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Active set algorithms for estimating shape-constrained density ratios

Author

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  • Dümbgen, Lutz
  • Mösching, Alexandre
  • Strähl, Christof

Abstract

In many instances, imposing a constraint on the shape of a density is a reasonable and flexible assumption. It offers an alternative to parametric models, which can be too rigid, and to other nonparametric methods, which require the choice of tuning parameters. The nonparametric estimation of log-concave or log-convex density ratios is treated by means of active set algorithms in a unified framework. In the setting of log-concave densities, the new algorithm is similar to, but substantially faster than, previously considered active set methods. Log-convexity, on the other hand, is a less common shape-constraint, described by some authors as “tail inflation”. The active set method proposed here is novel in this context. As a by-product, new goodness-of-fit tests of single hypotheses are formulated and are shown to be more powerful than higher criticism tests in a simulation study.

Suggested Citation

  • Dümbgen, Lutz & Mösching, Alexandre & Strähl, Christof, 2021. "Active set algorithms for estimating shape-constrained density ratios," Computational Statistics & Data Analysis, Elsevier, vol. 163(C).
    • Handle: RePEc:eee:csdana:v:163:y:2021:i:c:s0167947321001341
    DOI: 10.1016/j.csda.2021.107300
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    References listed on IDEAS

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    1. Madeleine Cule & Richard Samworth & Michael Stewart, 2010. "Maximum likelihood estimation of a multi‐dimensional log‐concave density," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 72(5), pages 545-607, November.
    2. Piet Groeneboom & Geurt Jongbloed & Jon A. Wellner, 2008. "The Support Reduction Algorithm for Computing Non‐Parametric Function Estimates in Mixture Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 35(3), pages 385-399, September.
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    4. Groeneboom,Piet & Jongbloed,Geurt, 2014. "Nonparametric Estimation under Shape Constraints," Cambridge Books, Cambridge University Press, number 9780521864015, Enero-Abr.
    5. Dümbgen, Lutz & Rufibach, Kaspar, 2011. "logcondens: Computations Related to Univariate Log-Concave Density Estimation," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 39(i06).
    Full references (including those not matched with items on IDEAS)

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