Large deviation results for the nonparametric regression function estimator on functional data

This paper is devoted to the study of large deviation behavior in the setting of the estimation of the regression function on functional data. A large deviation principle is stated for a process Z _n , defined below, allowing to derive a pointwise large deviation principle for the Nadaraya- Watson-type l-indexed regression function estimator as a by-product. Moreover, a uniform over VC-classes Chernoff type large deviation result is stated for the deviation of the l-indexed regression estimator.     

Optimal and superoptimal convergence rate of the local linear estimator of nonparametric regression function in continuous time

We consider the estimation problem of the nonparametric regression in continuous time by the local linear estimator in the asymptotic quadratic error sense. In suitable conditions of strongly mixing and that of irregularity, we obtained optimal and superoptimal convergence rate of the estimator.     

Bandwidth selection for a data sharpening estimator in nonparametric regression

This paper is concerned with data-based selection of the bandwidth for a data sharpening estimator in nonparametric regression. Two kinds of bandwidths are considered: a bandwidth vector which has a different bandwidth for each covariate, and a scalar bandwidth that is common for all covariates. A plug-in method is developed and its theoretical performance is fully investigated. The proposed plug-in method works efficiently in our simulation study.     

Adaptive nonparametric estimation of a multivariate regression function

We consider the kernel estimation of a multivariate regression function at a point. Theoretical choices of the bandwidth are possible for attaining minimum mean squared error or for local scaling, in the sense of asymptotic distribution. However, these choices are not available in practice. We follow the approach of Krieger and Pickands (Ann. Statist.9 (1981) 1066–1078) and Abramson (J. Multivariate Anal.12 (1982), 562–567) in constructing adaptive estimates after demonstrating the weak convergence of some error process. As consequences, efficient data-driven consistent estimation is feasible, and data-driven local scaling is also feasible. In the latter instance, nearest-neighbor-type estimates and variance-stabilizing estimates are obtained as special cases.     

Plug-in method for nonparametric regression

The problem of bandwidth selection for non-parametric kernel regression is considered. We will follow the Nadaraya–Watson and local linear estimator especially. The circular design is assumed in this work to avoid the difficulties caused by boundary effects. Most of bandwidth selectors are based on the residual sum of squares (RSS). It is often observed in simulation studies that these selectors are biased toward undersmoothing. This leads to consideration of a procedure which stabilizes the RSS by modifying the periodogram of the observations. As a result of this procedure, we obtain an estimation of unknown parameters of average mean square error function (AMSE). This process is known as a plug-in method. Simulation studies suggest that the plug-in method could have preferable properties to the classical one. Supported by the MSMT: LC 06024.     

On the estimation of a monotone conditional variance in nonparametric regression

A monotone estimate of the conditional variance function in a heteroscedastic, nonparametric regression model is proposed. The method is based on the application of a kernel density estimate to an unconstrained estimate of the variance function and yields an estimate of the inverse variance function. The final monotone estimate of the variance function is obtained by an inversion of this function. The method is applicable to a broad class of nonparametric estimates of the conditional variance and particularly attractive to users of conventional kernel methods, because it does not require constrained optimization techniques. The approach is also illustrated by means of a simulation study.     

Robust estimation for nonparametric generalized regression

This paper focuses on nonparametric regression estimation for the parameters of a discrete or continuous distribution, such as the Poisson or Gamma distributions, when anomalous data are present. The proposal is a natural extension of robust methods developed in the setting of parametric generalized linear models. Robust estimators bounding either large values of the deviance or of the Pearson residuals are introduced and their asymptotic behaviour is derived. Through a Monte Carlo study, for the Poisson and Gamma distributions, the finite properties of the proposed procedures are investigated and their performance is compared with that of the classical ones. A resistant cross-validation method to choose the smoothing parameter is also considered.     

A nonparametric estimator of a nonlinear functional of regression under a given plan

A lower bound for the estimation quality is found and on a set of sufficiently smooth regression fucntions one constructs an estimator that attains asymptotically this bound.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 166, pp. 143–154, 1988.     

A predictive estimator of finite population mean using nonparametric regression

The paper considers the problem of estimating the population mean using auxiliary information. We propose a new model-based estimator of the population mean, based on local polynomial regression. This estimator exhibits several attractive properties under the model-based approach. The estimator is compared to a number of methods which have been proposed in the literature via a simulation study based on several populations.     

The Berry–Esseen bounds of the weighted estimator in a nonparametric regression model

Consider the following nonparametric model: \(Y_{ni}=g(x_{ni})+ \varepsilon _{ni},1\le i\le n,\) where \(x_{ni}\in {\mathbb {A}}\) are the nonrandom design points and \({\mathbb {A}}\) is a compact set of \({\mathbb {R}}^{m}\) for some \(m\ge 1\), \(g(\cdot )\) is a real valued function defined on \({\mathbb {A}}\), and \(\varepsilon _{n1},\ldots ,\varepsilon _{nn}\) are \(\rho ^{-}\)-mixing random errors with zero mean and finite variance. We obtain the Berry–Esseen bounds of the weighted estimator of \(g(\cdot )\). The rate can achieve nearly \(O(n^{-1/4})\) when the moment condition is appropriate. Moreover, we carry out some simulations to verify the validity of our results.

     

On a characterization of the exponential distribution by spacings

Summary LetX be a non-negative random variable with probability distribution functionF. SupposeX _i,n (i=1,…,n) is theith smallest order statistics in a random sample of sizen fromF. A necessary and sufficient condition forF to be exponential is given which involves the identical distribution of the random variables (n−i)(X _i+1,n−X_i,n) and (n−j)(X _j+1,n−X_j,n) for somei, j andn, (1≦i<j<n). The work was partly completed when the author was at the Dept. of Statistics, University of Brasilia, Brazil.     

Stahel-Donoho kernel estimation for fixed design nonparametric regression models

This paper reports a robust kernel estimation for fixed design nonparametric regression models. A Stahel-Donoho kernel estimation is introduced, in which the weight functions depend on both the depths of data and the distances between the design points and the estimation points. Based on a local approximation, a computational technique is given to approximate to the incomputable depths of the errors. As a result the new estimator is computationally efficient. The proposed estimator attains a high breakdown point and has perfect asymptotic behaviors such as the asymptotic normality and convergence in the mean squared error. Unlike the depth-weighted estimator for parametric regression models, this depth-weighted nonparametric estimator has a simple variance structure and then we can compare its efficiency with the original one. Some simulations show that the new method can smooth the regression estimation and achieve some desirable balances between robustness and efficiency.     

A robust alternative to the ratio estimator under non-normality

In sampling theory, the traditional ratio estimator is the most common estimator of the population mean when the correlation between study and auxiliary variables is positively high. We introduce a new ratio-type estimator based on the order statistics of a simple random sample. We show that this new estimator is considerably more efficient than the traditional ratio estimator under non-normality, and remarkably robust to data anomalies such as presence of outliers in data sets.     

Probabilities of maximal deviations for nonparametric regression function estimates

Let (X, Y) have regression function m(x) = E(Y | X = x), and let X have a marginal density f₁(x). We consider two nonparameteric estimates of m(x): the Watson estimate when f₁ is known and the Yang estimate when f₁ is known or unknown. For both estimates the asymptotic distribution of the maximal deviation from m(x) is proved, thus extending results of Bickel and Rosenblatt for the estimation of density functions.     

Bias of a nonparametric entropy estimator for Markov measures

We consider the problem of nonparametric entropy (entropy rate) estimation. We study the technique of nonparametric entropy estimation based on the so-called “nearest neighbor distances” and obtain a closed-form expression of the bias for Markov measures. This bias is a discontinuous function of transition probabilities. Bibliography: 20 titles.     

A penalty method for nonparametric estimation of the logarithmic derivative of a density function

Summary Given a random sample of sizen from a densityf ₀ on the real line satisfying certain regularity conditions, we propose a nonparametric estimator forψ ₀=−f ₀ ^′ /f⁰. The estimate is the minimizer of a quadratic functional of the formλJ(ψ)+∫[ψ ²−2ψ′]dF_n where λ>0 is a smoothing parameter,J(·) is a roughness penalty, andF _n is the empirical c.d.f. of the sample. A characterization of the estimate (useful for computational purposes) is given which is related to spline functions. A more complete study of the caseJ(ψ)=∫[d ²ψ/dx²]² is given, since it has the desirable property of giving the maximum likelihood normal estimate in the infinite smoothness limit (λ→∞). Asymptotics under somewhat restrictive assumptions (periodicity) indicate that the estimator is asymptotically consistent and achieves the optimal rate of convergence. This type of estimator looks promising because the minimization problem is simple in comparison with the analogous penalized likelihood estimators. This research was supported by the Office of Naval Research under Grant Number N00014-82-C-0062.     

Asymptotic expansion for distribution function of moment estimator for the extreme-value index

Asymptotic expansion for distribution function of the moment estimator for the extreme-value index is obtained under reasonable conditions of second order regular variation.     

Asymptotically optimal selection of a piecewise polynomial estimator of a regression function

Let (X, Y) be a pair of random variables such that X = (X₁,…, X_d) ranges over a nondegenerate compact d-dimensional interval C and Y is real-valued. Let the conditional distribution of Y given X have mean θ(X) and satisfy an appropriate moment condition. It is assumed that the distribution of X is absolutely continuous and its density is bounded away from zero and infinity on C. Without loss of generality let C be the unit cube. Consider an estimator of θ having the form of a piecewise polynomial of degree k_n based on m_n^d cubes of length 1/m_n, where the m_n^d(_d^k_n+d) coefficients are chosen by the method of least squares based on a random sample of size n from the distribution of (X, Y). Let (k_n, m_n) be chosen by the FPE procedure. It is shown that the indicated estimator has an asymptotically minimal squared error of prediction if θ is not of the form of piecewise polynomial.     

Upper bounds for the L 1-risk of the minimum L 1-distance regression estimator

A new estimator of a regression function is introduced via minimizing the L ₁-distance between some empirical function and its theoretical counterpart plus penalty for the roughness. The L ₁-risk of the estimator is bounded from above for every sample size no matter what the dependence structure of the observed random variables is. In the case of independent errors of measurement with a common variance the estimator is shown to achieve the optimal L ₁-rate of convergence within the class of m-times differentiable functions with bounded derivatives.