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Estimation of the Spectral Density with Assigned Risk
Author(s) -
Efromovich Sam
Publication year - 2016
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12165
Subject(s) - mathematics , minimax , estimator , mean squared error , smoothness , minimax estimator , autoregressive model , spectral density , density estimation , spectral density estimation , kernel density estimation , probability density function , differentiable function , nonparametric statistics , statistics , mathematical optimization , minimum variance unbiased estimator , mathematical analysis , fourier transform
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.