Deconvolution Density Estimation Research Articles

Density estimation for mixed Euclidean and non-Euclidean data in the presence of measurement error

In this paper, we study density estimation for mixed Euclidean and non-Euclidean variables that are subject to measurement errors. This problem is largely unexplored in statistics. We develop a new deconvolution density estimator and derive its finite-sample properties. We also derive its asymptotic properties including the rate of convergence in various modes and the asymptotic distribution. For the derivation, we apply Fourier analysis on topological groups, which has not been well used in statistics. We provide full practical details on the implementation of the estimator as well as several simulation studies and real data analysis.

Nonparametric regression on Lie groups with measurement errors

This paper develops a foundation of methodology and theory for nonparametric regression with Lie group-valued predictors contaminated by measurement errors. Our methodology and theory are based on harmonic analysis on Lie groups, which is largely unknown in statistics. We establish a novel deconvolution regression estimator, and study its rate of convergence and asymptotic distribution. We also provide asymptotic confidence intervals based on the asymptotic distribution of the estimator and on the empirical likelihood technique. Several theoretical properties are also studied for a deconvolution density estimator, which is necessary to construct our regression estimator. The case of unknown measurement error distribution is also covered. We present practical details on implementation as well as the results of simulation studies for several Lie groups. A real data example is also provided.

Nonparametric tests of tail behavior in stochastic frontier models

SummaryThis article studies tail behavior for the error components in the stochastic frontier model, where one component has bounded support on one side, and the other has unbounded support on both sides. Under weak assumptions on the error components, we derive nonparametric tests for thin‐tailed distributional assumptions imposed on these two components. The tests are useful diagnostic tools for stochastic frontier analysis and kernel deconvolution density estimation. A simulation study and applications to four previously studied datasets are provided. In two of these applications, the new tests reject the thin‐tailed distributional assumptions such as normal or Laplace, which are commonly imposed in the existing literature.

Bivariate Kernel Deconvolution with Panel Data

We consider estimation of the density of a multivariate response, that is not observed directly but only through measurements contaminated by additive error. Our focus is on the realistic sampling case of bivariate panel data (repeated contaminated bivariate measurements on each sample unit) with an unknown error distribution. Several factors can affect the performance of kernel deconvolution density estimators, including the choice of the kernel and the estimation approach of the unknown error distribution. As the choice of the kernel function is critically important, the class of flat-top kernels can have advantages over more commonly implemented alternatives. We describe different approaches for density estimation with multivariate panel responses, and investigate their performance through simulation. We examine competing kernel functions and describe a flat-top kernel that has not been used in deconvolution problems. Moreover, we study several nonparametric options for estimating the unknown error distribution. Finally, we also provide guidelines to the numerical implementation of kernel deconvolution in higher sampling dimensions.

On the Study of Multiwavelet Deconvolution Density Estimators

In this paper, multiwavelet deconvolution density estimators are presented by a linear multiwavelet expansion and a nonlinear multiwavelet expansion, respectively. Moreover, the unbiased estimation is shown, and asymptotic normality is discussed for the multiwavelet deconvolution density estimators. Finally, a numerical example is given for our discussion.

A pointwise lower bound for generalized deconvolution density estimation

This current paper considers a pointwise lower bound for generalized deconvolution density estimation under the local Hölder condition. Our result can be considered as extensions of the deconvolution density estimation and the classical density estimation with no errors. In addition, we do not assume that the noises are identically distributed in this paper.

Density estimation in the presence of heteroscedastic measurement error of unknown type using phase function deconvolution.

It is important to properly correct for measurement error when estimating density functions associated with biomedical variables. These estimators that adjust for measurement error are broadly referred to as density deconvolution estimators. While most methods in the literature assume the distribution of the measurement error to be fully known, a recently proposed method based on the empirical phase function (EPF) can deal with the situation when the measurement error distribution is unknown. The EPF density estimator has only been considered in the context of additive and homoscedastic measurement error; however, the measurement error of many biomedical variables is heteroscedastic in nature. In this paper, we developed a phase function approach for density deconvolution when the measurement error has unknown distribution and is heteroscedastic. A weighted EPF (WEPF) is proposed where the weights are used to adjust for heteroscedasticity of measurement error. The asymptotic properties of the WEPF estimator are evaluated. Simulation results show that the weighting can result in large decreases in mean integrated squared error when estimating the phase function. The estimation of the weights from replicate observations is also discussed. Finally, the construction of a deconvolution density estimator using the WEPF is compared with an existing deconvolution estimator that adjusts for heteroscedasticity but assumes the measurement error distribution to be fully known. The WEPF estimator proves to be competitive, especially when considering that it relies on minimal assumption of the distribution of measurement error.

Asymptotic normality for wavelet deconvolution density estimators

This current paper shows the asymptotic normality for wavelet deconvolution density estimators, when a density function belongs to some L p ( R ) ( p > 2 ) and the noises are moderately ill-posed with the index β . The estimators include both the linear and non-linear wavelet ones. It turns out that the situation for 0 < β ≤ 1 is more complicated than that for β > 1 .

On the performance of weighted bootstrapped kernel deconvolution density estimators

We propose a weighted bootstrap approach that can improve on current methods to approximate the finite sample distribution of normalized maximal deviations of kernel deconvolution density estimators in the case of ordinary smooth errors. Using results from the approximation theory for weighted bootstrap empirical processes, we establish an unconditional weak limit theorem for the corresponding weighted bootstrap statistics. Because the proposed method uses weights that are not necessarily confined to be uniform (as in Efron’s original bootstrap), it provides the practitioner with additional flexibility for choosing the weights. As an immediate consequence of our results, one can construct uniform confidence bands, or perform goodness-of-fit tests, for the underlying density. We have also carried out some numerical examples which show that, depending on the bootstrap weights chosen, the proposed method has the potential to perform better than the current procedures in the literature.

Strong Lp convergence of wavelet deconvolution density estimators

Using compactly supported wavelets, Giné and Nickl [Uniform limit theorems for wavelet density estimators, Ann. Probab. 37(4) (2009) 1605–1646] obtain the optimal strong [Formula: see text] convergence rates of wavelet estimators for a fixed noise-free density function. They also study the same problem by spline wavelets [Adaptive estimation of a distribution function and its density in sup-norm loss by wavelet and spline projections, Bernoulli 16(4) (2010) 1137–1163]. This paper considers the strong [Formula: see text] convergence of wavelet deconvolution density estimators. We first show the strong [Formula: see text] consistency of our wavelet estimator, when the Fourier transform of the noise density has no zeros. Then strong [Formula: see text] convergence rates are provided, when the noises are severely and moderately ill-posed. In particular, for moderately ill-posed noises and [Formula: see text], our convergence rate is close to Giné and Nickl’s.

A note on SVM estimators in RKHS for the deconvolution problem

In this paper we discuss a deconvolution density estimator obtained using the support vector machines (SVM) and Tikhonov’s regularization method solving ill-posed problems in reproducing kernel Hilbert space (RKHS).A remarkable property of SVM is that the SVM leads to sparse solutions, but the support vector deconvolution density estimator does not preserve sparsity as well as we expected. Thus, in section 3, we propose another support vector deconvolution estimator (method II) which leads to a very sparse solution. The performance of the deconvolution density estimators based on the support vector method is compared with the classical kernel deconvolution density estimator for important cases of Gaussian and Laplacian measurement error by means of a simulation study. In the case of Gaussian error, the proposed support vector deconvolution estimator shows the same performance as the classical kernel deconvolution density estimator.

Supersmooth testing on the sphere over analytic classes

We consider the nonparametric goodness-of-fit test of the uniform density on the sphere when we have observations whose density is the convolution of an error density and the true underlying density. We will deal specifically with the supersmooth error case which includes the Gaussian distribution. Similar to deconvolution density estimation, the smoother the error density the harder is the rate recovery of the test problem. When considering nonparametric alternatives expressed over analytic classes, we show that it is possible to obtain original separation rates much faster than any logarithmic power of the sample size according to the ratio of the regularity index of the analytic class and the smoothness degree of the error. Furthermore, we show that our fully data-driven statistical procedure attains these optimal rates.

Adaptive quantile estimation in deconvolution with unknown error distribution

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.

A Note on the Asymptotic Normality of the Kernel Deconvolution Density Estimator with Logarithmic Chi-Square Noise

This paper studies the asymptotic normality for the kernel deconvolution estimator when the noise distribution is logarithmic chi-square; both identical and independently distributed observations and strong mixing observations are considered. The dependent case of the result is applied to obtain the pointwise asymptotic distribution of the deconvolution volatility density estimator in discrete-time stochastic volatility models.

Nonparametric specification tests for stochastic volatility models based on volatility density

This paper develops a specification test for stochastic volatility models by comparing the nonparametric kernel deconvolution density estimator of an integrated volatility density with its parametric counterpart. L2 distance is used to measure the discrepancy. The asymptotic null distributions of the test statistics are established and the asymptotic power functions are computed. Through Monte Carlo simulations, the size and power properties of the test statistics are studied. The tests are applied to an empirical example.

On a Plug-In Wavelet Estimator for Convolutions of Densities

The nonparametric estimation of the m-fold convolution power of an unknown function f is considered. We introduce an estimator based on a plug-in approach and a wavelet hard thresholding estimator. We explore its theoretical asymptotic performances via the mean integrated squared error, assuming that f has a certain degree of smoothness. Applications and numerical examples are given for the standard density estimation problem and the deconvolution density estimation problem.

A General Result on the Mean Integrated Squared Error of the Hard Thresholding Wavelet Estimator underα-Mixing Dependence

We consider the estimation of an unknown functionffor weakly dependent data (α-mixing) in a general setting. Our contribution is theoretical: we prove that a hard thresholding wavelet estimator attains a sharp rate of convergence under the mean integrated squared error (MISE) over Besov balls without imposing too restrictive assumptions on the model. Applications are given for two types of inverse problems: the deconvolution density estimation and the density estimation in a GARCH-type model, both improve existing results in this dependent context. Another application concerns the regression model with random design.

Deconvolution on the Euclidean motion group

Deconvolution is an important class of inverse problems. The goal of this paper is to develop the optimality properties of deconvolution density estimators on the noncommutative and noncompact six-dimensional Euclidean motion group. The idea is that if the underlying random object is a composition of several independent objects, then the underlying density is a convolution of densities of those independent objects. Therefore, deconvolution allows one to recover the main components of the mixture. We illustrate this with some numerical work.

Global uniform risk bounds for wavelet deconvolution estimators

We consider the statistical deconvolution problem where one observes n replications from the model Y = X + ϵ, where X is the unobserved random signal of interest and ϵ is an independent random error with distribution φ. Under weak assumptions on the decay of the Fourier transform of φ, we derive upper bounds for the finite-sample sup-norm risk of wavelet deconvolution density estimators fn for the density f of X, where f : ℝ → ℝ is assumed to be bounded. We then derive lower bounds for the minimax sup-norm risk over Besov balls in this estimation problem and show that wavelet deconvolution density estimators attain these bounds. We further show that linear estimators adapt to the unknown smoothness of f if the Fourier transform of φ decays exponentially and that a corresponding result holds true for the hard thresholding wavelet estimator if φ decays polynomially. We also analyze the case where f is a “supersmooth”/analytic density. We finally show how our results and recent techniques from Rademacher processes can be applied to construct global confidence bands for the density f.

Direct deconvolution density estimation of a mixture distribution motivated by mutation effects distribution

The mutation effect distribution is essential for understanding evolutionary dynamics. However, the existing studies on this problem have had limited resolution. So far, the most widely used method is to fit some parametric distribution, such as an exponential distribution whose validity has not been checked. In this paper, we propose a nonparametric density estimator for the mutation effect distribution, based on a deconvolution approach. Consistency of the estimator is also established. Unlike the existing deconvolution estimators, we cover the case that the target variable has a mixture structure with a pointmass and a continuous component. To study the property of the proposed estimator, several simulation studies are performed. In addition, an application for modelling virus mutation effects is provided.