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MM for penalized estimation

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  • Zhu Wang

    (UT Health San Antonio)

Abstract

Penalized estimation can conduct variable selection and parameter estimation simultaneously. The general framework is to minimize a loss function subject to a penalty designed to generate sparse variable selection. The majorization–minimization (MM) algorithm is a computational scheme for stability and simplicity, and the MM algorithm has been widely applied in penalized estimation. Much of the previous work has focused on convex loss functions such as generalized linear models. When data are contaminated with outliers, robust loss functions can generate more reliable estimates. Recent literature has witnessed a growing impact of nonconvex loss-based methods, which can generate robust estimation for data contaminated with outliers. This article investigates MM algorithm for penalized estimation, provides innovative optimality conditions and establishes convergence theory with both convex and nonconvex loss functions. With respect to applications, we focus on several nonconvex loss functions, which were formerly studied in machine learning for regression and classification problems. Performance of the proposed algorithms is evaluated on simulated and real data including cancer clinical status. Efficient implementations of the algorithms are available in the R package mpath in CRAN.

Suggested Citation

  • Zhu Wang, 2022. "MM for penalized estimation," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 31(1), pages 54-75, March.
    • Handle: RePEc:spr:testjl:v:31:y:2022:i:1:d:10.1007_s11749-021-00770-2
    DOI: 10.1007/s11749-021-00770-2
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    References listed on IDEAS

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    1. Fan J. & Li R., 2001. "Variable Selection via Nonconcave Penalized Likelihood and its Oracle Properties," Journal of the American Statistical Association, American Statistical Association, vol. 96, pages 1348-1360, December.
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    3. Alexander Hanbo Li & Jelena Bradic, 2018. "Boosting in the Presence of Outliers: Adaptive Classification With Nonconvex Loss Functions," Journal of the American Statistical Association, Taylor & Francis Journals, vol. 113(522), pages 660-674, April.
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