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Density estimation and comparison with a penalized mixture approach


  • Christian Schellhase
  • Göran Kauermann



The paper presents smooth estimation of densities utilizing penalized splines. The idea is to represent the unknown density by a convex mixture of basis densities, where the weights are estimated in a penalized form. The proposed method extends the work of Komárek and Lesaffre (Comput Stat Data Anal 52(7):3441–3458, 2008 ) and allows for general density estimation. Simulations show a convincing performance in comparison to existing density estimation routines. The idea is extended to allow the density to depend on some (factorial) covariate. Assuming a binary group indicator, for instance, we can test on equality of the densities in the groups. This provides a smooth alternative to the classical Kolmogorov-Smirnov test or an Analysis of Variance and it shows stable and powerful behaviour. Copyright Springer-Verlag 2012

Suggested Citation

  • Christian Schellhase & Göran Kauermann, 2012. "Density estimation and comparison with a penalized mixture approach," Computational Statistics, Springer, vol. 27(4), pages 757-777, December.
  • Handle: RePEc:spr:compst:v:27:y:2012:i:4:p:757-777
    DOI: 10.1007/s00180-011-0289-6

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    References listed on IDEAS

    1. Benaglia, Tatiana & Chauveau, Didier & Hunter, David R. & Young, Derek S., 2009. "mixtools: An R Package for Analyzing Mixture Models," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 32(i06).
    2. Simon N. Wood, 2011. "Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 73(1), pages 3-36, January.
    3. Gilles Celeux & Gilda Soromenho, 1996. "An entropy criterion for assessing the number of clusters in a mixture model," Journal of Classification, Springer;The Classification Society, vol. 13(2), pages 195-212, September.
    4. Håvard Rue & Sara Martino & Nicolas Chopin, 2009. "Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 319-392.
    5. Göran Kauermann & Jean D. Opsomer, 2011. "Data-driven selection of the spline dimension in penalized spline regression," Biometrika, Biometrika Trust, vol. 98(1), pages 225-230.
    6. Göran Kauermann & Tatyana Krivobokova & Ludwig Fahrmeir, 2009. "Some asymptotic results on generalized penalized spline smoothing," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 487-503.
    7. Ja-Yong Koo, 1999. "Logspline Density Estimation under Censoring and Truncation," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 26(1), pages 87-105.
    8. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521785167.
    9. Yingxing Li & David Ruppert, 2008. "On the asymptotics of penalized splines," Biometrika, Biometrika Trust, vol. 95(2), pages 415-436.
    10. Gerda Claeskens & Tatyana Krivobokova & Jean D. Opsomer, 2009. "Asymptotic properties of penalized spline estimators," Biometrika, Biometrika Trust, vol. 96(3), pages 529-544.
    11. Komárek, Arnost & Lesaffre, Emmanuel, 2008. "Generalized linear mixed model with a penalized Gaussian mixture as a random effects distribution," Computational Statistics & Data Analysis, Elsevier, vol. 52(7), pages 3441-3458, March.
    12. Ruppert,David & Wand,M. P. & Carroll,R. J., 2003. "Semiparametric Regression," Cambridge Books, Cambridge University Press, number 9780521780506.
    13. Philip T. Reiss & R. Todd Ogden, 2009. "Smoothing parameter selection for a class of semiparametric linear models," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 71(2), pages 505-523.
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    Cited by:

    1. Roland Langrock & Thomas Kneib & Alexander Sohn & Stacy L. DeRuiter, 2015. "Nonparametric inference in hidden Markov models using P-splines," Biometrics, The International Biometric Society, vol. 71(2), pages 520-528, June.
    2. Dalla Valle, Luciana & De Giuli, Maria Elena & Tarantola, Claudia & Manelli, Claudio, 2016. "Default probability estimation via pair copula constructions," European Journal of Operational Research, Elsevier, vol. 249(1), pages 298-311.
    3. Shang, Han Lin, 2013. "Bayesian bandwidth estimation for a nonparametric functional regression model with unknown error density," Computational Statistics & Data Analysis, Elsevier, vol. 67(C), pages 185-198.
    4. repec:spr:astaws:v:11:y:2017:i:2:d:10.1007_s11943-017-0205-9 is not listed on IDEAS
    5. Jaspers, Stijn & Aerts, Marc & Verbeke, Geert & Beloeil, Pierre-Alexandre, 2014. "A new semi-parametric mixture model for interval censored data, with applications in the field of antimicrobial resistance," Computational Statistics & Data Analysis, Elsevier, vol. 71(C), pages 30-42.
    6. Alexander Munteanu & Max Wornowizki, 2016. "Correcting statistical models via empirical distribution functions," Computational Statistics, Springer, vol. 31(2), pages 465-495, June.
    7. Roland Langrock & Théo Michelot & Alexander Sohn & Thomas Kneib, 2015. "Semiparametric stochastic volatility modelling using penalized splines," Computational Statistics, Springer, vol. 30(2), pages 517-537, June.


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