Testing Monotonicity of Regression Functions – An Empirical Process Approach

Birke, Melanie; Neumeyer, Natalie

doi:10.1111/j.1467-9469.2012.00820.x

Cited by 7 publications

(6 citation statements)

References 28 publications

(37 reference statements)

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“…We are not aware of hypothesis tests for monotonicity or other shape constraints in the context of boundary regression, but would like to mention Gijbels' (2005) review on testing for monotonicity in mean regression. Tests similar in spirit to the one we are suggesting here were considered by Birke and Neumeyer (2013) and Birke et al (2016+) for mean and quantile regression models, respectively.…”

Section: Introductionmentioning

confidence: 99%

Estimation and hypotheses testing in boundary regression models

Drees¹,

Neumeyer²,

Selk³

2019

Bernoulli

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Consider a nonparametric regression model with one-sided errors and regression function in a general Hölder class. We estimate the regression function via minimization of the local integral of a polynomial approximation. We show uniform rates of convergence for the simple regression estimator as well as for a smooth version. These rates carry over to mean regression models with a symmetric and bounded error distribution. In such a setting, one obtains faster rates for irregular error distributions concentrating sufficient mass near the endpoints than for the usual regular distributions. The results are applied to prove asymptotic √ n-equivalence of a residual-based (sequential) empirical distribution function to the (sequential) empirical distribution function of unobserved errors in the case of irregular error distributions. This result is remarkably different from corresponding results in mean regression with regular errors. It can readily be applied to develop goodness-of-fit tests for the error distribution. We present some examples and investigate the small sample performance in a simulation study. We further discuss asymptotically distribution-free hypotheses tests for independence of the error distribution from the points of measurement and for monotonicity of the boundary function as well.

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Section: Introductionmentioning

confidence: 99%

Estimation and hypotheses testing in boundary regression models

Drees¹,

Neumeyer²,

Selk³

2019

Bernoulli

Self Cite

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“…Then exactly as in the last part of the proof of Lemma B.1 in the supporting information to Birke and Neumeyer (2013)…”

Section: Now Let Eithermentioning

confidence: 80%

Heteroscedastic semiparametric transformation models: estimation and testing for validity

Neumeyer¹,

Noh²,

Keilegom³

2017

STAT SINICA

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In this paper we consider a heteroscedastic transformation model of the form Λ ϑ (Y ) = m(X) + σ(X)ε, where Λ ϑ belongs to a parametric family of monotone transformations, m(·) and σ(·) are unknown but smooth functions, ε is independent of the d-dimensional vector of covariates X, E(ε) = 0 and Var(ε) = 0. In this model, we first consider the estimation of the unknown components of the model, namely ϑ, m(·), σ(·) and the distribution of ε, and we show the asymptotic normality of the proposed estimators. Second, we propose tests for the validity of the model, and establish the limiting distribution of the test statistics under the null hypothesis. A bootstrap procedure is proposed to approximate the critical values of the tests. Finally, we carry out a simulation study to verify the small sample behavior of the proposed estimators and tests.

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“…, where D is defined in (7), g (n) (t) → g(t), as defined in (5), and b −1 k t−u b l(u) du → l(t), we find that…”

Section: Proofs For Sectionmentioning

confidence: 83%

“…The problem of testing a nonparametric null hypothesis of monotonicity has gained a lot of interest in the literature (see for example [29] for the density setting, [26], [23] for the hazard rate, [1], [4], [5], [18] for the regression function).…”

Section: Testingmentioning

confidence: 99%

Central limit theorems for the $L_{p}$-error of smooth isotonic estimators

Lopuhaä¹,

Musta²

2019

Electron. J. Statist.

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We investigate the asymptotic behavior of the Lp-distance between a monotone function on a compact interval and a smooth estimator of this function. Our main result is a central limit theorem for the Lp-error of smooth isotonic estimators obtained by smoothing a Grenander-type estimator or isotonizing the ordinary kernel estimator. As a preliminary result we establish a similar result for ordinary kernel estimators. Our results are obtained in a general setting, which includes estimation of a monotone density, regression function and hazard rate. We also perform a simulation study for testing monotonicity on the basis of the L2-distance between the kernel estimator and the smoothed Grenander-type estimator.MSC 2010 subject classifications: Primary 62G20; secondary 62G10.

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Testing Monotonicity of Regression Functions – An Empirical Process Approach

Cited by 7 publications

References 28 publications

Estimation and hypotheses testing in boundary regression models

Estimation and hypotheses testing in boundary regression models

Heteroscedastic semiparametric transformation models: estimation and testing for validity

Central limit theorems for the $L_{p}$-error of smooth isotonic estimators

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