Multivariate Local Polynomial Kernel Estimators: Leading Bias and Asymptotic Distribution

Gu, Jingping; Li, Qi; Yang, Jui‐Chung

doi:10.1080/07474938.2014.956615

Cited by 20 publications

(13 citation statements)

References 10 publications

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“…An advantage of using local polynomial estimators, compared for example to spline or wavelet estimators, is that the bias and variance can be derived analytically. For the univariate case these results can be found in Fan and Gijbels (1996) and for the multivariate case in Masry (1996) and Gu et al (2015). We summarize them in terms of order of convergence below…”

Section: Estimating the Basis Functions -We Keep Notations ν = D To Rmentioning

confidence: 81%

Functional Principal Component Analysis for Derivatives of Multivariate Curves

et al. 2016

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We present two methods based on functional principal component analysis (FPCA) for the estimation of smooth derivatives of a sample of random functions, which are observed in a more than one-dimensional domain. We apply eigenvalue decomposition to a) the dual covariance matrix of the derivatives, and b) the dual covariance matrix of the observed curves. To handle noisy data from discrete observations, we rely on local polynomial regressions. If curves are contained in a finite-dimensional function space, the second method performs better asymptotically. We apply our methodology in a simulation and empirical study, in which we estimate state price density (SPD) surfaces from call option prices. We identify three main components, which can be interpreted as volatility, skewness and tail factors. We also find evidence for term structure variation.

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Section: Estimating the Basis Functions -We Keep Notations ν = D To Rmentioning

confidence: 81%

Functional Principal Component Analysis for Derivatives of Multivariate Curves

et al. 2016

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“…, E( F D (x)) T and the expectation taken in each component is with respect to the marginal distribution of P (dy|x). Then by Theorem 1 of Gu et al (2014), the following holds:…”

Section: Resultsmentioning

confidence: 99%

“…Given the higher order smoothness assumption on µ(x), one can make higher order approximations and using a local polynomials regression estimate would result in the reduction of bias term in estimating µ(x). The asymptotic distribution for multivariate local regression estimator for Euclidean responses has been derived (Gu et al, 2014;Ruppert and Wand, 1994;Masry, 1996), and we leverage on some of their results in our proof.…”

Section: Resultsmentioning

confidence: 99%

Extrinsic Local Regression on Manifold-Valued Data

Lin

Thomas

Zhu

et al. 2017

Journal of the American Statistical Association

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We propose an extrinsic regression framework for modeling data with manifold valued responses and Euclidean predictors. Regression with manifold responses has wide applications in shape analysis, neuroscience, medical imaging and many other areas. Our approach embeds the manifold where the responses lie onto a higher dimensional Euclidean space, obtains a local regression estimate in that space, and then projects this estimate back onto the image of the manifold. Outside the regression setting both intrinsic and extrinsic approaches have been proposed for modeling i.i.d manifold-valued data. However, to our knowledge our work is the first to take an extrinsic approach to the regression problem. The proposed extrinsic regression framework is general, computationally efficient and theoretically appealing. Asymptotic distributions and convergence rates of the extrinsic regression estimates are derived and a large class of examples are considered indicating the wide applicability of our approach.

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“…where K(•) is an univariate kernel function, for example, the Epanechnikov kernel used in kernel smoothing (Fan & Gijbels, 1996). Recent literature on multivariate kernel estimation can be found in Gu, Li & Yang (2015) and the references therein. Let H = diag{h 1 , .…”

Section: A Dynamic Linear Programming Discriminant Rulementioning

confidence: 99%

Dynamic linear discriminant analysis in high dimensional space

Jiang¹,

Chen²,

Leng³

2020

Bernoulli

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High-dimensional data that evolve dynamically feature predominantly in the modern data era. As a partial response to this, recent years have seen increasing emphasis to address the dimensionality challenge. However, the non-static nature of these datasets is largely ignored. This paper addresses both challenges by proposing a novel yet simple dynamic linear programming discriminant (DLPD) rule for binary classification. Different from the usual static linear discriminant analysis, the new method is able to capture the changing distributions of the underlying populations by modeling their means and covariances as smooth functions of covariates of interest. Under an approximate sparse condition, we show that the conditional misclassification rate of the DLPD rule converges to the Bayes risk in probability uniformly over the range of the variables used for modeling the dynamics, when the dimensionality is allowed to grow exponentially with the sample size. The minimax lower bound of the estimation of the Bayes risk is also established, implying that the misclassification rate of our proposed rule is minimax-rate optimal. The promising performance of the DLPD rule is illustrated via extensive simulation studies and the analysis of a breast cancer dataset.

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Multivariate Local Polynomial Kernel Estimators: Leading Bias and Asymptotic Distribution

Cited by 20 publications

References 10 publications

Functional Principal Component Analysis for Derivatives of Multivariate Curves

Functional Principal Component Analysis for Derivatives of Multivariate Curves

Extrinsic Local Regression on Manifold-Valued Data

Dynamic linear discriminant analysis in high dimensional space

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