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Penalized least squares approximation methods and their applications to stochastic processes

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Abstract

We construct an objective function that consists of a quadratic approximation term and an \(L^q\) penalty \((0<q\le 1)\) term. Thanks to the quadratic approximation, we can deal with various kinds of loss functions into a unified way, and by taking advantage of the \(L^q\) penalty term, we can simultaneously execute variable selection and parameter estimation. In this article, we show that our estimator has oracle properties, and even better property. We also treat stochastic processes as applications.

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Notes

  1. The prime denotes the matrix transpose.

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Authors and Affiliations

  1. Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo, 153-8914, Japan

    Takumi Suzuki & Nakahiro Yoshida

  2. CREST, Japan Science and Technology Agency, Kawaguchi, Japan

    Takumi Suzuki & Nakahiro Yoshida

Authors
  1. Takumi Suzuki
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  2. Nakahiro Yoshida
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Correspondence to Takumi Suzuki.

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This work was in part supported by Japan Science and Technology Agency CREST JPMJCR14D7; Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research No. 17H01702 (Scientific Research); and by a Cooperative Research Program of the Institute of Statistical Mathematics.

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Suzuki, T., Yoshida, N. Penalized least squares approximation methods and their applications to stochastic processes. Jpn J Stat Data Sci 3, 513–541 (2020). https://doi.org/10.1007/s42081-019-00064-w

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  • DOI: https://doi.org/10.1007/s42081-019-00064-w

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