementable procedures are constructed and are shown to enjoy simultaneously a high degree of robustness and adaptivity. Other problems such as construction of confidence sets and nonparametric hypothesis testing can be handled in a similar fashion.Comment: Published in at http://dx.doi.org/10.1214/08-AOS681 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Minimax estimation with thresholding and its application to wavelet analysis

Many statistical practices involve choosing between a full model and reduced models where some coefficients are reduced to zero. Data were used to select a model with estimated coefficients. Is it possible to do so and still come up with an estimator always better than the traditional estimator based on the full model? The James-Stein estimator is such an estimator, having a property called minimaxity. However, the estimator considers only one reduced model, namely the origin. Hence it reduces no coefficient estimator to zero or every coefficient estimator to zero. In many applications including wavelet analysis, what should be more desirable is to reduce to zero only the estimators smaller than a threshold, called thresholding in this paper. Is it possible to construct this kind of estimators which are minimax? In this paper, we construct such minimax estimators which perform thresholding. We apply our recommended estimator to the wavelet analysis and show that it performs the best among the well-known estimators aiming simultaneously at estimation and model selection. Some of our estimators are also shown to be asymptotically optimal.Comment: Published at http://dx.doi.org/10.1214/009053604000000977 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Law of Log Determinant of Sample Covariance Matrix and Optimal Estimation of Differential Entropy for High-Dimensional Gaussian Distributions

Differential entropy and log determinant of the covariance matrix of a multivariate Gaussian distribution have many applications in coding, communications, signal processing and statistical inference. In this paper we consider in the high dimensional setting optimal estimation of the differential entropy and the log-determinant of the covariance matrix. We first establish a central limit theorem for the log determinant of the sample covariance matrix in the high dimensional setting where the dimension $p(n)$ can grow with the sample size $n$. An estimator of the differential entropy and the log determinant is then considered. Optimal rate of convergence is obtained. It is shown that in the case $p(n)/n \rightarrow 0$ the estimator is asymptotically sharp minimax. The ultra-high dimensional setting where $p(n) > n$ is also discussed.Comment: 19 page

Optimal rates of convergence for covariance matrix estimation

Covariance matrix plays a central role in multivariate statistical analysis. Significant advances have been made recently on developing both theory and methodology for estimating large covariance matrices. However, a minimax theory has yet been developed. In this paper we establish the optimal rates of convergence for estimating the covariance matrix under both the operator norm and Frobenius norm. It is shown that optimal procedures under the two norms are different and consequently matrix estimation under the operator norm is fundamentally different from vector estimation. The minimax upper bound is obtained by constructing a special class of tapering estimators and by studying their risk properties. A key step in obtaining the optimal rate of convergence is the derivation of the minimax lower bound. The technical analysis requires new ideas that are quite different from those used in the more conventional function/sequence estimation problems.Comment: Published in at http://dx.doi.org/10.1214/09-AOS752 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Nonparametric regression in exponential families

Most results in nonparametric regression theory are developed only for the case of additive noise. In such a setting many smoothing techniques including wavelet thresholding methods have been developed and shown to be highly adaptive. In this paper we consider nonparametric regression in exponential families with the main focus on the natural exponential families with a quadratic variance function, which include, for example, Poisson regression, binomial regression and gamma regression. We propose a unified approach of using a mean-matching variance stabilizing transformation to turn the relatively complicated problem of nonparametric regression in exponential families into a standard homoscedastic Gaussian regression problem. Then in principle any good nonparametric Gaussian regression procedure can be applied to the transformed data. To illustrate our general methodology, in this paper we use wavelet block thresholding to construct the final estimators of the regression function. The procedures are easily implementable. Both theoretical and numerical properties of the estimators are investigated. The estimators are shown to enjoy a high degree of adaptivity and spatial adaptivity with near-optimal asymptotic performance over a wide range of Besov spaces. The estimators also perform well numerically.Comment: Published in at http://dx.doi.org/10.1214/09-AOS762 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robust nonparametric estimation via wavelet median regression

In this paper we develop a nonparametric regression method that is simultaneously adaptive over a wide range of function classes for the regression function and robust over a large collection of error distributions, including those that are heavy-tailed, and may not even possess variances or means. Our approach is to first use local medians to turn the problem of nonparametric regression with unknown noise distribution into a standard Gaussian regression problem and then apply a wavelet block thresholding procedure to construct an estimator of the regression function. It is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. A key technical result in our development is a quantile coupling theorem which gives a tight bound for the quantile coupling between the sample medians and a normal variable. This median coupling inequality may be of independent interest.Comment: Published in at http://dx.doi.org/10.1214/07-AOS513 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Clear air turbulence

Research on forecasting, detection, and incidents of clear air turbulenc

Absence of Hybridization Gap in Heavy Electron Systems and Analysis of YbAl3 in terms of Nearly Free Electron Conduction Band

In the analysis of the heavy electron systems, theoretical models with c-f hybridization gap are often used. We point out that such a gap does not exist and the simple picture with the hybridization gap is misleading in the metallic systems, and present a correct picture by explicitly constructing an effective band model of YbAl_3. Hamiltonian consists of a nearly free electron model for conduction bands which hybridize with localized f-electrons, and includes only a few parameters. Density of states, Sommerfeld coefficient, f-electron number and optical conductivity are calculated and compared with the band calculations and the experiments.Comment: 9 pages, 9 figures, submitted to J. Phys. Soc. Jp

S=1/2 Kagome antiferromagnets Cs2_2Cu3MF_3MF_{12}$ with M=Zr and Hf

Magnetization and specific heat measurements have been carried out on Cs$_2$Cu$_3$ZrF$_{12}$ and Cs$_2$Cu$_3$HfF$_{12}$ single crystals, in which Cu$^{2+}$ ions with spin-1/2 form a regular Kagom\'{e} lattice. The antiferromagnetic exchange interaction between neighboring Cu$^{2+}$ spins is $J/k_{\rm B}\simeq 360$ K and 540 K for Cs$_2$Cu$_3$ZrF$_{12}$ and Cs$_2$Cu$_3$HfF$_{12}$, respectively. Structural phase transitions were observed at $T_{\rm t}\simeq 210$ K and 175 K for Cs$_2$Cu$_3$ZrF$_{12}$ and Cs$_2$Cu$_3$HfF$_{12}$, respectively. The specific heat shows a small bend anomaly indicative of magnetic ordering at $T_\mathrm{N}= 23.5$ K and 24.5 K in Cs$_2$Cu$_3$ZrF$_{12}$ and Cs$_2$Cu$_3$HfF$_{12}$, respectively. Weak ferromagnetic behavior was observed below $T_\mathrm{N}$. This weak ferromagnetism should be ascribed to the antisymmetric interaction of the Dzyaloshinsky-Moriya type that are generally allowed in the Kagom\'{e} lattice.Comment: 6 pages, 4 figure. Conference proceeding of Highly Frustrated Magnetism 200

Angle-resolved photoemission spectroscopy of perovskite-type transition-metal oxides and their analyses using tight-binding band structure

Nowadays it has become feasible to perform angle-resolved photoemission spectroscopy (ARPES) measurements of transition-metal oxides with three-dimensional perovskite structures owing to the availability of high-quality single crystals of bulk and epitaxial thin films. In this article, we review recent experimental results and interpretation of ARPES data using empirical tight-binding band-structure calculations. Results are presented for SrVO$_3$ (SVO) bulk single crystals, and La$_{1-x}$Sr$_x$FeO$_3$ (LSFO) and La$_{1-x}$Sr$_x$MnO$_3$ (LSMO) thin films. In the case of SVO, from comparison of the experimental results with calculated surface electronic structure, we concluded that the obtained band dispersions reflect the bulk electronic structure. The experimental band structures of LSFO and LSMO were analyzed assuming the G-type antiferromagnetic state and the ferromagnetic state, respectively. We also demonstrated that the intrinsic uncertainty of the electron momentum perpendicular to the crystal surface is important for the interpretation of the ARPES results of three-dimensional materials.Comment: 25 pages, 12 figure