Fixed Design Regression Research Articles

Estimation and testing of crossing‐points in fixed design regression

Many phenomena in the life sciences can be analyzed by using a fixed design regression model with a regression function m that exhibits a crossing‐point in the following sense: the regression function runs below or above its mean level, respectively, according as the input variable lies to the left or to the right of that crossing‐point, or vice versa. We propose a non‐parametric estimator and show weak and strong consistency as long as the crossing‐point is unique. It is defined as maximizing point arg max of a certain marked empirical process. For testing the hypothesis H0 that the regression function m actually is constant (no crossing‐point), a decision rule is designed for the specific alternative H1 that m possesses a crossing‐point. The pertaining test‐statistic is the ratio max/argmax of the maximum value and the maximizing point of the marked empirical process. Under the hypothesis the ratio converges in distribution to the corresponding ratio of a reflected Brownian bridge, for which we derive the distribution function. The test is consistent on the whole alternative and superior to the corresponding Kolmogorov–Smirnov test, which is based only on the maximal value max. Some practical examples of possible applications are given where a certain study about dental phobia is discussed in more detail.

Using randomization to improve performance of a variance estimator of strongly dependent errors

We consider a fixed-design regression model with long-range dependent errors which form a moving average or Gaussian process. We introduce an artificial randomization of grid points at which observations are taken in order to diminish the impact of strong

On Fixed Design Regression for General Linear Processes

This article is concerned with recovering a regression function g(x) on the basis of noisy observations taken at design points x i . The corresponding observations are corrupted by additive dependent noise induced by a general linear process. The regression function is estimated by a smoother, which is shown to have an asymptotic multivariate normal distribution at multiple points. The problem of finding confidence bands for g(x) is discussed. An illustrative example is also exhibited. The results for finite samples are evaluated by computer simulations.

Model checks for parametric regression models

In the paper we deal with the problem of model selection among fixed-design regression models. We establish a new test that indicates whether or not the model fits the data. The test statistic is based on the difference between a parametric estimator for the model variance and a nonparametric difference-based estimator, see Hall et al. (Biometrika 77:521–528, 1990). The weights in the nonparametric estimator depend on n, and they are chosen by solving an optimisation problem in order to obtain a test with high power.

Nonparametric variance function estimation with missing data

In this paper, a fixed design regression model where the errors follow a strictly stationary process is considered. In this model the conditional mean function and the conditional variance function are unknown curves. Correlated errors when observations are missing in the response variable are assumed. Four nonparametric estimators of the conditional variance function based on local polynomial fitting are proposed. Expressions of the asymptotic bias and variance of these estimators are obtained. A simulation study illustrates the behavior of the proposed estimators.

On optimal estimation of a non-smooth mode in a nonparametric regression model with [formula omitted]-mixing errors

We consider the problem of mode estimation in the fixed-design regression model, the regression function having a unique non-smooth mode. We estimate the mode by maximization over the curve estimator, which is given as a weighted mean of the observations, including most of the common kernel estimators, such as Gasser–Müller, Priestley–Chao and Nadaraya–Watson. To obtain optimal rates of convergence of the mode estimator, we first derive upper bounds, where we benefit from the contrast of the curve at the mode rather than taking into account the rate of uniform convergence of the curve estimator. In a next step we show that these rates are also optimal. We prove our results for α -mixing observations, and a non-smooth regression function that is only assumed to be Hölder continuous. Our method consists in a rather direct evaluation of the mean squared error of the empirical mode, using a recent moment inequality of Rosenthal type due to Yang [2007. Maximal moment inequality for partial sums of strong mixing sequences and application. Acta Math. Sinica 23, 1013–1024] for mixing random variables.

Fixed design regression for negatively associated random fields

Data collected on the surface of the earth at different sites often have two- or three-dimensional coordinates associated with it. We assume a simple setting where these sites are integer lattice points, say, 𝒵 N , N ≥ 1, in the N-dimensional Euclidean space R N . Denote n = (n 1, …, n N )∈𝒵 N and I n = {i: i∈𝒵 N , 1 ≤ i k ≤n k , k = 1, …, N}. Consider a simple regression model where the design points x ni 's and the responses Y ni 's are related as follows: Y ni = g(x ni )+ϵ ni , i∈I n , where x ni 's are fixed design points taking values in a compact subset of R d and where g is a bounded real-valued function defined on R d and ϵ ni are negatively associated random disturbances with zero means and finite variances. The function g(x) is estimated by a general linear smoother g n (x). The asymptotic normality of the estimate g n (x) is established under weak conditions, and general conditions under which the bias g n (x) tends to zero are also determined.

Asymptotic normality of wavelet estimator for strong mixing errors

In this paper, we consider a non-parametric fixed design regression model based on strong mixing sequences. We use a new tool presented in the literature recently for the weighted sum estimate, and establish the asymptotic normality of a wavelet estimator for an unknown regression function.

Consistency and uniformly asymptotic normality of wavelet estimator in regression model with associated samples

Abstract Consider the wavelet estimator of a nonparametric fixed design regression function when errors are strictly stationary and associated random variables. We establish pointwise weak consistency and uniformly asymptotic normality of wavelet estimator of regression function. We give rates of uniformly asymptotic normality for associated samples. The rates are near n − 1 / 4 and n − 1 / 6 when their third moments are finite for NA and PA cases, respectively.

Randomized Fixed Design Regression under Long-Range-Dependent Errors

We consider a fixed-design regression model with long-range-dependent errors and introduce an artificial randomization of grid points at which observations are taken in order to diminish the impact of strong dependence. The resulting estimator is shown to exhibit smoothing dichotomy with the variance in both cases, diminishing more quickly than in the fixed-design case.

Conditional density estimation in a regression setting

Regression problems are traditionally analyzed via univariate characteristics like the regression function, scale function and marginal density of regression errors. These characteristics are useful and informative whenever the association between the predictor and the response is relatively simple. More detailed information about the association can be provided by the conditional density of the response given the predictor. For the first time in the literature, this article develops the theory of minimax estimation of the conditional density for regression settings with fixed and random designs of predictors, bounded and unbounded responses and a vast set of anisotropic classes of conditional densities. The study of fixed design regression is of special interest and novelty because the known literature is devoted to the case of random predictors. For the aforementioned models, the paper suggests a universal adaptive estimator which (i) matches performance of an oracle that knows both an underlying model and an estimated conditional density; (ii) is sharp minimax over a vast class of anisotropic conditional densities; (iii) is at least rate minimax when the response is independent of the predictor and thus a bivariate conditional density becomes a univariate density; (iv) is adaptive to an underlying design (fixed or random) of predictors.

On the convergence rate of fixed design regression estimators for negatively associated random variables

Sharp convergence rates for fixed design regression estimators for negatively associated random variables are established. The rates are obtained by means of exponential inequalities, which hold under general conditions.

Maximal Moment Inequality for Partial Sums of Strong Mixing Sequences and Application

Some maximal moment inequalities for partial sums of the strong mixing random variable sequence are established. These inequalities use moment sums as up-boundary and improve the corresponding ones obtained by Shao (1996). To show the application of the inequalities, we apply them to discuss the asymptotic normality of the weight function estimate for the fixed design regression model.

Regression quantiles under dependent censoring

In a recent paper, Braekers and Veraverbeke [Braekers, R. and Veraverbeke, N., 2005, A copula-graphic estimator for the conditional survival function under dependent censoring. Canadian Journal of Statistics, 33, 429–447.] obtained asymptotic results for an estimator of the conditional distribution function of the lifetime, in a fixed design regression model with dependence between lifetime and censoring time. The author complements these findings by corresponding results for the conditional quantile estimator. The main results are uniform strong consistency and a Bahadur-type representation. The latter is an invaluable tool for deriving further asymptotic results, such as the weak convergence of the quantile process.

Adaptive tests for periodic signal detection with applications to laser vibrometry

Initially motivated by a practical issue in target detection via laser vibrometry, we are interested in the problem of periodic signal detection in a Gaussian fixed design regression framework. Assuming that the signal belongs to some periodic Sobolev ball and that the variance of the noise is known, we first consider the problem from a minimax point of view: we evaluate the so-called minimax separation rate which corresponds to the minimal l 2-distance between the signal and zero so that the detection is possible with prescribed probabilities of error. Then, we propose a testing procedure which is available when the variance of the noise is unknown and which does not use any prior information about the smoothness degree or the period of the signal. We prove that it is adaptive in the sense that it achieves, up to a possible logarithmic factor, the minimax separation rate over various periodic Sobolev balls simultaneously. The originality of our approach as compared to related works on the topic of signal detection is that our testing procedure is sensitive to the periodicity assumption on the signal. A simulation study is performed in order to evaluate the effect of this prior assumption on the power of the test. We do observe the gains that we could expect from the theory. At last, we turn to the application to target detection by laser vibrometry that we had in view.

A new algorithm for fixed design regression and denoising

In this paper, we present a new algorithm to estimate a regression function in a fixed design regression model, by piecewise (standard and trigonometric) polynomials computed with an automatic choice of the knots of the subdivision and of the degrees of the polynomials on each sub-interval. First we give the theoretical background underlying the method: the theoretical performances of our penalized least-squares estimator are based on non-asymptotic evaluations of a mean-square type risk. Then we explain how the algorithm is built and possibly accelerated (to face the case when the number of observations is great), how the penalty term is chosen and why it contains some constants requiring an empirical calibration. Lastly, a comparison with some well-known or recent wavelet methods is made: this brings out that our algorithm behaves in a very competitive way in term of denoising and of compression.

Fixed design regression quantiles for time series

This paper studies nonparametric estimation of regression quantiles under the fixed design model. We suppose that the error random variables are coming from a strictly stationary stochastic process satisfying the strong mixing condition. The joint asymptotic normality for the estimators of several quantiles is given. The same property is established for the regression quantile estimator at different fixed design points.

LOCAL MEDIAN ESTIMATION OF VARIANCE FUNCTION

This paper considers local median estimation in fixed design regression problems. The proposed method is employed to estimate the median function and the variance function of a heteroscedastic regression model. Strong convergence rates of the proposed estimators are obtained. Simulation results are given to show the performance of the proposed methods.

Uniformly asymptotic normality of the regression weighted estimator for negatively associated samples

In this paper, we discuss the uniformly asymptotic normality of the weighted function estimate of the fixed design regression model for negatively associated samples. We give the rates of uniform asymptotic normality. The rate is near n −1/4 when the third moment is finite.

Fixed-design regression for Linear time series

This paper obtains asymptotic normality for double array sum of linear time series ξt, and gives its application in the regression model. This generalizes the main results in [1].