Semiparametric deconvolution with unknown error variance

Econometric Reviews, 2010

We reconsider Taupin's (2001) Integrated Nonlinear Regression (INLR) estimator for a nonlinear regression with a mismeasured covariate. We find that if we restrict the distribution of the measurement error to the class of range-restricted distributions, then weak smoothness assumptions suffice to ensure √ n consistency of the estimator. The restriction to such distributions is innocuous, because it does not affect the fit to the data. Our results show that deconvolution can be used in a nonparametric first step without imposing restrictive smoothness assumptions on the parametric model.

Statistica Neerlandica, 2003

We consider the problem of estimating a probability density function based on data that are corrupted by noise from a uniform distribution. The (nonparametric) maximum likelihood estimator for the corresponding distribution function is well defined. For the density function this is not the case. We study two nonparametric estimators for this density. The first is a type of kernel density estimate based on the empirical distribution function of the observable data. The second is a kernel density estimate based on the MLE of the distribution function of the unobservable (uncorrupted) data.

Statistics and Computing, 2009

Nonparametric density estimation in the presence of measurement error is considered. The usual kernel deconvolution estimator seeks to account for the contamination in the data by employing a modified kernel. In this paper a new approach based on a weighted kernel density estimator is proposed. Theoretical motivation is provided by the existence of a weight vector that perfectly counteracts the bias in density estimation without generating an excessive increase in variance. In practice a data driven method of weight selection is required. Our strategy is to minimize the discrepancy between a standard kernel estimate from the contaminated data on the one hand, and the convolution of the weighted deconvolution estimate with the measurement error density on the other hand. We consider a direct implementation of this approach, in which the weights are optimized subject to sum and non-negativity constraints, and a regularized version in which the objective function includes a ridge-type penalty. Numerical tests suggest that the weighted kernel estimation can lead to tangible improvements in performance over the usual kernel deconvolution estimator. Furthermore, weighted kernel estimates are free from the problem of neg-ative estimation in the tails that can occur when using modified kernels. The weighted kernel approach generalizes to the case of multivariate deconvolution density estimation in a very straightforward manner.

Arxiv preprint arXiv:0902.2117, 2009

Abstract: 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 ...

Journal of the Royal Statistical Society: Series B (Statistical Methodology), 2004

We suggest two new methods, which are applicable to both deconvolution and regression with errors in explanatory variables, for nonparametric inference.The two approaches involve kernel or orthogonal series methods. They are based on defining a low order approximation to the problem at hand, and proceed by constructing relatively accurate estimators of that quantity rather than attempting to estimate the true target functions consistently. Of course, both techniques could be employed to construct consistent estimators, but in many contexts of importance (e.g. those where the errors are Gaussian) consistency is, from a practical viewpoint, an unattainable goal. We rephrase the problem in a form where an explicit, interpretable, low order approximation is available. The information that we require about the error distribution (the error-in-variables distribution, in the case of regression) is only in the form of low order moments and so is readily obtainable by a rudimentary analysis of indirect measurements of errors, e.g. through repeated measurements. In particular, we do not need to estimate a function, such as a characteristic function, which expresses detailed properties of the error distribution. This feature of our methods, coupled with the fact that all our estimators are explicitly defined in terms of readily computable averages, means that the methods are particularly economical in computing time.

Statistics & Probability Letters, 2000

We propose two estimates of the mode of the probability density function for nonparametric deconvolution problems. In fact, we observe Y = X + , where is a measurement error with a known distribution f , and we are interesting by estimating the mode of fX the unknown probability density function of X , where Y1; : : : ; Yn are n i.i.d given observations of Y . We study the asymptotic properties of these mode estimates and the asymptotic normality of the two mode estimates is also given.

Statistica Neerlandica, 2011

We construct a density estimator and an estimator of the distribution function in the uniform deconvolution model. The estimators are based on inversion formulas and kernel estimators of the density of the observations and its derivative. Asymptotic normality and the asymptotic biases are derived.

Bernoulli, 2016

Quantile estimation in deconvolution problems is studied comprehensively. In particular, the more realistic setup of unknown error distributions is covered. Our plug-in method is based on a deconvolution density estimator and is minimax optimal under minimal and natural conditions. This closes an important gap in the literature. Optimal adaptive estimation is obtained by a data-driven bandwidth choice. As a side result, we obtain optimal rates for the plug-in estimation of distribution functions with unknown error distributions. The method is applied to a real data example.

Econometric Theory

We use mollification to regularize the problem of deconvolution of random variables. This regularization method offers a unifying and generalizing framework in order to compare the benefits of various filter-type techniques like deconvolution kernels, Tikhonov, or spectral cutoff methods. In particular, the mollifier approach allows to relax some restrictive assumptions required for the deconvolution kernels, and has better stabilizing properties compared with spectral cutoff or Tikhonov. We show that this approach achieves optimal rates of convergence for both finitely and infinitely smoothing convolution operators under Besov and Sobolev smoothness assumptions on the unknown probability density. The qualification can be arbitrarily high depending on the choice of the mollifier function. We propose an adaptive choice of the regularization parameter using the Lepskiĭ method, and we provide simulations to compare the finite sample properties of our estimator with respect to the well-...

Vietnam Journal of Mathematics, 2015

Our aim in this article is to estimate a density function f of i.i.d. random variables X1, . . . , Xn from a noise model Yj = Xj + Zj, j = 1, 2, . . . , n. Here (Zj) 1≤j≤n is independent of (Xj) 1≤j≤n and is a finite sequence of i.i.d. noise random variables distributed with an unknown density function g. This problem is known as the deconvolution problem in nonparametric statistics. The general case in which the error density function g is unknown and its Fourier transform g ft can vanish on a subset of R has still not been considered much. In the present article, we consider this case. Using direct i.i.d. data Z 1 , . . . , Z m which are collected in separate independent experiments, we propose an estimator g to the unknown density function g. After that, applying a ridge-parameter regularization method and an estimation of the Lebesgue measure of low level sets of g ft , we give an estimatorf to the target density function f and evaluate the rate of convergence of the quantity E f − f 2 2 .