This paper considers estimation ofa continuous bounded probability density when observations from the density are contaminated by additive measurement errors having a known distribution. Properties of the estimator obtained by deconvolving a kernel estimator of the observed data are investigated. When the kernel used is sufficiently smooth the deconvolved estimator is shown to be pointwise consistent and bounds on its integrated mean squared error are derived. Very weak assumptions are made on the measurement-error density thereby permitting a comparison of the effects of different types of measurement error on the deconvolved estimator.

The effect of errors in variables in nonparametric regression estimation is examined. To account for errors in covariates, deconvolution is involved in the construction ofa new class of kernel estimators. It is shown that optima/local and global rates of convergence of these kernel estimators can be characterized by the tail behavior of the characteristic function of the error distribution. In fact, there are two types of rates of convergence according to whether the error is ordinary smooth or super smooth. It is also shown that these results hold uniformly over a class of joint distributions of the response and the covariates, which includes ordinary smooth regression functions as well as covariates with distributions satisfying regularity conditions. Furthermore, to achieve optimality, we show that the convergence rates of all nonparametric estimators have a lower bound possessed by the kernel estimators. oAbbreviated title. Error-in-variable regression AMS 1980 subject classification. Primary 62G20. Secondary 62G05, 62J99. Key words and phrases. Nonparametric regression; Kernel estimator; Errors in variables; Optimal rates

We discuss the difficulties of estimating quadratic functionals based on observations Y (t) from the white noise model Y (t) = Jf (u)du + cr W (t), t E [0,1], o where W (t) is a standard Wiener process on [0, 1]. The optimal rates of convergence (as cr-> 0) for estimating quadratic functionals under certain geometric constraints are 1 found. Specially, the optimal rates of estimating J[f (k)(x)f dx under hyperrectangular o constraints r = (J: Xj(f)::; CFP) and weighted lp-body constraints r p = (J: "Lj ' IXj(f)IP::; C) are computed explicitly, where Xj(f) is the jth Fourier-1 Bessel coefficient of the unknown function f. We invent a new method for developing lower bounds based on testing two highly composite hypercubes, and address its advantages. The attainable lower bounds are found by applying the hardest I-dimensional approach as well as the hypercube method. We demonstrate that for estimating regular quadratic functionals (Le., the functionals which can be estimated at rate 0 (cr 2», the difficulties of the estimation are captured by the hardest one dimensional subproblems and for estimating nonregular quadratic functionals (i.e. no 0 (cr1-consistent estimator exists), the difficulties are captured at certain finite dimensional (the dimension goes to infinite as cr-> 0) hypercube subproblems.

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This paper studies the issue of optimal deconvolution density estimation using wavelets. The approach taken here can be considered as orthogonal series estimation in the more general context of the density estimation. We explore the asymptotic properties of estimators based on thresholding of estimated wavelet coefficients. Minimax rates of convergence under the integrated square loss are studied over Besov classes Bσpq of functions for both ordinary smooth and supersmooth convolution kernels. The minimax rates of convergence depend on the smoothness of functions to be deconvolved and the decay rate of the characteristic function of convolution kernels. It is shown that no linear deconvolution estimators can achieve the optimal rates of convergence in the Besov spaces with p < 2 when the convolution kernel is ordinary smooth and super smooth. If the convolution kernel is ordinary smooth, then linear estimators can be improved by using thresholding wavelet deconvolution estimators which are asymptotically minimax within logarithmic terms. Adaptive minimax properties of thresholding wavelet deconvolution estimators are also discussed. Keywords. Adaptive estimation, Besov spaces, Kullback-Leibler information, linear estimators, minimax estimation, thresholding, wavelet bases.

We consider estimating an unknown function f from indirect white noise observations with particular emphasis on the problem of nonparametric deconvolution. Nonparametric estimators that can adapt to unknown smoothness of f are developed. The adaptive estimators are specified under two sets of assumptions on the kernel of the convolution transform. In particular, kernels having the Fourier transform with polynomially and exponentially decaying tails are considered. It is shown that the proposed estimates possess, in a sense, the best possible abilities for pointwise adaptation. Keywords: adaptive estimation; deconvolution; rates of convergence Running title: Adaptive nonparametric deconvolution Department of Statistics, University of Haifa, Mount Carmel, 31905 Haifa, Israel y e-mail: goldensh@rstat.haifa.ac.il 1 Introduction This paper investigates the problem of pointwise adaptive nonparametric estimation from indirect white noise observations. Let f 2 L 2 (R) be an unknown func...

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This paper studies the problem of estimating the density of U when only independent copies of X = U + Z is observable where Z is an independent measurement error. Convergence rates of a family of deconvolved Kernel density estimators are obtained under different assumptions on the density of Z.

The accurate assessment of participants’ private information may critically affect policy recommendations in auction markets. In many auction environments estimation of the private information distribution may be complicated by the presence of unobserved heterogeneity. This problem arises when some of the information available to all bidders at the time of the auction is subsequently not observed by the researcher. This paper develops a semi-parametric method that allows a researcher to uncover the distribution of bidders’ private information in a standard First-Price procurement auction when unobserved auction heterogeneity is present. Sufficient identification conditions are derived and a two-stage estimation procedure to recover bidders’ private information is developed. The procedure is applied to data from Michigan highway procurement auctions and compared to the estimation procedures traditionally used in the context of highway procurement auctions. The estimation results suggest that ignoring unobserved auction heterogeneity is likely to result in substantially biased estimates and may lead to erroneous policy recommendations.

The desire to recover the unknown density when data are contaminated with errors leads to nonparametric deconvolution problems. Optimal global rates of convergence are found under the weighted Lp-loss (1 $ p $ 00). It appears that the optimal rates of convergence are extremely slow for supersmooth error distributions. To overcome the difficulty, we examine how large the noise level can be for deconvolution to be feasible, and for the deconvolution estimate to be as good as the ordinary density estimate. It is shown that if noise level is not too large, nonparametric Gaussian deconvolution can still be practical. Several simulation studies are also presented. oAbbreviated title. Supersmooth Deconvolution. AMS 1980 lubject clallijication. Primary 62G20. Secondary 62G05. Key wortU and phralel. Deconvolution, Fourier transforms, kernel density estimates, Lp-norm, global rates of convergence, minimax risks. 1 Section 4 examines how the theory works for moderate sample sizes via simulation studies. Futher remarks are given in section 5. Proofs are deferred in section 6. 2. Optimal Global Rates Let's give a global lower bound on rates for supersmooth error distributions. Let's assume that the second half inequality of (1.4) holds: (as t-+ 00), (2.1) for some constants /3, i> 0, d1 ~ 0, and /3I, and that (as x-+ ±oo), (2.2) for some 0 < Qo < 1 and a> 1 + Qo. Theorem 1. Suppose that the distribution of error variable E satisfies (2.1) and (2.2) and f E Cm,B. Then, no estimator can estimate f(l)(x) faster than the rate 0 (log n)-(m-l)/t3) in the sense that for any estimator Tn(x), e (2.3) for all 1::; p:S 00, provided that the weight function w(.) is positive continuous on some