Let $Y$ be a Gaussian vector of $\mathbb{R}^n$ of mean $s$ and diagonal covariance matrix $Γ$. Our aim is to estimate both $s$ and the entries $σ_i=Γ_{i,i}$, for $i=1,...,n$, on the basis of the observation of two independent copies of $Y$. Our approach is free of any prior assumption on $s$ but requires that we know some upper bound $γ$ on the ratio $\max_iσ_i/\min_iσ_i$. For example, the choice $γ=1$ corresponds to the homoscedastic case where the components of $Y$ are assumed to have common (unknown) variance. In the opposite, the choice $γ>1$ corresponds to the heteroscedastic case where the variances of the components of $Y$ are allowed to vary within some range. Our estimation strategy is based on model selection. We consider a family $\{S_m\timesΣ_m, m\in\mathcal{M}\}$ of parameter sets where $S_m$ and $Σ_m$ are linear spaces. To each $m\in\mathcal{M}$, we associate a pair of estimators $(\hat{s}_m,\hatσ_m)$ of $(s,σ)$ with values in $S_m\timesΣ_m$. Then we design a model selection procedure in view of selecting some $\hat{m}$ among $\mathcal{M}$ in such a way that the Kullback risk of $(\hat{s}_{\hat{m}},\hatσ_{\hat{m}})$ is as close as possible to the minimum of the Kullback risks among the family of estimators $\{(\hat{s}_m,\hatσ_m), m\in\mathcal{M}\}$. Then we derive uniform rates of convergence for the estimator $(\hat{s}_{\hat{m}},\hatσ_{\hat{m}})$ over Hölderian balls. Finally, we carry out a simulation study in order to illustrate the performances of our estimators in practice.
Despite its prevalence in statistical datasets, heteroscedasticity (non-constant sample variances) has been largely ignored in the high-dimensional statistics literature. Recently, studies have shown that the Lasso can accommodate heteroscedastic errors, with minor algorithmic modifications (Belloni et al., 2012; Gautier and Tsybakov, 2013). In this work, we study heteroscedastic regression with linear mean model and log-linear variances model with sparse high-dimensional parameters. In this work, we propose estimating variances in a post-Lasso fashion, which is followed by weighted-least squa…
We consider a wavelet thresholding approach to adaptive variance function estimation in heteroscedastic nonparametric regression. A data-driven estimator is constructed by applying wavelet thresholding to the squared first-order differences of the observations. We show that the variance function estimator is nearly optimally adaptive to the smoothness of both the mean and variance functions. The estimator is shown to achieve the optimal adaptive rate of convergence under the pointwise squared error simultaneously over a range of smoothness classes. The estimator is also adaptively within a log…
If uncorrelated random variables have a common expected value and decreasing variances then the variance of a sample mean is decreasing with the number of observations. Unfortunately, this natural and desirable Variance Reduction Property (VRP) by augmenting data is not automatically inherited by Ordinary Least Squares (OLS) estimators of parameters. In the paper we find conditions for the OLS to have the VRP. In the case of a straight line regression we show that the OLS estimators of intercept and slope have the VRP if the design points are increasing. This also holds true for alternating tw…
Modern data are increasingly both high-dimensional and heteroscedastic. This paper considers the challenge of estimating underlying principal components from high-dimensional data with noise that is heteroscedastic across samples, i.e., some samples are noisier than others. Such heteroscedasticity naturally arises, e.g., when combining data from diverse sources or sensors. A natural way to account for this heteroscedasticity is to give noisier blocks of samples less weight in PCA by using the leading eigenvectors of a weighted sample covariance matrix. We consider the problem of choosing weigh…
Heteroscedasticity is common in real world applications and is often handled by incorporating case weights into a modeling procedure. Intuitively, models fitted with different weight schemes would have a different level of complexity depending on how well the weights match the inverse of error variances. However, existing statistical theories on model complexity, also known as model degrees of freedom, were primarily established under the assumption of equal error variances. In this work, we focus on linear regression procedures and seek to extend the existing measures to a heteroscedastic set…
An adaptive nonparametric estimation procedure is constructed for heteroscedastic regression when the noise variance depends on the unknown regression. A non-asymptotic upper bound for a quadratic risk (oracle inequality) is obtained
An adaptive nonparametric estimation procedure is constructed for the estimation problem of heteroscedastic regression when the noise variance depends on the unknown regression. A non-asymptotic upper bound for a quadratic risk (an oracle inequality) is constructed.
Principal Component Analysis (PCA) is a classical method for reducing the dimensionality of data by projecting them onto a subspace that captures most of their variation. Effective use of PCA in modern applications requires understanding its performance for data that are both high-dimensional and heteroscedastic. This paper analyzes the statistical performance of PCA in this setting, i.e., for high-dimensional data drawn from a low-dimensional subspace and degraded by heteroscedastic noise. We provide simplified expressions for the asymptotic PCA recovery of the underlying subspace, subspace a…
We consider the common nonlinear regression model where the variance, as well as the mean, is a parametric function of the explanatory variables. The $c$-optimal design problem is investigated in the case when the parameters of both the mean and the variance function are of interest. A geometric characterization of $c$-optimal designs in this context is presented, which generalizes the classical result of Elfving [Ann. Math. Statist. 23 (1952) 255--262] for $c$-optimal designs. As in Elfving's famous characterization, $c$-optimal designs can be described as representations of boundary points o…
In this paper we consider a heteroscedastic transformation model, where the transformation belongs to a parametric family of monotone transformations, the regression and variance function are modelled nonparametrically and the error is independent of the multidimensional covariates. In this model, we first consider the estimation of the unknown components of the model, namely the transformation parameter, regression and variance function and the distribution of the error. We show the asymptotic normality of the proposed estimators. Second, we propose tests for the validity of the model, and es…
In many real applications, the distribution of measurement error could vary with each subject or even with each observation so the errors are heteroscedastic. In this paper, we propose a fast algorithm using a simulation-extrapolation (SIMEX) method to recover the unknown density in the case of heteroscedastic contamination. We show the consistency of the estimator and obtain its asymptotic variance and then address the practical selection of the smoothing parameter. We demonstrate that, through a finite sample simulation study, the proposed method performs better than the Fourier-type deconvo…
We propose a new asymptotic test to assess the stationarity of a time series' mean that is applicable in the presence of both heteroscedasticity and short-range dependence. Our test statistic is composed of Gini's mean difference of local sample means. To analyse its asymptotic behaviour, we develop new limit theory for U-statistics of strongly mixing triangular arrays under non-stationarity. Most importantly, we show asymptotic normality of the test statistic under the hypothesis of a constant mean and prove the test's consistency against a very general class of alternatives, including both s…
Consider the problem of estimating a multivariate normal mean with a known variance matrix, which is not necessarily proportional to the identity matrix. The coordinates are shrunk directly in proportion to their variances in Efron and Morris' (J. Amer. Statist. Assoc. 68 (1973) 117-130) empirical Bayes approach, whereas inversely in proportion to their variances in Berger's (Ann. Statist. 4 (1976) 223-226) minimax estimators. We propose a new minimax estimator, by approximately minimizing the Bayes risk with a normal prior among a class of minimax estimators where the shrinkage direction is o…
We consider the problem of designing experiments to detect the presence of a specified heteroscedastity in a non-linear Gaussian regression model. In this framework, we focus on the ${\rm D}_s$- and KL-criteria and study their relationship with the noncentrality parameter of the asymptotic chi-squared distribution of a likelihood-based test, for local alternatives. Specifically, we found that when the variance function depends just on one parameter, the two criteria coincide asymptotically and in particular, the ${\rm D}_1$-criterion is proportional to the noncentrality parameter. Differently,…
In clinical trials, the response of a given subject often depends on the selected treatment as well as on some covariates. We study optimal approximate designs of experiments in the models with treatment and covariate effects. We allow for the variances of the responses to depend on the chosen treatments, which introduces heteroscedasticity into the models. For estimating systems of treatment contrasts and linear functions of the covariates, we extend known results on D-optimality of product designs by providing product designs that are optimal with respect to general eigenvalue-based criteria…
This paper discusses asymptotic distributions of various estimators of the underlying parameters in some regression models with long memory (LM) Gaussian design and nonparametric heteroscedastic LM moving average errors. In the simple linear regression model, the first-order asymptotic distribution of the least square estimator of the slope parameter is observed to be degenerate. However, in the second order, this estimator is $n^{1/2}$-consistent and asymptotically normal for $h+H<3/2$; nonnormal otherwise, where $h$ and $H$ are LM parameters of design and error processes, respectively. The f…
This article discusses estimation of a multivariate normal mean based on heteroscedastic observations. Under heteroscedasticity, estimators shrinking more on the coordinates with larger variances, seem desirable. Although they are not necessarily minimax in the ordinary sense, we show that such James-Stein type estimators can be ensemble minimax, minimax with respect to the ensemble risk, related to empirical Bayes perspective of Efron and Morris.
We introduce a robust and fully adaptive method for pointwise estimation in heteroscedastic regression. We allow for noise and design distributions that are unknown and fulfill very weak assumptions only. In particular, we do not impose moment conditions on the noise distribution. Moreover, we do not require a positive density for the design distribution. In a first step, we study the consistency of locally polynomial M-estimators that consist of a contrast and a kernel. Afterwards, minimax results are established over unidimensional Hölder spaces for degenerate design. We then choose the cont…
Carnegie Mellon University（卡内基梅隆大学）；University of Florida（佛罗里达大学）
We consider least squares estimation in a general nonparametric regression model. The rate of convergence of the least squares estimator (LSE) for the unknown regression function is well studied when the errors are sub-Gaussian. We find upper bounds on the rates of convergence of the LSE when the errors have uniformly bounded conditional variance and have only finitely many moments. We show that the interplay between the moment assumptions on the error, the metric entropy of the class of functions involved, and the "local" structure of the function class around the truth drives the rate of con…
In this paper, a new family of resampling-based penalization procedures for model selection is defined in a general framework. It generalizes several methods, including Efron's bootstrap penalization and the leave-one-out penalization recently proposed by Arlot (2008), to any exchangeable weighted bootstrap resampling scheme. In the heteroscedastic regression framework, assuming the models to have a particular structure, these resampling penalties are proved to satisfy a non-asymptotic oracle inequality with leading constant close to 1. In particular, they are asympotically optimal. Resampling…
We present a novel approach to test for heteroscedasticity of a non-stationary time series that is based on Gini's mean difference of logarithmic local sample variances. In order to analyse the large sample behaviour of our test statistic, we establish new limit theorems for U-statistics of dependent triangular arrays. We derive the asymptotic distribution of the test statistic under the null hypothesis of a constant variance and show that the test is consistent against a large class of alternatives, including multiple structural breaks in the variance. Our test is applicable even in the case …
This study examines effects of calibration errors on model assumptions and data--analytic tools in direct calibration assays. These effects encompass induced dependencies, inflated variances, and heteroscedasticity among the calibrated measurements, whose distributions arise as mixtures. These anomalies adversely affect conventional inferences, to include the inconsistency of sample means; the underestimation of measurement variance; and the distributions of sample means, sample variances, and Student's t as mixtures. Inferences in comparative experiments remain largely intact, although error …
We consider heteroscedastic nonparametric regression models, when both the mean function and variance function are unknown and to be estimated with nonparametric approaches. We derive convergence rates of posterior distributions for this model with different priors, including splines and Gaussian process priors. The results are based on the general ones on the rates of convergence of posterior distributions for independent, non-identically distributed observations, and are established for both of the cases with random covariates, and deterministic covariates. We also illustrate that the result…
Estimation for the log-logistic and Weibull distributions can be performed by using the equations used for probability plotting. The equations leads to highly heteroscedastic regression. Exact expressions for the variances of the residuals are derived which can be used to perform weighted regression. In large samples maximum likelihood performs best, but it is shown that in smaller samples the weighted regression outperforms maximum likelihood estimation with respect to bias and mean square error.
Consider the heteroscedastic nonparametric regression model with random design \begin{align*} Y_i = f(X_i) + V^{1/2}(X_i)\varepsilon_i, \quad i=1,2,\ldots,n, \end{align*} with $f(\cdot)$ and $V(\cdot)$ $α$- and $β$-Hölder smooth, respectively. We show that the minimax rate of estimating $V(\cdot)$ under both local and global squared risks is of the order \begin{align*} n^{-\frac{8αβ}{4αβ+ 2α+ β}} \vee n^{-\frac{2β}{2β+1}}, \end{align*} where $a\vee b := \max\{a,b\}$ for any two real numbers $a,b$. This result extends the fixed design rate $n^{-4α} \vee n^{-2β/(2β+1)}$ derived in Wang et al. [2…
We propose a new method of estimation in high-dimensional linear regression model. It allows for very weak distributional assumptions including heteroscedasticity, and does not require the knowledge of the variance of random errors. The method is based on linear programming only, so that its numerical implementation is faster than for previously known techniques using conic programs, and it allows one to deal with higher dimensional models. We provide upper bounds for estimation and prediction errors of the proposed estimator showing that it achieves the same rate as in the more restrictive si…
We propose simultaneous mean-variance regression for the linear estimation and approximation of conditional mean functions. In the presence of heteroskedasticity of unknown form, our method accounts for varying dispersion in the regression outcome across the support of conditioning variables by using weights that are jointly determined with the mean regression parameters. Simultaneity generates outcome predictions that are guaranteed to improve over ordinary least-squares prediction error, with corresponding parameter standard errors that are automatically valid. Under shape misspecification o…
