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We study estimation of a multivariate function $f:\mathbf{R}^d\to\mathbf{R}$ when the observations are available from the function $Af$, where $A$ is a known linear operator. Both the Gaussian white noise model and density estimation are studied. We define an $L_2$-empirical risk functional which is used to define a $\delta$-net minimizer and a dense empirical risk minimizer. Upper bounds for the mean integrated squared error of the estimators are given. The upper bounds show how the difficulty of the estimation depends on the operator through the norm of the adjoint of the inverse of the operator and on the underlying function class through the entropy of the class. Corresponding lower bounds are also derived. As examples, we consider convolution operators and the Radon transform. In these examples, the estimators achieve the optimal rates of convergence. Furthermore, a new type of oracle inequality is given for inverse problems in additive models., Comment: Published in at http://dx.doi.org/10.1214/09-AOS726 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

The removal of blur from a signal, in the presence of noise, is readily accomplished if the blur can be described in precise mathematical terms. However, there is growing interest in problems where the extent of blur is known only approximately, for example in terms of a blur function which depends on unknown parameters that must be computed from data. More challenging still is the case where no parametric assumptions are made about the blur function. There has been a limited amount of work in this setting, but it invariably relies on iterative methods, sometimes under assumptions that are mathematically convenient but physically unrealistic (e.g., that the operator defined by the blur function has an integrable inverse). In this paper we suggest a direct, noniterative approach to nonparametric, blind restoration of a signal. Our method is based on a new, ridge-based method for deconvolution, and requires only mild restrictions on the blur function. We show that the convergence rate of the method is close to optimal, from some viewpoints, and demonstrate its practical performance by applying it to real images., Comment: Published in at http://dx.doi.org/10.1214/009053606000001442 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

With modern technology development, functional data are being observed frequently in many scientific fields. A popular method for analyzing such functional data is ``smoothing first, then estimation.'' That is, statistical inference such as estimation and hypothesis testing about functional data is conducted based on the substitution of the underlying individual functions by their reconstructions obtained by one smoothing technique or another. However, little is known about this substitution effect on functional data analysis. In this paper this problem is investigated when the local polynomial kernel (LPK) smoothing technique is used for individual function reconstructions. We find that under some mild conditions, the substitution effect can be ignored asymptotically. Based on this, we construct LPK reconstruction-based estimators for the mean, covariance and noise variance functions of a functional data set and derive their asymptotics. We also propose a GCV rule for selecting good bandwidths for the LPK reconstructions. When the mean function also depends on some time-independent covariates, we consider a functional linear model where the mean function is linearly related to the covariates but the covariate effects are functions of time. The LPK reconstruction-based estimators for the covariate effects and the covariance function are also constructed and their asymptotics are derived. Moreover, we propose a $L^2$-norm-based global test statistic for a general hypothesis testing problem about the covariate effects and derive its asymptotic random expression. The effect of the bandwidths selected by the proposed GCV rule on the accuracy of the LPK reconstructions and the mean function estimator is investigated via a simulation study. The proposed methodologies are illustrated via an application to a real functional data set collected in climatology., Comment: Published at http://dx.doi.org/10.1214/009053606000001505 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)