Consistency of the regression estimator with functional data under long memory conditions

We study the nonparametric regression estimation when the explanatory variable takes values in some abstract functional space. We establish some asymptotic results and we give the (pointwise and uniform) convergence of the kernel type estimator constructed from functional data under long memory conditions.     

Inequalities for convex sequences and nondecreasing convex functions

In this paper, by using Wu–Debnath’s method, we establish inequalities of Hammer–Bullen type for convex sequences and nondecreasing conve     

Consistency of the least squares estimator of an infinite-dimensional parameter

Kiev. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 4, pp. 65–69, July–August, 1992.     

Inequality constrained ridge regression estimator

We carry out the idea of inequality constrained least squares (ICLS) estimation of Liew (1976) to the inequality constrained ridge regression (ICRR) estimation. We propose ICRR estimator by reducing the primal–dual relation to the fundamental problem of Dantzig and Cottle, 1967, Cottle and Dantzig, 1974 with Lemke (1962) algorithm. Furthermore, we conduct a Monte Carlo experiment.     

Correction of nonlinear orthogonal regression estimator

For any nonlinear regression function, it is shown that the orthogonal regression procedure delivers an inconsistent estimator. A new technical approach to the proof of inconsistency based on the implicit-function theorem is presented. For small measurement errors, the leading term of the asymptotic expansion of the estimator is derived. We construct a corrected estimator, which has a smaller asymptotic deviation for small measurement errors.Published in Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1101–1118, August, 2004.     

A stochastic restricted ridge regression estimator

Groß [J. Groß, Restricted ridge estimation, Statistics & Probability Letters 65 (2003) 57–64] proposed a restricted ridge regression estimator when exact restrictions are assumed to hold. When there are stochastic linear restrictions on the parameter vector, we introduce a new estimator by combining ideas underlying the mixed and the ridge regression estimators under the assumption that the errors are not independent and identically distributed. Apart from [J. Groß, Restricted ridge estimation, Statistics & Probability Letters 65 (2003) 57–64], we call this new estimator as the stochastic restricted ridge regression (SRRR) estimator. The performance of the SRRR estimator over the mixed estimator in respect of the variance and the mean square error matrices is examined. We also illustrate our findings with a numerical example. The shrinkage generalized least squares (GLS) and the stochastic restricted shrinkage GLS estimators are proposed.     

A bias corrected nonparametric regression estimator

In this article, we propose a new method of bias reduction in nonparametric regression estimation. The proposed new estimator has asymptotic bias order h⁴, where h is a smoothing parameter, in contrast to the usual bias order h² for the local linear regression. In addition, the proposed estimator has the same order of the asymptotic variance as the local linear regression. Our proposed method is closely related to the bias reduction method for kernel density estimation proposed by Chung and Lindsay (2011). However, our method is not a direct extension of their density estimate, but a totally new one based on the bias cancelation result of their proof.     

Consistency of spike and slab regression

Spike and slab models are a popular and attractive variable selection approach in regression settings. Applications for these models have blossomed over the last decade and they are increasingly being used in challenging problems. At the same time, theory for spike and slab models has not kept pace with the applications. There are many gaps in what we know about their theoretical properties. An important property known to hold in these models is selective shrinkage: a unique property whereby the posterior mean is shrunk toward zero for non-informative variables only. This property has been shown to hold under orthogonality for continuous priors under the modified class of rescaled spike and slab models. In this paper, we extend this result to the general case and prove an oracle property for the posterior mean under a discrete two-component prior. An immediate consequence is that a strong selective shrinkage property holds. Interestingly, the conditions needed for our result to hold in the non-orthogonal setting are more stringent than in the orthogonal case and amount to a type of enforced sparsity condition that must be met by the prior.     

Asymptotic normality of a projective estimator of an infinite-dimensional parameter of nonlinear regression

A model of nonlinear regression is studied in infinite-dimensional space. Observation errors are equally distributed and have the identity correlation operator. A projective estimator of a parameter is constructed, and the conditions under which it is true are established. For a parameter that belongs to an ellipsoid in a Hilbert space, we prove that the estimators are asymptotically normal; for this purpose, the representation of the estimator in terms of the Lagrange factor is used and the asymptotics of this factor are studied. An example of the nonparametric estimator of a signal is examined for iterated observations under an additive noise.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 9, pp. 1205–1214, September, 1993.     

When is the inverse regression estimator MSE-superior to the standard regression estimator in multivariate controlled calibration situations?

We assume as model a standard multivariate regression of y on x, fitted to a controlled calibration sample and used to estimate unknown x′s from observed y-values. The standard weighted least squares estimator (‘classical’, regress y on x and ‘solve’ for x) and the biased inverse regression estimator (regress x on y) are compared with respect to mean squared error. The regions are derived where the inverse regression estimator yields the smaller MSE. For any particular component of x this region is likely to contain ‘most’ future values in usual practice. For simultaneous estimation this needs not be true, however.     

Consistency of a nonparametric conditional quantile estimator for random fields

Given a stationary multidimensional spatial process (Z _i = (X _i , Y _i ) ∈ ℝ ^d × ℝ, i ∈ ℤ ^N ), we investigate a kernel estimate of the spatial conditional quantile function of the response variable Y _i given the explicative variable X _i . Almost complete convergence and consistency in L ^2r norm (r ∈ ℕ*) of the kernel estimate are obtained when the sample considered is an α-mixing sequence.     

A bivariate histogram density estimator: Consistency and asymptotic normality

This paper presents a nonparametric histogram density estimator based on the spacings of order statistics. This estimator generalizes to the bivariate case the univariate histogram estimator proposed by Van Ryzin (1973). The first of the two theorems in this paper gives conditions under which the estimator is pointwise strongly consistent. The second theorem provides conditions for the asymptotic normality of the estimator for points at which the density function possesses continuous partial derivatives of second order.     

Non-crossing quantile regression via doubly penalized kernel machine

Quantile regression provides a more complete statistical analysis of the stochastic relationships among random variables. Sometimes quantile regression functions estimated at different orders can cross each other. We propose a new non-crossing quantile regression method using doubly penalized kernel machine (DPKM) which uses heteroscedastic location-scale model as basic model and estimates both location and scale functions simultaneously by kernel machines. The DPKM provides the satisfying solution to estimating non-crossing quantile regression functions when multiple quantiles for high-dimensional data are needed. We also present the model selection method that employs cross validation techniques for choosing the parameters which affect the performance of the DPKM. One real example and two synthetic examples are provided to show the usefulness of the DPKM.     

Modified restricted Liu estimator in logistic regression model

?iray et al. (Commun Stat Simul Comput, 2014) proposed a restricted Liu estimator in logistic regression model with linear restrictions. However, this estimator did not satisfy the linear restrictions. In this paper, we introduce a modified restricted Liu estimator in logistic regression model with linear restrictions. Our results show that the new estimator satisfies the linear restrictions. We also discuss the properties of the new estimator under the matrix mean squared error criterion. Finally, a Monte Carlo study and a numerical example are given to show the performances of the new estimator.     

Asymptotic normality of wavelet estimator in heteroscedastic regression model

The following heteroscedastic regression model Y_i=g(x_i) σ_ie_i(1≤i≤n)is considered,where it is assumed thatσ_i~2=f(u_i),the design points(x_i,u_i)are known and nonrandom,g and f are unknown functions.Under the unobservable disturbance e_i form martingale differences,the asymptotic normality of wavelet estimators of g with f being known or unknown function is studied.     

Consistency of kernel‐based quantile regression

Quantile regression is used in many areas of applied research and business. Examples are actuarial, financial or biometrical applications. We show that a non‐parametric generalization of quantile regression based on kernels shares with support vector machines the property of consistency to the Bayes risk. We further use this consistency to prove that the non‐parametric generalization approximates the conditional quantile function which gives the mathematical justification for kernel‐based quantile regression. Copyright © 2008 John Wiley & Sons, Ltd.     

Efficiency of a Liu-type estimator in semiparametric regression models

In this paper we consider the semiparametric regression model, y=Xβ+f+ε. Recently, Hu [11] proposed ridge regression estimator in a semiparametric regression model. We introduce a Liu-type (combined ridge-Stein) estimator (LTE) in a semiparametric regression model. Firstly, Liu-type estimators of both β and f are attained without a restrained design matrix. Secondly, the LTE estimator of β is compared with the two-step estimator in terms of the mean square error. We describe the almost unbiased Liu-type estimator in semiparametric regression models. The almost unbiased Liu-type estimator is compared with the Liu-type estimator in terms of the mean squared error matrix. A numerical example is provided to show the performance of the estimators.     

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.     

Best estimator of the regression parameter in a reversing model

A mathematical model of data processing for frequency sensors is proposed. Minimum-variance linear unbiased estimator of the constant signal is determined in the presence of white noise and unknown nonstochastic drift by reversal of observations. The reversal time is taken into account. It is shown that if the drift is from a finite-dimensional space, the estimation process reduces to solving a system of linear equations. An example is considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 94–99, 1986.