Comparison of error distributions in nonparametric regression
Section snippets
Introduction and statistical model
The equality of the error distributions is a common assumption in many statistical methods involving several regression models, such as some tests of equality of regression curves (see for example Hall and Hart, 1990; Kulasekera and Wang, 2001, or Neumeyer and Dette, 2003). In this paper we consider a procedure to test this hypothesis in a nonparametric framework. It can be seen as an extension of the comparison of several populations, called ‘-samples problem’, where the variables are not
Testing procedure
Let (respectively, ) be the common error distribution function (respectively, density) under the null hypothesis. Firstly, consider the following estimator of the error distribution in population , for :whereandare kernel estimators of the regression curves and conditional variances in each population, where are Nadaraya–Watson-type
Asymptotic results
Let us firstly state some notation and list the regularity assumptions needed in order to prove our main results. For , let and be the distribution of the covariate and the conditional distribution of the response given the covariate, respectively, and let be the density corresponding to .
(A1): For ,
- (i)
is absolutely continuous with compact support and density .
- (ii)
, and are two times continuously differentiable.
- (iii)
and
Bootstrap, simulations and real-data analysis
Bootstrap approximation: The asymptotic distributions of the test statistics and given in Corollary 2 are complicated. In practical applications, the critical values of the test can be approximated by the bootstrap procedure described below.
Let be the standardized version of the empirical distribution function of the joint sample of estimated residuals considered in (6). This standardization is performed in order to obtain samples of residuals from a distribution verifying the
Proofs
Proof of Theorem 1 From Theorem 1 in Akritas and Van Keilegom (2001) it is easy to obtain the following representation: uniformly in , where Note that . By simply writing , we have To proof the weak convergence of the -dimensional process
References (12)
- et al.
Nonparametric estimation of the residual distribution
Scand. J. Statist.
(2001) Visualizing Data
(1993)Bootstrapping regression models
Ann. Statist.
(1981)- et al.
Bootstrap test for difference between means in nonparametric regression
J. Amer. Statist. Assoc.
(1990) - et al.
A test of equality of regression curves using Gâteaux scores
Austral. N. Z. J. Statist.
(2001) - Mora, J., Neumeyer, N., 2005. The two-sample problem with regression errors: an empirical process approach...
Cited by (10)
On nonparametric comparison of images and regression surfaces
2010, Journal of Statistical Planning and InferenceEstimating the error distribution in nonparametric multiple regression with applications to model testing
2010, Journal of Multivariate AnalysisCitation Excerpt :In this context [3] consider goodness-of-fit tests for the regression function and Dette et al. [4] propose a goodness-of-fit test for the variance function. For comparison of several independent regression models, [5,6] investigated tests for equality of regression functions and tests for equality of error distributions, respectively. Neumeyer et al. [7] suggested a goodness-of-fit test for the error distribution.
A weighted bootstrap approximation for comparing the error distributions in nonparametric regression
2017, Journal of Statistical Computation and SimulationTests for the equality of conditional variance functions in nonparametric regression
2015, Electronic Journal of StatisticsTesting the difference between two sets of data using comparison two linear regression functions
2014, Electronic Journal of Applied Statistical Analysis
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Financial support from the Spanish Ministerio de Educación y Ciencia (with additional European FEDER support) through the project MTM2005-00820.