Julia Dony, Uwe Einmahl, David M. Mason
We generalize a method for proving uniform in bandwidth consistency results for kernel type estimators developed by the two last named authors. Such results are shown to be useful in establishing consistency of local polynomial estimators of the regression function.
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We are interested in the rate of consistency of kernel density estimators with respect to the weighted sup-norm determined by some unbounded weight function. This problem has been considered by Gine, Koltchinskii and Zinn (2004) for a deterministic bandwidth sequence. We provide "uniform in h" versions of some of their results, allowing us to determine the corresponding rates of consistency for kernel density estimators where the bandwidth sequences may depend on the data and/or the location.
We investigate the asymptotic mean squared error of kernel estimators of the intensity function of a spatial point process. We show that when $n$ independent copies of a point process in $\mathbb R^d$ are superposed, the optimal bandwidth $h_n$ is of the order $n^{-1/(d+4)}$ under appropriate smoothness conditions on the kernel and true intensity function. We apply the Abramson principle to define adaptive kernel estimators and show that asymptotically the optimal adaptive bandwidth is of the order $n^{-1/(d+8)}$ under appropriate smoothness conditions.
In this paper we study the ideal variable bandwidth kernel density estimator introduced by McKay (1993) and Jones, McKay and Hu (1994) and the plug-in practical version of the variable bandwidth kernel estimator with two sequences of bandwidths as in Giné and Sang (2013). Based on the bias and variance analysis of the ideal and true variable bandwidth kernel density estimators, we study the central limit theorems for each of them.
Nonparametric density and regression estimators commonly depend on a bandwidth. The asymptotic properties of these estimators have been widely studied when bandwidths are nonstochastic. In practice, however, in order to improve finite sample performance of these estimators, bandwidths are selected by data driven methods, such as cross-validation or plug-in procedures. As a result nonparametric estimators are usually constructed using stochastic bandwidths. In this paper we establish the asymptotic equivalence in probability of local polynomial regression estimators under stochastic and nonstoc…
We establish uniform-in-bandwidth consistency for kernel-type estimators of the differential entropy. We consider two kernel-type estimators of Shannon's entropy. As a consequence, an asymptotic 100% confidence interval of entropy is provided.
In a regression model, we write the Nadaraya-Watson estimator of the regression function as the quotient of two kernel estimators, and propose a bandwidth selection method for both the numerator and the denominator. We prove risk bounds for both data driven estimators and for the resulting ratio. The simulation study confirms that both estimators have good performances, compared to the ones obtained by cross-validation selection of the bandwidth. However, unexpectedly, the single-bandwidth cross-validation estimator is found to be much better than the ratio of the previous two good estimators,…
In this paper we establish the uniform in bandwidth consistency for the transformation kernel estimator of copulas introduced in [Omelka et al.(2009)]. To this end, we first prove a uniform in bandwidth law of the iterated logarithm for the maximal deviation of this estimator from its expectation. We then show that, as n goes to infinity, the bias of the estimator converges to zero uniformly in the bandwidth h, varying over a suitable interval. A practical method of selecting the optimal bandwidth is also presented. Finally, we make conclusive simulation experiments showing the performance of …
We introduce a general method to prove uniform in bandwidth consistency of kernel-type function estimators. Examples include the kernel density estimator, the Nadaraya-Watson regression estimator and the conditional empirical process. Our results may be useful to establish uniform consistency of data-driven bandwidth kernel-type function estimators.
On large bandwidth matrix values kernel smoothed estimators for multi-index models
本研究探讨了多索引模型中大带宽矩阵值核平滑估计量的渐近性质，发现其能自然减少维度灾难，通过数值实验和案例研究验证了其有效性。
For $n$ independent random variables having the same Hölder continuous density, this paper deals with controls of the Wolverton-Wagner's estimator MSE and MISE. Then, for a bandwidth $h_n(β)$, estimators of $β$ are obtained by a Goldenshluger-Lepski type method and a Lacour-Massart-Rivoirard type method. Some numerical experiments are provided for this last method.
In this paper we propose an automatic selection of the bandwidth of the semi-recursive kernel estimators of a regression function defined by the stochastic approximation algorithm. We showed that, using the selected bandwidth and some special stepsizes, the proposed semi-recursive estimators will be very competitive to the nonrecursive one in terms of estimation error but much better in terms of computational costs. We corroborated these theoretical results through simulation study and a real dataset.
It is shown that the Hall, Hu and Marron [Hall, P., Hu, T., and Marron J.S. (1995), Improved Variable Window Kernel Estimates of Probability Densities, {\it Annals of Statistics}, 23, 1--10] modification of Abramson's [Abramson, I. (1982), On Bandwidth Variation in Kernel Estimates - A Square-root Law, {\it Annals of Statistics}, 10, 1217--1223] variable bandwidth kernel density estimator satisfies the optimal asymptotic properties for estimating densities with four uniformly continuous derivatives, uniformly on bounded sets where the preliminary estimator of the density is bounded away from z…
In this paper we consider the kernel estimators of a distribution function defined by the stochastic approximation algorithm when the observation are contamined by measurement errors. It is well known that this estimators depends heavily on the choice of a smoothing parameter called the bandwidth. We propose a specific second generation plug-in method of the deconvolution kernel distribution estimators defined by the stochastic approximation algorithm. We show that, using the proposed bandwidth selection and the stepsize which minimize the MISE (Mean Integrated Squared Error), the proposed est…
A continuous-time nonlinear regression model with Lévy-driven linear noise process is considered. Sufficient conditions of consistency and asymptotic normality of the Whittle estimator for the parameter of the noise spectral density are obtained in the paper.
In 1991 Stute introduced a class of estimators called conditional U-statistics. They can be seen as a generalization of the Nadaraya-Watson estimator, and their strong pointwise consistency to the general regression function has been obtained in the same paper by Stute. Very recently, Gine and Mason introduced the notion of a local U-process, which generalizes that of a local empirical process, and obtained central limit theorems and laws of the iterated logarithm for this class. We apply the methods developed by Einmahl and Mason (2005) and Gine and Mason (2007a,b) to establish uniform in ban…
A note on kernel density estimators with optimal bandwidths
本文研究了最优带宽核密度估计器的累积分布函数，证明其在样本量增加时几乎必然位于经验分布函数的置信区间外。
New local linear estimators are proposed for a wide class of nonparametric regression models. The estimators are uniformly consistent regardless of satisfying traditional conditions of depen\-dence of design elements. The estimators are the solutions of a specially weighted least-squares method. The design can be fixed or random and does not need to meet classical regularity or independence conditions. As an application, several estimators are constructed for the mean of dense functional data. The theoretical results of the study are illustrated by simulations. An example of processing real me…
In this paper, we study the asymptotic normality of the conditional maximum likelihood (ML) estimators for the truncated regression model and the Tobit model. We show that under the general setting assumed in his book, the conjectures made by Hayashi (2000) \footnote{see page 516, and page 520 of Hayashi (2000).} about the asymptotic normality of the conditional ML estimators for both models are true, namely, a sufficient condition is the nonsingularity of $\mathbf{x_tx'_t}$.
We consider the derivation of data-dependent simultaneous bandwidths for double kernel heteroskedasticity and autocorrelation consistent (DK-HAC) estimators. In addition to the usual smoothing over lagged autocovariances for classical HAC estimators, the DK-HAC estimator also applies smoothing over the time direction. We obtain the optimal bandwidths that jointly minimize the global asymptotic MSE criterion and discuss the trade-off between bias and variance with respect to smoothing over lagged autocovariances and over time. Unlike the MSE results of Andrews (1991), we establish how nonstatio…
This study investigates the effect of bandwidth selection via a plug-in method on the asymptotic structure of the nonparametric kernel density estimator. We generalise the result of Hall and Kang (2001) and find that the plug-in method has no effect on the asymptotic structure of the estimator up to the order of $O\{(nh_0)^{-1/2}+h_0^L\}=O(n^{-L/(2L+1)})$ for a bandwidth $h_0$ and any kernel order $L$ when the kernel order for pilot estimation $L_p$ is high enough. We also provide the valid Edgeworth expansion up to the order of $O\{(nh_0)^{-1}+h_0^{2L}\}$ and find that, as long as the $L_p$ i…
In this paper, based on the kernel estimator proposed by Ould-Said and Lemdani (Ann. Instit. Statist. Math. 2006), we develop some new generalized M-estimator procedures for single index regression models with left-truncated responses. The consistency and asymptotic normality of our estimators are also established. Some simulation studies are given to investigate the finite sample performance of the proposed estimators.
In this paper we propose a variable bandwidth kernel regression estimator for $i.i.d.$ observations in $\mathbb{R}^2$ to improve the classical Nadaraya-Watson estimator. The bias is improved to the order of $O(h_n^4)$ under the condition that the fifth order derivative of the density function and the sixth order derivative of the regression function are bounded and continuous. We also establish the central limit theorems for the proposed ideal and true variable kernel regression estimators. The simulation study confirms our results and demonstrates the advantage of the variable bandwidth kerne…
We propose an adaptive estimator for the stationary distribution of a bifurcating Markov Chain on $\mathbb R^d$. Bifurcating Markov chains (BMC for short) are a class of stochastic processes indexed by regular binary trees. A kernel estimator is proposed whose bandwidth is selected by a method inspired by the works of Goldenshluger and Lepski [18]. Drawing inspiration from dimension jump methods for model selection, we also provide an algorithm to select the best constant in the penalty.
Three common classes of kernel regression estimators are considered: the Nadaraya--Watson (NW) estimator, the Priestley--Chao (PC) estimator, and the Gasser--Müller (GM) estimator. It is shown that (i) the GM estimator has a certain monotonicity preservation property for any kernel $K$, (ii) the NW estimator has this property if and only the kernel $K$ is log concave, and (iii) the PC estimator does not have this property for any kernel $K$. Other related properties of these regression estimators are discussed.
Strong consistency of the Bayes estimator of a regression curve for the $L^1$-squared loss function is proved. It is also shown the convergence to 0 of the Bayes risk of this estimator both for the $L^1$ and $L^1$-squared loss functions. The Bayes estimator of a regression curve is the regression curve with respect to the posterior predictive distribution.
We establish asymptotic normality for estimators of the additive regression components under random censorship. To build our estimators, we couple the marginal integration method (Newey (1994)) with an initial Inverse Probability of Censoring Weighted estimator of the multivariate censored regression function introduced by Carbonez et al. (1995) and Kohler et al. (2002). Asymptotic confidence bands are derived from our result.
This paper discusses minimum distance estimation method in the linear regression model with dependent errors which are strongly mixing. The regression parameters are estimated through the minimum distance estimation method, and asymptotic distributional properties of the estimators are discussed. A simulation study compares the performance of the minimum distance estimator with other well celebrated estimator. This simulation study shows the superiority of the minimum distance estimator over another estimator. KoulMde (R package) which was used for the simulation study is available online. See…
We study the isotonic regression estimator over a general countable pre-ordered set. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under some general assumptions, the limiting distribution of the isotonized estimator is given by the concatenation of the separate isotonic regressions of the certain subvectors of an unrestrecred estimator's asymptotic distribution. Also, we show that the isotonization preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression fu…
We study the least square estimator, in the framework of simple linear regression, when the deviance term $\varepsilon$ with respect to the linear model is modeled by a uniform distribution. In particular, we give the law of this estimator, and prove some convergence properties.
This paper concerns the estimation of the regression function at a given point in nonparametric heteroscedastic models with Gaussian noise or with noise having unknown distribution. In the two cases an asymptotically efficient kernel estimator is constructed for the minimax absolute error risk.
Bias correction for quantile regression estimators
本文提出了一种用于分位数回归估计量的二阶偏倚校正方法，通过有限差分估计偏倚组件，提高了估计精度，并通过实证数据验证了其有效性。
A new nonparametric estimator of a convex regression function in any dimension is proposed and its convergence properties are studied. We start by using any estimator of the regression function and we \emph{convexify} it by taking the convex envelope of a sample of the approximation obtained. We prove that the uniform rate of convergence of the estimator is maintained after the convexification is applied. The finite sample properties of the new estimator are investigated by means of a simulation study and the application of the new method is demonstrated in examples.
Consider a linear regression model. Fan and Li (2001) describe the smoothly clipped absolute deviation (SCAD) point estimator of the regression parameter vector. To gain insight into the properties of this estimator, they consider an orthonormal design matrix and focus on the estimation of a specified component of this vector. They show that the SCAD point estimator has three attractive properties. We answer the question: To what extent can an interval estimator, centred on the SCAD estimator, have similar attractive properties?
We derive minimax generalized Bayes estimators of regression coefficients in the general linear model with spherically symmetric errors under invariant quadratic loss for the case of unknown scale. The class of estimators generalizes the class considered in Maruyama and Strawderman (2005) to include non-monotone shrinkage functions.
We present a greedy method for simultaneously performing local bandwidth selection and variable selection in nonparametric regression. The method starts with a local linear estimator with large bandwidths, and incrementally decreases the bandwidth of variables for which the gradient of the estimator with respect to bandwidth is large. The method--called rodeo (regularization of derivative expectation operator)--conducts a sequence of hypothesis tests to threshold derivatives, and is easy to implement. Under certain assumptions on the regression function and sampling density, it is shown that t…
We study asymptotic behavior of one-step $M$-estimators based on samples from arrays of not necessarily identically distributed random variables and representing explicit approximations to the corresponding consistent $M$-estimators. These estimators generalize Fisher's one-step approximations to consistent maximum likelihood estimators. As a consequence, we consider some nonlinear regression problems where the procedure mentioned allow us to construct explicit asymptotically optimal estimators. We also consider the problem of constructing initial estimators which are needed for one-step estim…
Penalized regression estimators are a popular tool for the analysis of sparse and high-dimensional data sets. However, penalized regression estimators defined using an unbounded loss function can be very sensitive to the presence of outlying observations, especially high leverage outliers. Moreover, it can be particularly challenging to detect outliers in high-dimensional data sets. Thus, robust estimators for sparse and high-dimensional linear regression models are in need. In this paper, we study the robust and asymptotic properties of MM-Bridge and adaptive MM-Bridge estimators: $\ell_q$-pe…
To address the problem of multicollinearity in the logistic regression model, in this paper we propose a new estimator called Stochastic restricted almost unbiased logistic Liu-estimator (SRAULLE) when the prior information is available in the form of stochastic linear restrictions. A Monte Carlo simulation study was carried out to compare the performance of the proposed estimator with some existing estimators in the scalar mean squared error (SMSE) sense. Finally, a real data example was given to appraise the performance of the estimators.
It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, in the context of biased shrinkage ridge estimation, Chang (2015) introduced an almost unbiased Liu estimator in the logistic regression model. Making use of his approach, when some prior knowledge in the form of linear restrictions are also available, we introduce a restricted almost unbiased Liu estimator in the logistic regression model. Statistical properties of this newly defined estimator are derived and some comparison result are also pr…
We study asymptotic behavior of one-step weighted $M$-estimators based on samples from arrays of not necessarily identically distributed random variables and representing explicit approximations to the corresponding consistent weighted $M$-estimators. Sufficient conditions are presented for asymptotic normality of the one-step weighted $M$-estimators under consideration. As a consequence, we consider some well-known nonlinear regression models where the procedure mentioned allow us to construct explicit asymptotically optimal estimators.
In this paper, we prove large deviations principle for the Nadaraya-Watson estimator and for the semi-recursive kernel estimator of the regression in the multidimensional case. Under suitable conditions, we show that the rate function is a good rate function. We thus generalize the results already obtained in the unidimensional case for the Nadaraya-Watson estimator. Moreover, we give a moderate deviations principle for these two estimators. It turns out that the rate function obtained in the moderate deviations principle for the semi-recursive estimator is larger than the one obtained for the…
Recently, the well known Liu estimator (Liu, 1993) is attracted researcher's attention in regression parameter estimation for an ill conditioned linear model. It is also argued that imposing sub-space hypothesis restriction on parameters improves estimation by shrinking toward non-sample information. Chang (2015) proposed the almost unbiased Liu estimator (AULE) in the binary logistic regression. In this article, some improved unbiased Liu type estimators, namely, restricted AULE, preliminary test AULE, Stein-type shrinkage AULE and its positive part for estimating the regression parameters in…
In this paper, a lower bound is determined in the minimax sense for change point estimators of the first derivative of a regression function in the fractional white noise model. Similar minimax results presented previously in the area focus on change points in the derivatives of a regression function in the white noise model or consider estimation of the regression function in the presence of correlated errors.
Quantile regression predicts the $τ$-quantile of the conditional distribution of a response variable given the explanatory variable for $τ\in(0,1)$. The aim of this paper is to establish the asymptotic distribution of the quantile estimator obtained by penalized spline method. A simulation and an exploration of real data are performed to validate our results.
Higher-order Refinements of Small Bandwidth Asymptotics for Density-Weighted Average Derivative Estimators
本文利用Edgeworth展开证明了小带宽渐近近似导出的密度加权平均导数推断程序具有更优的高阶分布性质，优于基于渐近线性近似的程序，为模拟中的有限样本性能提供了理论依据。
The main Theorem of Jain et al.[Jain, K., Singh, S., and Sharma, S. (2011), Re- stricted estimation in multivariate measurement error regression model; JMVA, 102, 2, 264-280] is established in its full generality. Namely, we derive the joint asymp- totic normality of the unrestricted estimator (UE) and the restricted estimators of the matrix of the regression coefficients. The derived result holds under the hypothesized restriction as well as under the sequence of alternative restrictions. In addition, we establish Asymptotic Distributional Risk for the estimators and compare their relative pe…
We consider the high order moments estimator of the frontier of a random pair introduced by Girard, S., Guillou, A., Stupfler, G. (2012). {\it Frontier estimation with kernel regression on high order moments}. In the present paper, we show that this estimator is strongly uniformly consistent on compact sets and its rate of convergence is given when the conditional cumulative distribution function belongs to the Hall class of distribution functions.
We propose a class of weighted least squares estimators for the tail index of a distribution function with a regularly varying upper tail. Our approach is based on the method developed by \cite{Holan2010} for the Parzen tail index. Asymptotic normality of the estimators is proved. Through a simulation study, these and earlier estimators are compared in the Pareto and Hall models using the mean squared error as criterion. The results show that the weighted least squares estimator is better than the other estimators investigated.
Rates of strong uniform consistency for the $k$-nearest neighbors kernel estimators of density and regression function
本文探讨了多元情形下密度与回归函数的 $k$-最近邻核估计量的一致性问题，并在特定假设下推导出这些估计量在整个空间 $\mathbb{R}^p$ 上的强一致一致性的收敛速率。
In this article, basing on NQD samples, we investigate the fixed design nonparametric regression model, where the errors are pairwise NQD random errors, with fixed design points, and an unknown function. Nonparametric weighted estimator will be introduced and its consistency is studied. As special case, the consistency result for weighted kernel estimators of the model is obtained. This extends the earlier work on independent random and dependent random errors to NQD case.
Robust Inference for Convex Pairwise Difference Estimators
针对凸成对差分估计量，本文发展了分布理论和基于自助法的推断方法，通过小带宽渐近理论、广义刀切去偏和调整带宽方差扭曲的自助法，实现了在更弱带宽条件下的有效推断。
An effective estimation of multivariate density functions using extended-beta kernels with Bayesian adaptive bandwidths
针对多元密度函数估计问题，提出带贝叶斯自适应带宽的多重扩展β核（MEBK）估计器，建立其渐近性质并提供带宽选择方法，经模拟与真实数据验证，在灵活性和通用性上优于高斯、伽马核估计器。
In this paper, we establish uniform-in-bandwidth limit laws of the logarithm for nonparametric Inverse Probability of Censoring Weighted (I.P.C.W.) estimators of the multivariate regression function under random censorship. A similar result is deduced for estimators of the conditional distribution function. The uniform-in-bandwidth consistency for estimators of the conditional density and the conditional hazard rate functions are also derived from our main result. Moreover, the logarithm laws we establish are shown to yield almost sure simultaneous asymptotic confidence bands for the functions…
The multivariate errors-in-variables regression model is applicable when both dependent and independent variables in a multivariate regression are subject to measurement errors. In such a scenario it is long established that the traditional least squares approach to estimating the model parameters is biased and inconsistent. The generalized least squares, ordinary least squares and maximum likelihood estimators (under the assumption of Gaussian errors) were derived in the seminal paper of Gleser (1981). However, the ordinary least squares and maximum likelihood estimators for the mean vectors …
Under a partially linear models we study a family of robust estimates for the regression parameter and the regression function when some of the predictor variables take values on a Riemannian manifold. We obtain the consistency and the asymptotic normality of the proposed estimators. Also, we consider a robust cross validation procedure to select the smoothing parameter. Simulations and application to real data show the performance of our proposal under small samples and contamination.
We generalize the naïve estimator of a Poisson regression model with measurement errors as discussed in Kukush et al. [1]. The explanatory variable is not always normally distributed as they assume. In this study, we assume that the explanatory variable and measurement error are not limited to a normal distribution. We clarify the requirements for the existence of the naïve estimator and derive its asymptotic bias and asymptotic mean squared error (MSE). In addition, we propose a consistent estimator of the true parameter by correcting the bias of the naïve estimator. As illustrative examples,…
In this article, we consider an imputation method to handle missing response values based on semiparametric quantile regression estimation. In the proposed method, the missing response values are generated using the estimated conditional quantile regression function at given values of covariates. We adopt the generalized method of moments for estimation of parameters defined through a general estimation equation. We demonstrate that the proposed estimator, which combines both semiparametric quantile regression imputation and generalized method of moments, has competitive edge against some of t…
It has been recently shown that nonparametric estimators of the additive regression function could be obtained in the presence of right censoring by coupling the marginal integration method with initial kernel-type Inverse Probability of Censoring Weighted estimators of the multivariate regression function \citeDebbarhViallon3. In this paper, we get the exact rate of strong uniform consistency for such estimators. Our uniform limit laws especially lead to the construction of asymptotic simultaneous 100% confidence bands for the true regression function.
In this paper the asymptotic distribution of estimators is derived in a general regression setting where rank restrictions on a submatrix of the coefficient matrix are imposed and the regressors can include stationary or I(1) processes. Such a setting occurs e.g. in factor models. Rates of convergence are derived and the asymptotic distribution is given for least squares estimators as well as fully-modified estimators. The gains in imposing the rank restrictions are investigated. A number of special cases are discussed including the Johansen results in the case of cointegrated VAR(p) processes…
This paper considers M-estimation of a nonlinear regression model with multiple change-points occuring at unknown times. The multi-phase random design regression model, discontinuous in each change-point, have an arbitrary error $ε$. In the case when the number of jumps is known, the M-estimator of locations of breaks and of regression parameters are studied. These estimators are consistent and the distribution of the regression parameter estimators is Gaussian. The estimator of each change-point converges, with the rate $n^{-1}$, to the smallest minimizer of the independent compound Poisson p…
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