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

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

It is known that both Watson-Nadaraya and Gasser-Muller types of regression estimators have some disadvantages. A smooth version of local polynomial regression estimators are proposed to remedy the disadvantages. The mean squared error and mean integrated squared errors are computed explicitly. It turns out that by suitably selecting a kernel and a bandwidth, the proposed estimator has at least asymptotic minimax efficiency 89.6%-proposed estimator is efficient in rates and nearly efficient in constant factors! In nonparametric regression context, the asymptotic minimax lower bound is developed via the heuristic of the "hardest 1-dimensional subproblem". The explicit connections of minimax risks with modulus of continuity are made. Normal submodels are used to avoid the technical difficulty of Le Cam's theory of convergence of experiments. The lower bound is applicable for estimating conditional mean (regression) and conditional quantiles (including median) for both fixed design and random design regression problems. Abbreviated title. Minimax nonparametric regression. AMS 1980 subject classification. Primary 62G20. Secondary 62G05, 62F35. Key words and phrases. Hardest I-dimensional subproblem; Local polynomials; Minimax risk; Modulus

The desire to recover the unknown density when data are contaminated with errors leads to nonparametric deconvolution problems. The difficulty of deconvolution depends on both the smoothness of error distribution and the smoothness of the priori. Under a general class of smoothness constraints, we show that deconvolution kernel density k-l estimates achieve the best attainable global rates of convergence n- 2(kH)+1 under L p (1 ~ p < 00) norm, where I is the order of the derivative function of the unknown density to be estimated, k is the degrees of smoothness constraints, and {3 is the degree of the smoothness of the error distribution. Our results indicate that in present of errors, the bandwidth should be chosen larger than the ordinary density estimate. These results also constitute an extension of the ordinary kernel density estimates. oAbbreviated title: global rates of deconvolution. AMS 1980 subject classification. Primary 62G20. Secondary 62G05. Key words and phrases. Deconvolution, Fourier transforms, kernel density estimates, Lp-norm, global