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Nonparametric function estimation with infinite-order kernels and applications


Improved performance in higher-order spectral density estimation (polyspectral estimation) and density estimation of censored data is achieved using a general class of infinite-order kernels. These estimates are asymptotically less biased but with the same order of variance as compared to the classical estimators with second-order kernels. A simple, data-dependent algorithm for selecting the bandwidth is introduced and is shown to be consistent with estimating the optimal bandwidth for the infinite-order kernels. The combination of the specialized family of kernels with the new bandwidth selection algorithm yields a considerably improved density estimation procedure surpassing the performances of existing estimators using second-order kernels. Infinite- order estimators are also utilized in a secondary manner as pilot estimators in the plug-in approach for bandwidth choice in second-order kernels. Simulations illustrate the improved accuracy of the proposed estimator against other nonparametric estimators of the density, bispectrum, and hazard function. Symmetries of the auto-cumulant function of a kth-order stationary time series play an important role in polyspectral estimation, and these symmetries are derived through a connection with the symmetric group of degree k. Using theory of group representations, these symmetries are demystified and lag-window functions are symmetrized to satisfy these symmetries. A generalized Gabr-Rao optimal kernel, used to estimate general kth- order spectra, is also derived through the developed theory

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