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Lijian Yang - One of the best experts on this subject based on the ideXlab platform.

  • smooth simultaneous Confidence Band for the error distribution function in nonparametric regression
    Computational Statistics & Data Analysis, 2021
    Co-Authors: Suojin Wang, Lijian Yang
    Abstract:

    Abstract A smooth simultaneous Confidence Band (SCB) is constructed for the distribution of unobserved errors in a nonparametric regression model based on a plug-in kernel distribution estimator. The normalized estimation error process is shown to converge to a Gaussian process. Simulation experiments indicate that the proposed SCB not only strikes an intelligent balance between coverage probability and precision, but also achieves surprisingly as much as double efficiency of the classical infeasible SCB. Furthermore, extensive empirical studies are carried out to compare the proposed method with the smooth residual bootstrap method in order to demonstrate the usefulness of each of these methods. As an illustration, the proposed SCB is applied to the Old Faithful geyser data for testing the error distribution.

  • simultaneous Confidence Band for the difference of regression functions of two samples
    Communications in Statistics-theory and Methods, 2020
    Co-Authors: Jiakun Jiang, Li Cai, Lijian Yang
    Abstract:

    This paper concerns the comparison of two sample non parametric regression. An asymptotically correct simultaneous Confidence Band (SCB) is proposed for the difference of two-sample non parametric ...

  • simultaneous Confidence Band for stationary covariance function of dense functional data
    Journal of Multivariate Analysis, 2020
    Co-Authors: Jiangyan Wang, Guanqun Cao, Li Wang, Lijian Yang
    Abstract:

    Abstract The inference via simultaneous Confidence Band is studied for stationary covariance function of dense functional data. A two-stage estimation procedure is proposed based on spline approximation, the first stage involving estimation of all the individual trajectories and the second stage involving estimation of the covariance function through smoothing the empirical covariance function. The proposed covariance estimator is smooth and as efficient as the oracle estimator when all individual trajectories are known. An asymptotic simultaneous Confidence Band (SCB) is developed for the true covariance function, and the coverage probabilities are shown to be asymptotically correct. Intensive simulation experiments are conducted to demonstrate the performance of the proposed estimator and SCB. The proposed method is also illustrated with a real data example.

  • simultaneous Confidence Band for stationary covariance function of dense functional data
    arXiv: Methodology, 2019
    Co-Authors: Jiangyan Wang, Guanqun Cao, Li Wang, Lijian Yang
    Abstract:

    Inference via simultaneous Confidence Band is studied for stationary covariance function of dense functional data. A two-stage estimation procedure is proposed based on spline approximation, the first stage involving estimation of all the individual trajectories and the second stage involving estimation of the covariance function through smoothing the empirical covariance function. The proposed covariance estimator is smooth and as efficient as the oracle estimator when all individual trajectories are known. An asymptotic simultaneous Confidence Band (SCB) is developed for the true covariance function, and the coverage probabilities are shown to be asymptotically correct. Simulation experiments are conducted on the numerical performance of the proposed estimator and SCB. The proposed method is also illustrated by two real data examples.

  • a smooth simultaneous Confidence Band for correlation curve
    Test, 2018
    Co-Authors: Yuanyuan Zhang, Lijian Yang
    Abstract:

    A plug-in estimator is proposed for a local measure of variance explained by regression, termed correlation curve in Doksum et al. (J Am Stat Assoc 89:571–582, 1994), consisting of a two-step spline–kernel estimator of the conditional variance function and local quadratic estimator of first derivative of the mean function. The estimator is oracally efficient in the sense that it is as efficient as an infeasible correlation estimator with the variance function known. As a consequence of the oracle efficiency, a smooth simultaneous Confidence Band (SCB) is constructed around the proposed correlation curve estimator and shown to be asymptotically correct. Simulated examples illustrate the versatility of the proposed oracle SCB which confirms the asymptotic theory. Application to a 1995 British Family Expenditure Survey data has found marginally significant evidence for a local version of Engel’s law, i.e., food budget share and household real income are inversely related (Hamilton in Am Econ Rev 91:619–630, 2001).

Song Song - One of the best experts on this subject based on the ideXlab platform.

  • Bootstrap Confidence Bands and partial linear quantile regression
    Journal of Multivariate Analysis, 2012
    Co-Authors: Song Song, Ya'acov Ritov, Wolfgang Karl Härdle
    Abstract:

    AbstractIn this paper bootstrap Confidence Bands are constructed for nonparametric quantile estimates of regression functions, where resampling is done from a suitably estimated empirical distribution function (edf) for residuals. It is known that the approximation error for the Confidence Band by the asymptotic Gumbel distribution is logarithmically slow. It is proved that the bootstrap approximation provides an improvement. The case of multidimensional and discrete regressor variables is dealt with using a partial linear model. An economic application considers the labor market differential effect with respect to different education levels

  • Partial Linear Quantile Regression and Bootstrap Confidence Bands
    2010
    Co-Authors: Wolfgang Karl Härdle, Ya'acov Ritov, Song Song
    Abstract:

    In this paper uniform Confidence Bands are constructed for nonparametric quantile estimates of regression functions. The method is based on the bootstrap, where resampling is done from a suitably estimated empirical density function (edf) for residuals. It is known that the approximation error for the uniform Confidence Band by the asymptotic Gumbel distribution is logarithmically slow. It is proved that the bootstrap approximation provides a substantial improvement. The case of multidimensional and discrete regressor variables is dealt with using a partial linear model. Comparison to classic asymptotic uniform Bands is presented through a simulation study. An economic application considers the labour market differential effect with respect to different education levels.

  • Confidence BandS IN QUANTILE REGRESSION
    Econometric Theory, 2009
    Co-Authors: Wolfgang Karl Härdle, Song Song
    Abstract:

    Let (X1, Y1), …, (Xn, Yn) be independent and identically distributed random variables and let l(x) be the unknown p-quantile regression curve of Y conditional on X. A quantile smoother ln(x) is a localized, nonlinear estimator of l(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {ln(x) – l(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1 |ln(x) − l(x)|. The derived result helps in the construction of a uniform Confidence Band for the quantile curve l(x). This Confidence Band can be applied as a econometric model check. An economic application considers the relation between age and earnings in the labor market by means of parametric model specification tests, which presents a new framework to describe trends in the entire wage distribution in a parsimonious way.

  • The Stochastic Fluctuation of the Quantile Regression Curve
    Social Science Research Network, 2008
    Co-Authors: Wolfgang Karl Härdle, Song Song
    Abstract:

    Let (X1, Y1), . . ., (Xn, Yn) be i.i.d. rvs and let l(x) be the unknown p-quantile regression curve of Y on X. A quantile-smoother ln(x) is a localised, nonlinear estimator of l(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {ln(x) - l(x)}. Using strong approximations of the empirical process and extreme value theory allows us to consider the asymptotic maximal deviation sup06x61 |ln(x)-l(x)|. The derived result helps in the construction of a uniform Confidence Band for the quantile curve l(x). This Confidence Band can be applied as a model check, e.g. in econometrics. An application considers a labour market discrimination effect.

A J Hayter - One of the best experts on this subject based on the ideXlab platform.

  • a ray method of Confidence Band construction for multiple linear regression models
    Journal of Statistical Planning and Inference, 2009
    Co-Authors: A J Hayter, Wei Liu, P Ahkine
    Abstract:

    This paper addresses the problem of Confidence Band construction for a standard multiple linear regression model. A “ray” method of construction is developed which generalizes the method of Graybill and Bowden [1967. Linear segment Confidence Bands for simple linear regression models. J. Amer. Statist. Assoc. 62, 403–408] for a simple linear regression model to a multiple linear regression model. By choosing suitable directions for the rays this method requires only critical points from t-distributions so that the Confidence Bands are easy to construct. Both one-sided and two-sided Confidence Bands can be constructed using this method. An illustration of the new method is provided.

  • minimum area Confidence set optimality for Confidence Bands in simple linear regression
    Journal of the American Statistical Association, 2007
    Co-Authors: A J Hayter
    Abstract:

    The average width of a simultaneous Confidence Band has been used by several authors (e.g., Naiman and Piegorsch) as a criterion for the comparison of different Confidence Bands. In this article the area of the Confidence set that corresponds to a Confidence Band is used as a new criterion. For simple linear regression, comparisons have been carried out under this new criterion between hyperbolic Bands, two-segment Bands, and three-segment Bands, which include constant width Bands as special cases. It is found that if one requires a Confidence Band over the whole range of the covariate, then the best Confidence Band is given by the Working and Hotelling hyperbolic Band. Furthermore, if one needs a Confidence Band over a finite interval of the covariate, then a restricted hyperbolic Band can again be recommended, although a three-segment Band may be very slightly superior in certain cases.

Wolfgang Karl Härdle - One of the best experts on this subject based on the ideXlab platform.

  • Bootstrap Confidence Bands and partial linear quantile regression
    Journal of Multivariate Analysis, 2012
    Co-Authors: Song Song, Ya'acov Ritov, Wolfgang Karl Härdle
    Abstract:

    AbstractIn this paper bootstrap Confidence Bands are constructed for nonparametric quantile estimates of regression functions, where resampling is done from a suitably estimated empirical distribution function (edf) for residuals. It is known that the approximation error for the Confidence Band by the asymptotic Gumbel distribution is logarithmically slow. It is proved that the bootstrap approximation provides an improvement. The case of multidimensional and discrete regressor variables is dealt with using a partial linear model. An economic application considers the labor market differential effect with respect to different education levels

  • Partial Linear Quantile Regression and Bootstrap Confidence Bands
    2010
    Co-Authors: Wolfgang Karl Härdle, Ya'acov Ritov, Song Song
    Abstract:

    In this paper uniform Confidence Bands are constructed for nonparametric quantile estimates of regression functions. The method is based on the bootstrap, where resampling is done from a suitably estimated empirical density function (edf) for residuals. It is known that the approximation error for the uniform Confidence Band by the asymptotic Gumbel distribution is logarithmically slow. It is proved that the bootstrap approximation provides a substantial improvement. The case of multidimensional and discrete regressor variables is dealt with using a partial linear model. Comparison to classic asymptotic uniform Bands is presented through a simulation study. An economic application considers the labour market differential effect with respect to different education levels.

  • Uniform Confidence Bands for Pricing Kernels
    SSRN Electronic Journal, 2010
    Co-Authors: Wolfgang Karl Härdle, Yarema Okhrin, Weining Wang
    Abstract:

    Pricing kernels implicit in option prices play a key role in assessing the risk aversion over equity returns. We deal with non-parametric estimation of the pricing kernel (Empirical Pricing Kernel) given by the ratio of the risk-neutral density estimator and the subjective density estimator. The former density can be represented as the second derivative w.r.t. the European call option price function, which we estimate by non-parametric regression. The subjective density is estimated non-parametrically too. In this framework, we develop the asymptotic distribution theory of the EPK in the L1 sense. Particularly, to evaluate the overall variation of the pricing kernel, we develop a uniform Confidence Band of the EPK. Furthermore, as an alternative to the asymptotic approach, we propose a bootstrap Confidence Band. The developed theory is helpful for testing parametric specifications of pricing kernels and has a direct extension to estimating risk aversion patterns. The established results are assessed and compared in a Monte-Carlo study. As a real application, we test risk aversion over time induced by the EPK.

  • Confidence BandS IN QUANTILE REGRESSION
    Econometric Theory, 2009
    Co-Authors: Wolfgang Karl Härdle, Song Song
    Abstract:

    Let (X1, Y1), …, (Xn, Yn) be independent and identically distributed random variables and let l(x) be the unknown p-quantile regression curve of Y conditional on X. A quantile smoother ln(x) is a localized, nonlinear estimator of l(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {ln(x) – l(x)}. Using strong approximations of the empirical process and extreme value theory, we consider the asymptotic maximal deviation sup0≤x≤1 |ln(x) − l(x)|. The derived result helps in the construction of a uniform Confidence Band for the quantile curve l(x). This Confidence Band can be applied as a econometric model check. An economic application considers the relation between age and earnings in the labor market by means of parametric model specification tests, which presents a new framework to describe trends in the entire wage distribution in a parsimonious way.

  • The Stochastic Fluctuation of the Quantile Regression Curve
    Social Science Research Network, 2008
    Co-Authors: Wolfgang Karl Härdle, Song Song
    Abstract:

    Let (X1, Y1), . . ., (Xn, Yn) be i.i.d. rvs and let l(x) be the unknown p-quantile regression curve of Y on X. A quantile-smoother ln(x) is a localised, nonlinear estimator of l(x). The strong uniform consistency rate is established under general conditions. In many applications it is necessary to know the stochastic fluctuation of the process {ln(x) - l(x)}. Using strong approximations of the empirical process and extreme value theory allows us to consider the asymptotic maximal deviation sup06x61 |ln(x)-l(x)|. The derived result helps in the construction of a uniform Confidence Band for the quantile curve l(x). This Confidence Band can be applied as a model check, e.g. in econometrics. An application considers a labour market discrimination effect.

Julian J. Faraway - One of the best experts on this subject based on the ideXlab platform.

  • Confidence Bands for smoothness in nonparametric regression
    Stat, 2016
    Co-Authors: Julian J. Faraway
    Abstract:

    The choice of the smoothing parameter in nonparametric regression is critical to the form of the estimated curve and any inference that follows. Many methods are available that will generate a single choice for this parameter. Here, we argue that the considerable uncertainty in this choice should be explicitly represented. The construction of standard simultaneous Confidence Bands in nonparametric regression often requires difficult mathematical arguments. We question their practical utility, presenting several deficiencies. We propose a new kind of Confidence Band that reflects the uncertainty regarding the smoothness of the estimate. Copyright © 2016 John Wiley & Sons, Ltd.

  • Confidence Bands for smoothness in nonparametric regression
    Stat, 2016
    Co-Authors: Julian J. Faraway
    Abstract:

    The choice of the smoothing parameter in nonparametric regression is critical to the form of the estimated curve and any inference that follows. Many methods are available that will generate a single choice for this parameter. Here we argue that the considerable uncertainty in this choice should be explicitly represented. The construction of standard simultaneous Confidence Bands in nonparametric regression often requires difficult mathematical arguments. We question their practical utility, presenting several deficiencies. We propose a new kind of Confidence Band that reflects the uncertainty regarding the smoothness of the estimate