We study the Cramér type moderate deviation for partial sums of random fields by applying the conjugate method. The results are applicable to the partial sums of linear random fields with short or long memory and to nonparametric regression with random field errors.

By extending the methods of Peligrad et al. (2014), we establish exact moderate and large deviation asymptotics for linear random fields with independent innovations. These results are useful for studying nonparametric regression with random field errors and strong limit theorems.

Two methods were used to estimate the cost of forward contracting hard red winter wheat. One hundred days before delivery, the estimated cost of forward contracting ranged from six cents/bu. to eight cents/bu. Thus, further evidence is provided that the cost of forward contracting grain is not zero.

This paper presents a practical and simple fully nonparametric multivariate smoothing procedure that adapts to the underlying smoothness of the true regression function. Our estimator is easily computed by successive application of existing base smoothers (without the need of selecting an optimal smoothing parameter), such as thin-plate spline or kernel smoothers. The resulting smoother has better out of sample predictive capabilities than the underlying base smoother, or competing structurally constrained models (MARS, GAM) for small dimension (3 ≤ d ≤ 7) and moderate sample size n ≤ 1000. Moreover our estimator is still useful when d > 10 and to our knowledge, no other adaptive fully nonparametric regression estimator is available without constrained assumption such as additivity for example. On a real example, the Boston Housing Data, our method reduces the out of sample prediction error by 20%. An R package ibr, available at CRAN, implements the proposed multivariate nonparametric method in R.

We consider the nonparametric regression estimation problem of recovering an unknown response function f on the basis of spatially inhomogeneous data when the design points follow a known density g with a finite number of well-separated zeros. In particular, we consider two different cases: when g has zeros of a polynomial order and when g has zeros of an exponential order. These two cases correspond to moderate and severe data losses, respectively. We obtain asymptotic (as the sample size increases) minimax lower bounds for the L2-risk when f is assumed to belong to a Besov ball, and construct adaptive wavelet thresholding estimators of f that are asymptotically optimal (in the minimax sense) or near-optimal within a logarithmic factor (in the case of a zero of a polynomial order), over a wide range of Besov balls. The spatially inhomogeneous ill-posed problem that we investigate is inherently more difficult than spatially homogeneous ill-posed problems like, e.g., deconvolution. In particular, due to spatial irregularity, assessment of asymptotic minimax global convergence rates is a much harder task than the derivation of asymptotic minimax local convergence rates studied recently in the literature. Furthermore, the resulting estimators exhibit very different behavior and asymptotic minimax global convergence rates in comparison with the solution of spatially homogeneous ill-posed problems. For example, unlike in the deconvolution problem, the asymptotic minimax global convergence rates are greatly influenced not only by the extent of data loss but also by the degree of spatial homogeneity of f. Specifically, even if 1/g is non-integrable, one can recover f as well as in the case of an equispaced design (in terms of asymptotic minimax global convergence rates) when it is homogeneous enough since the estimator is “borrowing strength” in the areas where f is adequately sampled.

Let Y ∈ ℝn be a random vector with mean s and covariance matrix σ2PntPn where Pn is some known n × n-matrix. We construct a statistical procedure to estimate s as well as under moment condition on Y or Gaussian hypothesis. Both cases are developed for known or unknown σ2. Our approach is free from any prior assumption on s and is based on non-asymptotic model selection methods. Given some linear spaces collection {Sm, m ∈ ℳ}, we consider, for any m ∈ ℳ, the least-squares estimator ŝm of s in Sm. Considering a penalty function that is not linear in the dimensions of the Sm’s, we select some m̂ ∈ ℳ in order to get an estimator ŝm̂ with a quadratic risk as close as possible to the minimal one among the risks of the ŝm’s. Non-asymptotic oracle-type inequalities and minimax convergence rates are proved for ŝm̂. A special attention is given to the estimation of a non-parametric component in additive models. Finally, we carry out a simulation study in order to illustrate the performances of our estimators in practice.

In this paper we address the general issue of estimating the sensitivity of the expectation of a random variable with respect to a parameter characterizing its evolution. In finance, for example, the sensitivities of the price of a contingent claim are called the Greeks. A new way of estimating the Greeks has recently been introduced in Elie, Fermanian and Touzi (2007) through a randomization of the parameter of interest combined with nonparametric estimation techniques. In this paper we study another type of estimator that turns out to be closely related to the score function, which is well known to be the optimal Greek weight. This estimator relies on the use of two distinct kernel functions and the main interest of this paper is to provide its asymptotic properties. Under a slightly more stringent condition, its rate of convergence is the same as the one of the estimator introduced in Elie, Fermanian and Touzi (2007) and outperforms the finite differences estimator. In addition to the technical interest of the proofs, this result is very encouraging in the dynamic of creating new types of estimator for the sensitivities.

There are standard modifications of certain compactly supported wavelets that yield orthonormal bases on a bounded interval. We extend one such construction to those wavelets, such as ‘coiflets', that may have fewer vanishing moments than had to be assumed previously. Our motivation lies in function estimation in statistics. We use these boundary-modified coiflets to show that the discrete wavelet transform of finite data from sampled regression models asymptotically provides a close approximation to the wavelet transform of the continuous Gaussian white noise model. In particular, estimation errors in the discrete setting of computational practice need not be essentially larger than those expected in the continuous setting of statistical theory.

Jump points in curves arise when the conditions under which data are generated change suddenly, for example because of an unplanned change in a treatment. This paper suggests bootstrap methods for quantifying the error in estimates of jump points, and for constructing confidence intervals for jump points and confidence bands for the curve. These problems have the unusual feature that the sampling error of the jump-point estimator often has a highly non-normal distribution, which depends intimately on the distribution of regression errors. The methods are illustrated by a simulation study as well as by an application to data on the annual flow volume of the Nile river.

We consider the problem of estimating an unknown regression function when the design is random with values in $\mathbb{R}^k$. Our estimation procedure is based on model selection and does not rely on any prior information on the target function. We start with a collection of linear functional spaces and build, on a data selected space among this collection, the least-squares estimator. We study the performance of an estimator which is obtained by modifying this least-squares estimator on a set of small probability. For the so-defined estimator, we establish nonasymptotic risk bounds that can be related to oracle inequalities. As a consequence of these, we show that our estimator possesses adaptive properties in the minimax sense over large families of Besov balls Bα,l,∞(R) with R>0, l ≥ 1 and α > α1 where α1 is a positive number satisfying 1/l - 1/2 ≤ α1 < 1/l. We also study the particular case where the regression function is additive and then obtain an additive estimator which converges at the same rate as it does when k=1.


In this paper, we study the problem of non parametric estimation of an unknown regression function from dependent data with sub-Gaussian errors. As a particular case, we handle the autoregressive framework. For this purpose, we consider a collection of finite dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on a possibly irregular grid) and we estimate the regression function by a least-squares estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization of a penalized criterion akin to the Mallows' Cp. We state non asymptotic risk bounds for our estimator in some ${\mathbb{L}}_2$-norm and we show that it is adaptive in the minimax sense over a large class of Besov balls of the form Bα,p,∞(R) with p ≥ 1.

We consider the problem of hypothesis testing within a monotone regression model. We propose a new test of the hypothesis H0: “ƒ = ƒ0” against the composite alternative Ha: “ƒ ≠ ƒ0” under the assumption that the true regression function f is decreasing. The test statistic is based on the ${\mathbb L}_{1}$-distance between the isotonic estimator of f and the function f0, since it is known that a properly centered and normalized version of this distance is asymptotically standard normally distributed under H0. We study the asymptotic power of the test under alternatives that converge to the null hypothesis.

The backfitting algorithm is an iterative procedure for fitting additive models in which, at each step, one component is estimated keeping the other components fixed, the algorithm proceeding component by component and iterating until convergence. Convergence of the algorithm has been studied by Buja, Hastie, and Tibshirani (1989). We give a simple, but more general, geometric proof of the convergence of the backfitting algorithm when the additive components are estimated by penalized least squares. Our treatment covers spline smoothers and structural time series models, and we give a full discussion of the degenerate case. Our proof is based on Halperin's (1962) generalization of von Neumann's alternating projection theorem.