Statistical Papers

, Volume 59, Issue 2, pp 423–447 | Cite as

Partitioning estimation of local variance based on nearest neighbors under censoring

  • Paola Gloria FerrarioEmail author
Regular Article


In a nonparametric and heteroscedastic setting, our primary interest is in the local variance estimation when the response variable is subject to right censoring. For the proposed partitioning local variance estimators, based on the first and second nearest neighbors, some transformations on the observed censoring times are involved, using their estimated survival functions. Proofs of consistency and rate of convergence for the presented estimators are given. Moreover, local variance estimation is demonstrated on the basis of real survival data.


Local variance Censoring Partitioning estimation Nearest neighbors Weak consistency Rate of convergence Survival data analysis 



The author gratefully acknowledges many helpful suggestions of two anonymous referees. Moreover, the author wishes to express her gratitude to Prof. em. Dr. Harro Walk and Dr. Maik Döring for essential help and support.


  1. Brown LD, Levine M (2007) Variance estimation in nonparametric regression via the difference sequence method. Ann Stat 35:2219–2232MathSciNetCrossRefzbMATHGoogle Scholar
  2. Buckley J, James I (1979) Linear regression with censored data. Biometrika 66:429–436CrossRefzbMATHGoogle Scholar
  3. Cai T, Levine M, Wang L (2009) Variance function estimation in multivariate nonparametric regression with fixed design. J Multivar Anal 100:126–136MathSciNetCrossRefzbMATHGoogle Scholar
  4. Cox DR (1972) Regression models and life-tables. J R Stat Soc B 34:187–220MathSciNetzbMATHGoogle Scholar
  5. Evans D (2005) Estimating the variance of multiplicative noise. In: 18th international conference on noise and fluctuations. ICNF, in AIP conference proceedings, vol 780, pp 99–102Google Scholar
  6. Evans D, Jones A (2008) Non-parametric estimation of residual moments and covariance. Proc R Soc Lond Ser A Math Phys Eng Sci 464:2831–2846MathSciNetCrossRefzbMATHGoogle Scholar
  7. Fan J, Gijbels I (1994) Censored regression: local linear approximations and their applications. J Am Stat Assoc 89(426):560–570MathSciNetCrossRefzbMATHGoogle Scholar
  8. Ferrario PG (2013) Local variance estimation for uncensored and censored observations. Springer Vieweg Verlag, WiesbadenCrossRefzbMATHGoogle Scholar
  9. Ferrario PG, Walk H (2012) Nonparametric partitioning estimation of residual and local variance based on first and second nearest neighbors. J Nonparametric Stat 24:1019–1039MathSciNetCrossRefzbMATHGoogle Scholar
  10. Györfi L, Kohler M, Krzyżak A, Walk H (2002) A distribution-free theory of nonparametric regression. Springer Series in Statistics, Springer, New YorkCrossRefzbMATHGoogle Scholar
  11. Hall P, Carroll PJ (1989) Variance function estimation in regression: the effect of estimating the mean. J R Stat Soc Ser B 51:3–14MathSciNetzbMATHGoogle Scholar
  12. Härdle W, Tsybakov A (1997) Local polynomial estimators of the volatility function in nonparametric autoregression. J Econom 81:223–242MathSciNetCrossRefzbMATHGoogle Scholar
  13. Kaplan EL, Meier P (1958) Nonparametric estimation from incomplete observations. J Am Stat Assoc 53:457–481MathSciNetCrossRefzbMATHGoogle Scholar
  14. Kohler M (2006) Nonparametric regression with additional measurement errors in the dependent variable. J Stat Plan Inference 136:3339–3361MathSciNetCrossRefzbMATHGoogle Scholar
  15. Kohler M, Krzyżak A, Walk H (2006) Rates of convergence for partitioning and nearest neighbor regression estimates with unbounded data. J Multivar Anal 97:311–323MathSciNetCrossRefzbMATHGoogle Scholar
  16. Koul H, Susarla V, Van Ryzin J (1981) Regression analysis with random rightly-censored data. Ann Stat 9:1276–1288CrossRefzbMATHGoogle Scholar
  17. Lemdani M, Saïd EU (2015) Nonparametric robust regression estimation for censored data. Stat Pap 1:1–21zbMATHGoogle Scholar
  18. Li J, Zheng M (2009) Robust estimation of multivariate regression model. Stat Papers 50:81–100MathSciNetCrossRefzbMATHGoogle Scholar
  19. Liitiäinen E, Corona F, Lendasse A (2007) Non-parametric residual variance estimation in supervised learning. In: IWANN’07 Proceedings of the 9th international work-conference on artificial neural networks. Lecture Notes in Computer Science: Computational and Ambient Intelligence, vol 4507, pp 63–71Google Scholar
  20. Liitiäinen E, Corona F, Lendasse A (2008) On nonparametric residual variance estimation. Neural Process Lett 28:155–167CrossRefzbMATHGoogle Scholar
  21. Liitiäinen E, Corona F, Lendasse A (2010) Residual variance estimation using a nearest neighbor statistic. J Multivar Anal 101:811–823MathSciNetCrossRefzbMATHGoogle Scholar
  22. Loprinzi CL, Laurie JA, Wieand HS, Krook JE, Novotny PJ, Kugler JW, Bartel J, Law M, Bateman M, Klatt NE et al, North Central Cancer Treatment Group (1994) Prospective evaluation of prognostic variables from patient-completed questionnaires. J Clin Oncol 12(3):601–607CrossRefGoogle Scholar
  23. Luo S, Zhang C-Y (2015) Nonparametric M-type regression estimation under missing response data. Stat Papers 1–23Google Scholar
  24. Mathe K (2006) Regressionanalyse mit zensierten Daten. PhD Thesis. Institute of Stochastics and Applications, Universität StuttgartGoogle Scholar
  25. Müller HG, Stadtmüller U (1987) Estimation of heteroscedasticity in regression analysis. Ann Stat 15:610–625MathSciNetCrossRefzbMATHGoogle Scholar
  26. Müller HG, Stadtmüller U (1993) On variance function estimation with quadratic forms. J Stat Plan Inference 35:213–231MathSciNetCrossRefzbMATHGoogle Scholar
  27. Munk A, Bissantz N, Wagner T, Freitag G (2005) On difference based variance estimation in nonparametric regression when the covariate is high dimensional. J R Stat Soc Ser B 67:19–41MathSciNetCrossRefzbMATHGoogle Scholar
  28. Neumann M (1994) Fully data-driven nonparametric variance estimators. Statistics 25:189–212MathSciNetCrossRefzbMATHGoogle Scholar
  29. Ruppert D, Wand M, Holst U, Hössjer O (1997) Local polynomial variance-function estimation. Technometrics 39:262–273MathSciNetCrossRefzbMATHGoogle Scholar
  30. Spokoiny V (2002) Variance estimation for high-dimensional regression models. J Multivar Anal 82:111–133MathSciNetCrossRefzbMATHGoogle Scholar
  31. Steele JM (1986) An Efron-Stein inequality for nonsymmetric statistics. Ann Stat 14:753–758MathSciNetCrossRefzbMATHGoogle Scholar
  32. Strobel M (2008) Estimation of minimum mean squared error with variable metric from censored observations. PhD Thesis. Institute of Stochastics and Applications, Universität StuttgartGoogle Scholar
  33. Wang L, Brown LD, Cai T, Levine M (2008) Effect of mean on variance function estimation in nonparametric regression. Ann Stat 36:646–664MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institut für Medizinische Biometrie und StatistikUniversität zu LübeckLübeckGermany
  2. 2.Institut für Stochastik und AnwendungenUnversität StuttgartStuttgartGermany

Personalised recommendations