Weak convergence of the empirical process of residuals in linear models with many parameters

Chen, Gemai; Lockhart, Richard

doi:10.1214/aos/1009210688

Cited by 13 publications

(6 citation statements)

References 11 publications

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“…Weak convergence results for F n play a central role in, for example, testing the goodness-of-fit of error distributions or in the derivation of the asymptotic behavior of more sophisticated estimators for F ; see Koul and Qian (2002) for an overview. First results on the asymptotic behavior of F n were derived in Koul (1969) and Loynes (1980) (in generalized regression models), and more recently those findings were extended in various directions such as, for instance, time series analysis [see Koul and Qian (2002), Engler and Nielsen (2009), and the references cited therein] or coefficient vectors of growing dimension [see Chen and Lockhart (2001), for an overview]. All of those extensions share the assumption that F has a continuous probability density function f .…”

Section: Error Distributions In Regression Models Consider a Linear R...mentioning

confidence: 99%

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

Bücher¹,

Segers²,

Volgushev³

2014

Ann. Statist.

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In the past decades, weak convergence theory for stochastic processes has become a standard tool for analyzing the asymptotic properties of various statistics. Routinely, weak convergence is considered in the space of bounded functions equipped with the supremum metric. However, there are cases when weak convergence in those spaces fails to hold. Examples include empirical copula and tail dependence processes and residual empirical processes in linear regression models in case the underlying distributions lack a certain degree of smoothness. To resolve the issue, a new metric for locally bounded functions is introduced and the corresponding weak convergence theory is developed. Convergence with respect to the new metric is related to epiand hypo-convergence and is weaker than uniform convergence. Still, for continuous limits, it is equivalent to locally uniform convergence, whereas under mild side conditions, it implies L p convergence. For the examples mentioned above, weak convergence with respect to the new metric is established in situations where it does not occur with respect to the supremum distance. The results are applied to obtain asymptotic properties of resampling procedures and goodness-of-fit tests.

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Section: Error Distributions In Regression Models Consider a Linear R...mentioning

confidence: 99%

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

Bücher¹,

Segers²,

Volgushev³

2014

Ann. Statist.

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show abstract

“…We do not permit such a large p as our hypothesis of interest concerns a p ‐dimensional restriction and the design matrix of time‐series data faces more difficulties in satisfying the rank condition. In this regard, Chen and Lockhart ( 2001 ) provide an interesting example from an ANOVA design where the weak convergence of the empirical distribution of residuals from the linear regression with growing dimension fails when the dimension p is of order

n^{1 false/ 3}

. They compare various growth conditions for p in the literature and conclude that

p^{3} {normallog}^{2} p = o false(n false)

is nearly necessary for a general stochastic design.…”

Section: Asymptotic Distribution Of Scriptqnmentioning

confidence: 99%

Robust Inference on Infinite and Growing Dimensional Time‐Series Regression

Gupta

Seo

2023

ECTA

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We develop a class of tests for time‐series models such as multiple regression with growing dimension, infinite‐order autoregression, and nonparametric sieve regression. Examples include the Chow test and general linear restriction tests of growing rank p. Employing such increasing p asymptotics, we introduce a new scale correction to conventional test statistics, which accounts for a high‐order long‐run variance (HLV), which emerges as p grows with sample size. We also propose a bias correction via a null‐imposed bootstrap to alleviate finite‐sample bias without sacrificing power unduly. A simulation study shows the importance of robustifying testing procedures against the HLV even when p is moderate. The tests are illustrated with an application to the oil regressions in Hamilton (2003).

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“…Another interesting issue is whether the rate p = o(n 1/3 / log 4/3 n) achieves the optimal in the leading term n 1/3 or not. Chen and Lockhart (2001) showed that for linear models with fixed design, the residual empirical process admits an uniformly asymptotic linear representation, and thus converges to a zero-mean Gaussian process provided p = o(n 1/3 / log 2/3 n). They further gave a counterexample to show that the leading term n 1/3 cannot be improved, in general.…”

Section: 3)mentioning

confidence: 99%

Weighted residual empirical processes, martingale transformations and model checking for regressions

Tan¹,

Xu²,

Zhu³

2022

Preprint

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In this paper we propose a new methodology for testing the parametric forms of the mean and variance functions based on weighted residual empirical processes and their martingale transformations in regression models. The dimensions of the parameter vectors can be divergent as the sample size goes to infinity. We then study the convergence of weighted residual empirical processes and their martingale transformation under the null and alternative hypotheses in the diverging dimension setting. The proposed tests based on weighted residual empirical processes can detect local alternatives distinct from the null at the fastest possible rate of order n −1/2 but are not asymptotically distribution-free. While the tests based on martingale transformed weighted residual empirical processes can be asymptotically distribution-free, yet, unexpectedly, can only detect the local alternatives converging to the null at a much slower rate of order n −1/4 , which is somewhat different from existing asymptotically distribution-free tests based on martingale transformations. As the tests based on the residual empirical process are not distribution-free, we propose a smooth residual bootstrap and verify the validity of its approximation in diverging dimension settings. Simulation studies and a real data example are conducted to illustrate the effectiveness of our tests.

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Weak convergence of the empirical process of residuals in linear models with many parameters

Cited by 13 publications

References 11 publications

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

When uniform weak convergence fails: Empirical processes for dependence functions and residuals via epi- and hypographs

Robust Inference on Infinite and Growing Dimensional Time‐Series Regression

Weighted residual empirical processes, martingale transformations and model checking for regressions

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