Local Polynomial Regression: Optimal Kernels and Asymptotic Minimax Efficiency

Fan, Jianqing; Gasser, Théo; Gijbels, Irène; Brockmann, Michael; Engel, Joachim

doi:10.1023/a:1003162622169

Cited by 102 publications

(77 citation statements)

References 32 publications

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“…., b g ) and zero elsewhere. As noted by Fan et al (1997) the solution of the minimization problem (15) can also be represented using a weight function W One is coming from smoothing noisy curves at a common grid, and has been analyzed in Lemma (2.1). The other one is from approximating the integral in (17) by a sum, see equation above.…”

Section: 4mentioning

confidence: 99%

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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: 4mentioning

confidence: 99%

“…As mentioned by Fan et al (1997), the design of the kernel automatically adapts to the boundary which gives the same order of convergence for the interior and boundary points, see Ruppert and Wand (1994). The estimator can be rewritten as…”

Section: 2mentioning

confidence: 99%

Functional Principal Component Analysis for Derivatives of Multivariate Curves

et al. 2016

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“…, ( )} (or ≡ ( )). Epanechnikov kernel is the optimal kernel for the kernel density estimation [40] and the local polynomial estimation [41] which minimizes the mean square error. Also, it is a truncated function which will help choose the neighborhood radius.…”

Section: The Klpp Modelmentioning

confidence: 99%

Local Prediction of Chaotic Time Series Based on Polynomial Coefficient Autoregressive Model

2015

Mathematical Problems in Engineering

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We apply the polynomial function to approximate the functional coefficients of the state-dependent autoregressive model for chaotic time series prediction. We present a novel local nonlinear model called local polynomial coefficient autoregressive prediction (LPP) model based on the phase space reconstruction. The LPP model can effectively fit nonlinear characteristics of chaotic time series with simple structure and have excellent one-step forecasting performance. We have also proposed a kernel LPP (KLPP) model which applies the kernel technique for the LPP model to obtain better multistep forecasting performance. The proposed models are flexible to analyze complex and multivariate nonlinear structures. Both simulated and real data examples are used for illustration.

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“…Moreover, it has been shown that the Epanechnikov kernel is optimal in minimizing the mean squared errors for local polynomial regression, see Fan et al (1997). The biweight and triweight kernel, which behave very similarly, could have also been chosen.…”

Section: Université Claude Bernard Lyon 1 Isfamentioning

confidence: 99%