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Register a new userBayesian High-dimensional Semi-parametric Inference beyond sub-Gaussian Errors
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Added by
Kyoungjae Lee
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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We consider a sparse linear regression model with unknown symmetric error under the high-dimensional setting. The true error distribution is assumed to belong to the locally $beta$-H{o}lder class with an exponentially decreasing tail, which does not need to be sub-Gaussian. We obtain posterior convergence rates of the regression coefficient and the error density, which are nearly optimal and adaptive to the unknown sparsity level. Furthermore, we derive the semi-parametric Bernstein-von Mises (BvM) theorem to characterize asymptotic shape of the marginal posterior for regression coefficients. Under the sub-Gaussianity assumption on the true score function, strong model selection consistency for regression coefficients are also obtained, which eventually asserts the frequentists validity of credible sets.
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We consider the semi-parametric estimation of a scale parameter of a one-dimensional Gaussian process with known smoothness. We suggest an estimator based on quadratic variations and on the moment method. We provide asymptotic approximations of the mean and variance of this estimator, together with asymptotic normality results, for a large class of Gaussian processes. We allow for general mean functions and study the aggregation of several estimators based on various variation sequences. In extensive simulation studies, we show that the asymptotic results accurately depict thefinite-sample situations already for small to moderate sample sizes. We also compare various variation sequences and highlight the efficiency of the aggregation procedure.
We discuss the possibilities and limitations of estimating the mean of a real-valued random variable from independent and identically distributed observations from a non-asymptotic point of view. In particular, we define estimators with a sub-Gaussian behavior even for certain heavy-tailed distributions. We also prove various impossibility results for mean estimators.
In this paper we discuss the estimation of a nonparametric component $f_1$ of a nonparametric additive model $Y=f_1(X_1) + ...+ f_q(X_q) + epsilon$. We allow the number $q$ of additive components to grow to infinity and we make sparsity assumptions about the number of nonzero additive components. We compare this estimation problem with that of estimating $f_1$ in the oracle model $Z= f_1(X_1) + epsilon$, for which the additive components $f_2,dots,f_q$ are known. We construct a two-step presmoothing-and-resmoothing estimator of $f_1$ and state finite-sample bounds for the difference between our estimator and some smoothing estimators $hat f_1^{text{(oracle)}}$ in the oracle model. In an asymptotic setting these bounds can be used to show asymptotic equivalence of our estimator and the oracle estimators; the paper thus shows that, asymptotically, under strong enough sparsity conditions, knowledge of $f_2,dots,f_q$ has no effect on estimation accuracy. Our first step is to estimate $f_1$ with an undersmoothed estimator based on near-orthogonal projections with a group Lasso bias correction. We then construct pseudo responses $hat Y$ by evaluating a debiased modification of our undersmoothed estimator of $f_1$ at the design points. In the second step the smoothing method of the oracle estimator $hat f_1^{text{(oracle)}}$ is applied to a nonparametric regression problem with responses $hat Y$ and covariates $X_1$. Our mathematical exposition centers primarily on establishing properties of the presmoothing estimator. We present simulation results demonstrating close-to-oracle performance of our estimator in practical applications.
Bayesian methods are actively used for parameter identification and uncertainty quantification when solving nonlinear inverse problems with random noise. However, there are only few theoretical results justifying the Bayesian approach. Recent papers, see e.g. cite{Nickl2017,lu2017bernsteinvon} and references therein, illustrate the main difficulties and challenges in studying the properties of the posterior distribution in the nonparametric setup. This paper offers a new approach for study the frequentist properties of the nonparametric Bayes procedures. The idea of the approach is to relax the nonlinear structural equation by introducing an auxiliary functional parameter and replacing the structural equation with a penalty and by imposing a prior on the auxiliary parameter. For the such extended model, we state sharp bounds on posterior concentration and on the accuracy of the penalized MLE and on Gaussian approximation of the posterior, and a number of further results. All the bounds are given in terms of effective dimension, and we show that the proposed calming device does not significantly affect this value.
Let $(Y,(X_i)_{iinmathcal{I}})$ be a zero mean Gaussian vector and $V$ be a subset of $mathcal{I}$. Suppose we are given $n$ i.i.d. replications of the vector $(Y,X)$. We propose a new test for testing that $Y$ is independent of $(X_i)_{iin mathcal{I}backslash V}$ conditionally to $(X_i)_{iin V}$ against the general alternative that it is not. This procedure does not depend on any prior information on the covariance of $X$ or the variance of $Y$ and applies in a high-dimensional setting. It straightforwardly extends to test the neighbourhood of a Gaussian graphical model. The procedure is based on a model of Gaussian regression with random Gaussian covariates. We give non asymptotic properties of the test and we prove that it is rate optimal (up to a possible $log(n)$ factor) over various classes of alternatives under some additional assumptions. Besides, it allows us to derive non asymptotic minimax rates of testing in this setting. Finally, we carry out a simulation study in order to evaluate the performance of our procedure.
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