Additive regression with Hilbertian responses

Jeon, Jeong Min; Park, Byeong U.

doi:10.1214/19-aos1902

Cited by 21 publications

(45 citation statements)

References 42 publications

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“…The notion of Bochner integral generalizes that of the conventional Lebesgue integral to maps taking values in Hilbert or more generally in Banach spaces. By Propositions 2.1 and 2.2 in [15], the expected values E(m j (Z j )) and also the conditional expected values E(m k (Z k )|Z j = z j ) for k = j may be written in Bochner integrals as…”

Section: General Bochner Sbf Estimationmentioning

confidence: 99%

“…It is a general belief that a political election is determined by population characteristics and underlying political orientation. Recently, [15] analyzed the 2017 Korea presidential election data to see the effects of these factors. In that study, however, the effect of underlying political orientation, which is believed to be one of the most important factors, could not be analyzed, because the earlier method can only deal with continuous predictors.…”

Section: Compositional Responsementioning

confidence: 99%

“…The latter two works are about nonparametric regression but based on full-dimensional (i.e., unstructured) modeling, so that their approaches suffer from the curse of dimensionality when the number of predictors increases. Recently, nonparametric additive regression has been developed for Hilbertian responses [15]. Additive modeling is known to be an efficient way of avoiding the dimensionality problem.…”

Section: Introductionmentioning

confidence: 99%

“…The predictors in our model can be general objects including nominal and ordinal discrete variables as well as continuous ones. Hence, our coverage includes the case where only continuous predictors are considered [15]. In addition, we allow for incompletely observed responses.…”

Section: Introductionmentioning

confidence: 99%

“…The theoretical developments here are based on kernel weighting schemes for the estimation of the marginal densities and the marginal regression maps. They are much more complex than in [15]. The latter work considers only continuous predictors and completely observed responses, and thus all stochastic terms from the asymptotic expansion of their estimator are based on kernels of the same type and bandwidths of the same size.…”

Section: Introductionmentioning

confidence: 99%

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Additive regression for predictors of various natures and possibly incomplete Hilbertian responses

Jeon

Park

Keilegom

2021

Electron. J. Statist.

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In this paper we consider a fully nonparametric additive regression model for responses and predictors of various natures. This includes the case of Hilbertian and incomplete (like censored or missing) responses, and continuous, nominal discrete and ordinal discrete predictors. We propose a backfitting technique that estimates this additive model, and establish the existence of the estimator and the convergence of the associated backfitting algorithm under minimal conditions. We also develop a general asymptotic theory for the estimator such as the rates of convergence and asymptotic distribution. We verify the practical performance of the proposed estimator in a simulation study. We also apply the method to various real data sets, including those for a density-valued response regressed on a mixture of continuous and nominal discrete predictors, for a compositional response regressed on a mixture of continuous and ordinal discrete predictors, and for a censored scalar response regressed on a mixture of continuous and nominal discrete predictors.

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Section: General Bochner Sbf Estimationmentioning

confidence: 99%

Section: Compositional Responsementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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