Estimation of the Spectral Density with Assigned Risk

Estimation of the Spectral Density with Assigned Risk

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Article ID: iaor2016655
Volume: 43
Issue: 1
Start Page Number: 70
End Page Number: 82
Publication Date: Mar 2016
Journal: Scandinavian Journal of Statistics
Authors:
Keywords: risk, statistics: regression
Abstract:

It is well known that adaptive sequential nonparametric estimation of differentiable functions with assigned mean integrated squared error and minimax expected stopping time is impossible. In other words, no sequential estimator can compete with an oracle estimator that knows how many derivatives an estimated curve has. Differentiable functions are typical in probability density and regression models but not in spectral density models, where considered functions are typically smoother. This paper shows that for a large class of spectral densities, which includes spectral densities of classical autoregressive moving average processes, an adaptive minimax sequential estimation with assigned mean integrated squared error is possible. Furthermore, a two‐stage sequential procedure is proposed, which is minimax and adaptive to smoothness of an underlying spectral density.

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