The Experts below are selected from a list of 360 Experts worldwide ranked by ideXlab platform
Michael Nussbaum - One of the best experts on this subject based on the ideXlab platform.
-
Asymptotic Equivalence of spectral density estimation and gaussian white noise
Annals of Statistics, 2010Co-Authors: Georgi K Golubev, Michael Nussbaum, Harrison H. ZhouAbstract:We consider the statistical experiment given by a sample y(1 ), ... , y(n) of a stationary Gaussian process with an unknown smooth spectral density f . Asymptotic Equivalence, in the sense of Le Cam's deficiency Δ-distance, to two Gaussian experiments with simpler structure is established. The first one is given by independent zero mean Gaussians with variance approximately f (ω i ), where ω i is a uniform grid of points in (-π, π) (nonparametric Gaussian scale regression). This approximation is closely related to well-known Asymptotic independence results for the periodogram and corresponding inference methods. The second Asymptotic Equivalence is to a Gaussian white noise model where the drift function is the log-spectral density. This represents the step from a Gaussian scale model to a location model, and also has a counterpart in established inference methods, that is, log-periodogram regression. The problem of simple explicit Equivalence maps (Markov kernels), allowing to directly carry over inference, appears in this context but is not solved here.
-
Asymptotic Equivalence of spectral density estimation and gaussian white noise
arXiv: Statistics Theory, 2009Co-Authors: Georgi K Golubev, Michael Nussbaum, Harrison H. ZhouAbstract:We consider the statistical experiment given by a sample of a stationary Gaussian process with an unknown smooth spectral density f. Asymptotic Equivalence, in the sense of Le Cam's deficiency Delta-distance, to two Gaussian experiments with simpler structure is established. The first one is given by independent zero mean Gaussians with variance approximately the value of f in points of a uniform grid (nonparametric Gaussian scale regression). This approximation is closely related to well-known Asymptotic independence results for the periodogram and corresponding inference methods. The second Asymptotic Equivalence is to a Gaussian white noise model where the drift function is the log-spectral density. This represents the step from a Gaussian scale model to a location model, and also has a counterpart in established inference methods, i.e. log-periodogram regression. The problem of simple explicit Equivalence maps (Markov kernels), allowing to directly carry over inference, appears in this context but is not solved here.
-
Asymptotic Equivalence for nonparametric generalized linear models
Probability Theory and Related Fields, 1998Co-Authors: Ion Grama, Michael NussbaumAbstract:We establish that a non-Gaussian nonparametric regression model is Asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then Asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value f(ti) of a regression function f at a grid point ti (nonparametric GLM). When f is in a Holder ball with exponent \(\) we establish global Asymptotic Equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process.
-
constructive Asymptotic Equivalence of density estimation and gaussian white noise
Research Papers in Economics, 1998Co-Authors: Michael Nussbaum, Jussi KlemelaAbstract:A recipe is provided for producing, from a sequence of procedures in the Gaussian regression model, an Asymptotically equivalent sequence in the density estimation model with i. i. d. observations. The recipe is, to put it roughly, to calculate square roots of normalised frequencies over certain intervals, add a small random distortion, and pretend these to be observations from a Gaussian discrete regression model.
-
Asymptotic Equivalence of density estimation and gaussian white noise
Annals of Statistics, 1996Co-Authors: Michael NussbaumAbstract:Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a pure form. The Equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance Δ would make it precise. The models are then Asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression. We consider the analogous problem for the experiment given by n i.i.d. observations having density f on the unit interval. Our basic result concerns the parameter space of densities which are in a Holder ball with exponent α > 1/2 and which are uniformly bounded away from zero. We show that an i. i. d. sample of size n with density f is globally Asymptotically equivalent to a white noise experiment with drift f l/2 and variance 1/4n -l . This represents a nonparametric analog of Le Cam's heteroscedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques related to the Hungarian construction. White noise models on f and log f are also considered, allowing for various automatic Asymptotic risk bounds in the i.i.d. model from white noise.
Harrison H. Zhou - One of the best experts on this subject based on the ideXlab platform.
-
Asymptotic Equivalence of spectral density estimation and gaussian white noise
Annals of Statistics, 2010Co-Authors: Georgi K Golubev, Michael Nussbaum, Harrison H. ZhouAbstract:We consider the statistical experiment given by a sample y(1 ), ... , y(n) of a stationary Gaussian process with an unknown smooth spectral density f . Asymptotic Equivalence, in the sense of Le Cam's deficiency Δ-distance, to two Gaussian experiments with simpler structure is established. The first one is given by independent zero mean Gaussians with variance approximately f (ω i ), where ω i is a uniform grid of points in (-π, π) (nonparametric Gaussian scale regression). This approximation is closely related to well-known Asymptotic independence results for the periodogram and corresponding inference methods. The second Asymptotic Equivalence is to a Gaussian white noise model where the drift function is the log-spectral density. This represents the step from a Gaussian scale model to a location model, and also has a counterpart in established inference methods, that is, log-periodogram regression. The problem of simple explicit Equivalence maps (Markov kernels), allowing to directly carry over inference, appears in this context but is not solved here.
-
Asymptotic Equivalence and Adaptive Estimation for Robust Nonparametric Regression
Annals of Statistics, 2009Co-Authors: Harrison H. ZhouAbstract: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.
-
Asymptotic Equivalence of spectral density estimation and gaussian white noise
arXiv: Statistics Theory, 2009Co-Authors: Georgi K Golubev, Michael Nussbaum, Harrison H. ZhouAbstract:We consider the statistical experiment given by a sample of a stationary Gaussian process with an unknown smooth spectral density f. Asymptotic Equivalence, in the sense of Le Cam's deficiency Delta-distance, to two Gaussian experiments with simpler structure is established. The first one is given by independent zero mean Gaussians with variance approximately the value of f in points of a uniform grid (nonparametric Gaussian scale regression). This approximation is closely related to well-known Asymptotic independence results for the periodogram and corresponding inference methods. The second Asymptotic Equivalence is to a Gaussian white noise model where the drift function is the log-spectral density. This represents the step from a Gaussian scale model to a location model, and also has a counterpart in established inference methods, i.e. log-periodogram regression. The problem of simple explicit Equivalence maps (Markov kernels), allowing to directly carry over inference, appears in this context but is not solved here.
Alex Samarov - One of the best experts on this subject based on the ideXlab platform.
-
Asymptotic Equivalence of nonparametric regression and white noise model has its limits
Statistics & Probability Letters, 1996Co-Authors: Sam Efromovich, Alex SamarovAbstract:Abstract One of the common themes of the modern nonparametric functional estimation folklore is that the signal-in-white-noise model can serve as a prototype for nonparametric regression. In this note we give an example showing that it may not always be the case
Yingying Fan - One of the best experts on this subject based on the ideXlab platform.
-
Asymptotic Equivalence of regularization methods in thresholded parameter space
Journal of the American Statistical Association, 2013Co-Authors: Yingying FanAbstract:High-dimensional data analysis has motivated a spectrum of regularization methods for variable selection and sparse modeling, with two popular methods being convex and concave ones. A long debate has taken place on whether one class dominates the other, an important question both in theory and to practitioners. In this article, we characterize the Asymptotic Equivalence of regularization methods, with general penalty functions, in a thresholded parameter space under the generalized linear model setting, where the dimensionality can grow exponentially with the sample size. To assess their performance, we establish the oracle inequalities—as in Bickel, Ritov, and Tsybakov (2009)—of the global minimizer for these methods under various prediction and variable selection losses. These results reveal an interesting phase transition phenomenon. For polynomially growing dimensionality, the L 1-regularization method of Lasso and concave methods are Asymptotically equivalent, having the same convergence rates in the...
-
Asymptotic Equivalence of regularization methods in thresholded parameter space
Journal of the American Statistical Association, 2013Co-Authors: Yingying FanAbstract:High-dimensional data analysis has motivated a spectrum of regularization methods for variable selection and sparse modeling, with two popular classes of convex ones and concave ones. A long debate has been on whether one class dominates the other, an important question both in theory and to practitioners. In this paper, we characterize the Asymptotic Equivalence of regularization methods, with general penalty functions, in a thresholded parameter space under the generalized linear model setting, where the dimensionality can grow up to exponentially with the sample size. To assess their performance, we establish the oracle inequalities, as in Bickel, Ritov and Tsybakov (2009), of the global minimizer for these methods under various prediction and variable selection losses. These results reveal an interesting phase transition phenomenon. For polynomially growing dimensionality, the $L_1$-regularization method of Lasso and concave methods are Asymptotically equivalent, having the same convergence rates in the oracle inequalities. For exponentially growing dimensionality, concave methods are Asymptotically equivalent but have faster convergence rates than the Lasso. We also establish a stronger property of the oracle risk inequalities of the regularization methods, as well as the sampling properties of computable solutions. Our new theoretical results are illustrated and justified by simulation and real data examples.
Markus Reis - One of the best experts on this subject based on the ideXlab platform.
-
Asymptotic Equivalence and sufficiency for volatility estimation under microstructure noise
arXiv: Statistics Theory, 2011Co-Authors: Markus ReisAbstract:The basic model for high-frequency data in finance is considered, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam's sense Asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function $\sigma$. As an application, simple rate-optimal estimators of the volatility and efficient estimators of the integrated volatility are constructed.
-
Asymptotic Equivalence for inference on the volatility from noisy observations
Annals of Statistics, 2011Co-Authors: Markus ReisAbstract:We consider discrete-time observations of a continuous martingale under measurement error. This serves as a fundamental model for high-frequency data in finance, where an efficient price process is observed under microstructure noise. It is shown that this nonparametric model is in Le Cam’s sense Asymptotically equivalent to a Gaussian shift experiment in terms of the square root of the volatility function σ and a nonstandard noise level. As an application, new rate-optimal estimators of the volatility function and simple efficient estimators of the integrated volatility are constructed.
-
Asymptotic Equivalence for nonparametric regression with non regular errors
arXiv: Statistics Theory, 2011Co-Authors: Alexander Meister, Markus ReisAbstract:Asymptotic Equivalence in Le Cam's sense for nonparametric regression experiments is extended to the case of non-regular error densities, which have jump discontinuities at their endpoints. We prove Asymptotic Equivalence of such regression models and the observation of two independent Poisson point processes which contain the target curve as the support boundary of its intensity function. The intensity of the point processes is of order of the sample size $n$ and involves the jump sizes as well as the design density. The statistical model significantly differs from regression problems with Gaussian or regular errors, which are known to be Asymptotically equivalent to Gaussian white noise models.
-
Asymptotic statistical Equivalence for scalar ergodic diffusions
Probability Theory and Related Fields, 2006Co-Authors: Arnak S Dalalyan, Markus ReisAbstract:For scalar diffusion models with unknown drift function Asymptotic Equivalence in the sense of Le Cam's deficiency between statistical experiments is considered under long-time Asymptotics. A local Asymptotic Equivalence result is established with an accompanying sequence of simple Gaussian shift experiments. Corresponding globally Asymptotically equivalent experiments are obtained as compound experiments. The results are extended in several directions including time discretisation. An explicit transformation of decision functions from the Gaussian to the diffusion experiment is constructed.
