Weighted angle Radon transform: Convergence rates and efficient estimation

Hohmann, Daniel; Holzmann, Hajo

doi:10.5705/ss.2014.064

Cited by 5 publications

(5 citation statements)

References 22 publications

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“…This happens if the normalized design variables Θ i = X i / X i systematically miss observations from some directions. In this case the problem becomes unevenly harder and only logarithmic convergence rates can be obtained (see Davison, 1983;Frikel, 2013;Hohmann and Holzmann, 2016). When the support of Θ i does not contain an open ball, f β might be non-identifiable.…”

Section: The Random Coefficients Model As An Inverse Problemmentioning

confidence: 99%

See 1 more Smart Citation

Tests for qualitative features in the random coefficients model

Dunker¹,

Eckle²,

Proksch³

et al. 2019

Electron. J. Statist.

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The random coefficients model is an extension of the linear regression model that allows for unobserved heterogeneity in the population by modeling the regression coefficients as random variables. Given data from this model, the statistical challenge is to recover information about the joint density of the random coefficients which is a multivariate and ill-posed problem. Because of the curse of dimensionality and the ill-posedness, pointwise nonparametric estimation of the joint density is difficult and suffers from slow convergence rates. Larger features, such as an increase of the density along some direction or a well-accentuated mode can, however, be much easier detected from data by means of statistical tests. In this article, we follow this strategy and construct tests and confidence statements for qualitative features of the joint density, such as increases, decreases and modes. We propose a multiple testing approach based on aggregating single tests which are designed to extract shape information on fixed scales and directions. Using recent tools for Gaussian approximations of multivariate empirical processes, we derive expressions for the critical value. We apply our method to simulated and real data.

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Section: The Random Coefficients Model As An Inverse Problemmentioning

confidence: 99%

“…. , n, and define S i := ζ i Y i / X i and Θ i : Davison, 1983;Frikel, 2013;Hohmann and Holzmann, 2016). When the support of Θ i does not contain an open ball, f β might be non-identifiable.…”

Section: Introductionmentioning

confidence: 99%

Tests for qualitative features in the random coefficients model

Dunker¹,

Eckle²,

Proksch³

et al. 2019

Electron. J. Statist.

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show abstract

“…Yet in contrast to Hoderlein et al [2010] (see their Section 4.3), no heavy-tailedness of covariates X is required for our convergence results. In particular, Hohmann and Holzmann [2016] showed under Gaussian white noise that the rate of convergence can be much slower, even severely ill-posed, under lighter tails of X. Also note that the condition (3.10) ensures that n −1 K dominates the squared bias of estimating h and is only a mild restriction.…”

Section: Rate Of Convergencementioning

confidence: 99%

Varying Random Coefficient Models

Breunig¹

2018

Preprint

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This paper provides a new methodology to analyze unobserved heterogeneity when observed characteristics are modeled nonlinearly. The proposed model builds on varying random coefficients (VRC) that are determined by nonlinear functions of observed regressors and additively separable unobservables. This paper proposes a novel estimator of the VRC density based on weighted sieve minimum distance. The main example of sieve bases are Hermite functions which yield a numerically stable estimation procedure. This paper shows inference results that go beyond what has been shown in ordinary RC models. We provide in each case rates of convergence and also establish pointwise limit theory of linear functionals, where a prominent example is the density of potential outcomes. In addition, a multiplier bootstrap procedure is proposed to construct uniform confidence bands. A Monte Carlo study examines finite sample properties of the estimator and shows that it performs well even when the regressors associated to RC are far from being heavy tailed. Finally, the methodology is applied to analyze heterogeneity in income elasticity of demand for housing.

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“…This is a much more useful and realistic framework for the random coefficients model. When p = 1, this is related to limited angle tomography (see, e.g., [20,32]). There, one has measurements over a subset of angles and the unknown density has support in the unit disk.…”

Section: Introductionmentioning

confidence: 99%

Adaptive estimation in the linear random coefficients model when regressors have limited variation

Gaillac

Gautier

2022

Bernoulli

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We consider a linear model where the coefficients -intercept and slopes -are random and independent from regressors which support is a proper subset. When the slopes do not have heavy tails, the joint density of the random coefficients is identified. Lower bounds on the supremum risk for the estimation of the density are derived for this model and a related white noise model. We present an estimator, its rates of convergence, and a data-driven rule which delivers adaptive estimators. The corresponding R package is RandomCoefficients.

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Weighted angle Radon transform: Convergence rates and efficient estimation

Cited by 5 publications

References 22 publications

Tests for qualitative features in the random coefficients model

Tests for qualitative features in the random coefficients model

Varying Random Coefficient Models

Adaptive estimation in the linear random coefficients model when regressors have limited variation

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