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Nonparametric imputation by data depth


  • Pavlo Mozharovskyi

    (CREST; ENSAI; Université Bretagne Loire)

  • Julie Josse

    (CMAP; Ecole polytechnique)

  • François Husson

    (IRMAR; Applied Mathematics Unit; Agrocampus Ouest)


The presented methodology for single imputation of missing values borrows the idea from data depth — a measure of centrality defined for an arbitrary point of the space with respect to a probability distribution or a data cloud. This consists in iterative maximization of the depth of each observation with missing values, and can be employed with any properly defined statistical depth function. On each single iteration, imputation is narrowed down to optimization of quadratic, linear, or quasiconcave function being solved analytically, by linear programming, or the Nelder-Mead method, respectively. Being able to grasp the underlying data topology, the procedure is distribution free, allows to impute close to the data, preserves prediction possibilities different to local imputation methods (k-nearest neighbors, random forest), and has attractive robustness and asymptotic properties under elliptical symmetry. It is shown that its particular case — when using Mahalanobis depth — has direct connection to well known treatments for multivariate normal model, such as iterated regression or regularized PCA. The methodology is extended to the multiple imputation for data stemming from an elliptically symmetric distribution. Simulation and real data studies positively contrast the procedure with existing popular alternatives. The method has been implemented as an R-package.

Suggested Citation

  • Pavlo Mozharovskyi & Julie Josse & François Husson, 2017. "Nonparametric imputation by data depth," Working Papers 2017-72, Center for Research in Economics and Statistics.
  • Handle: RePEc:crs:wpaper:2017-72

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    References listed on IDEAS

    1. Pavel Bazovkin & Karl Mosler, 2015. "A general solution for robust linear programs with distortion risk constraints," Annals of Operations Research, Springer, vol. 229(1), pages 103-120, June.
    2. Marc Hallin & Davy Paindaveine & Miroslav Siman, 2008. "Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth," Working Papers ECARES 2008_042, ULB -- Universite Libre de Bruxelles.
    3. Templ, Matthias & Kowarik, Alexander & Filzmoser, Peter, 2011. "Iterative stepwise regression imputation using standard and robust methods," Computational Statistics & Data Analysis, Elsevier, vol. 55(10), pages 2793-2806, October.
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    More about this item


    Elliptical symmetry; Outliers; Tukey depth; Zonoid depth; Nonparametric imputation; Convex optimization;
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