Summary
Local polynomial regression estimation has a number of advantages and might become a “golden standard” for curve fitting. The attractive theoretical features are in partial contradiction to variance properties for random design and to practical experience. The conditional variance is unbounded. The unconditional variance is infinite when using optimal (compact) weights. A tutorial illustration of construction of weights for kernel and local polynomial estimators clarifies the mechanism of these problems. Properties are better for Gaussian weights, which are, however, computationally slow. We show the connection between numerical and statistical instabilities and corresponding solutions. The k-nearest-neighbour rule is shown to be an inadequate tool in this context. We propose a refined local modulation of bandwidth using a variance-bias compromise and local polynomial ridge regression.
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Seifert, B., Gasser, T. (1996). Variance Properties of Local Polynomials and Ensuing Modifications. In: Härdle, W., Schimek, M.G. (eds) Statistical Theory and Computational Aspects of Smoothing. Contributions to Statistics. Physica-Verlag HD. https://doi.org/10.1007/978-3-642-48425-4_3
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DOI: https://doi.org/10.1007/978-3-642-48425-4_3
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