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Smooth Blockwise Iterative Thresholding: A Smooth Fixed Point Estimator Based on the Likelihood’s Block Gradient

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  • Sylvain Sardy

Abstract

The proposed smooth blockwise iterative thresholding estimator (SBITE) is a model selection technique defined as a fixed point reached by iterating a likelihood gradient-based thresholding function. The smooth James--Stein thresholding function has two regularization parameters λ and ν, and a smoothness parameter s . It enjoys smoothness like ridge regression and selects variables like lasso. Focusing on Gaussian regression, we show that SBITE is uniquely defined, and that its Stein unbiased risk estimate is a smooth function of λ and ν, for better selection of the two regularization parameters. We perform a Monte Carlo simulation to investigate the predictive and oracle properties of this smooth version of adaptive lasso. The motivation is a gravitational wave burst detection problem from several concomitant time series. A nonparametric wavelet-based estimator is developed to combine information from all captors by block-thresholding multiresolution coefficients. We study how the smoothness parameter s tempers the erraticity of the risk estimate, and derives a universal threshold, an information criterion, and an oracle inequality in this canonical setting.

Suggested Citation

  • Sylvain Sardy, 2012. "Smooth Blockwise Iterative Thresholding: A Smooth Fixed Point Estimator Based on the Likelihood’s Block Gradient," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 107(498), pages 800-813, June.
  • Handle: RePEc:taf:jnlasa:v:107:y:2012:i:498:p:800-813
    DOI: 10.1080/01621459.2012.664527
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