Function Estimation

Part of the Selected Works in Probability and Statistics book series (SWPS, volume 13)


The following is a brief review of three landmark papers of Peter Bickel on theoretical and methodological aspects of nonparametric density and regression estimation and the related topic of goodness-of-fit testing, including a class of semiparametric goodness-of-fit tests. We consider the context of these papers, their contribution and their impact. Bickel’s first work on density estimation was carried out when this area was still in its infancy and proved to be highly influential for the subsequent wide-spread development of density and curve estimation and goodness-of-fit testing.


Kernel Density Estimation Nonparametric Regression Optimal Bandwidth Kernel Density Estimator Mean Integrate Square Error 
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  1. Aswani A, Bickel P, Tomlin C (2011) Regression on manifolds: estimation of the exterior derivative. Ann Stat 39:48–81CrossRefMATHMathSciNetGoogle Scholar
  2. Bachmann, D, Dette H (2005) A note on the Bickel-Rosenblatt test in autoregressive time series. Stat Probab Lett 74:221–234CrossRefMATHMathSciNetGoogle Scholar
  3. Bickel P (1982) On adaptive estimation. Ann Stat 10:647–671CrossRefMATHMathSciNetGoogle Scholar
  4. Bickel P, Li B (2007). Local polynomial regression on unknown manifolds. In: Complex datasets and inverse problems: tomography, networks and beyond. IMS lecture notes-monograph series, vol 54. Institute of Mathematical Statistics, Beachwood, pp 177–186Google Scholar
  5. Bickel P, Ritov Y (1988) Estimating integrated squared densiuty derivatives: sharp best order of convergence estimates. Sankhya Indian J Stat Ser A 50:381–393MATHMathSciNetGoogle Scholar
  6. Bickel P, Rosenblatt M (1973) On some global measures of the deviations of density function estimates. Ann Stat 1:1071–1095CrossRefMATHMathSciNetGoogle Scholar
  7. Bickel P, Ritov Y, Stoker T (2006) Tailor-made tests for goodness of fit to semiparametric hypotheses. Ann Stat 34:721–741CrossRefMATHMathSciNetGoogle Scholar
  8. Birgé L, Massart P (1995) Estimation of integral functionals of a density. Ann Stat 23:11–29CrossRefMATHGoogle Scholar
  9. Cao R, Lugosi G (2005) Goodness-of-fit tests based on kernel density estimator. Scand J Stat 32:599–616CrossRefMATHMathSciNetGoogle Scholar
  10. Čencov N (1962) Evaluation of an unknown density from observations. Sov Math 3:1559–1562Google Scholar
  11. Daniell P (1946) Discussion of paper by M.S. Bartlett. J R Stat Soc Suppl 8:88–90Google Scholar
  12. Efromovich S, Low M (1996) On Bickel and Ritov’s conjecture about adaptive estimation of the integral of the square of density derivative. Ann Stat 24:682–686CrossRefMATHMathSciNetGoogle Scholar
  13. Efromovich S, Samarov A (2000) Adaptive estimation of the integral of squared regression derivatives. Scand J Stat 27:335–351CrossRefMATHMathSciNetGoogle Scholar
  14. Einstein A (1914) Méthode pour la détermination de valeurs statistiques d’observations concernant des grandeurs soumises à des fluctuations irrégulières. Arch Sci Phys et Nat Ser 4 37:254–256Google Scholar
  15. Hall P (1992) Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density. Ann Stat 20:675–694CrossRefMATHGoogle Scholar
  16. Hall P, Marron J (1987) Estimation of integrated squared density derivatives. Stat Probab Lett 6:109–115CrossRefMATHMathSciNetGoogle Scholar
  17. Koul H, Mimoto N (2010) A goodness-of-fit test for garch innovation density. Metrika 71:127–149MathSciNetGoogle Scholar
  18. Lee S, Na S (2002) On the Bickel-Rosenblatt test for first order autoregressive models. Stat Probab Lett 56:23–35CrossRefMATHMathSciNetGoogle Scholar
  19. Mukherjee S, Wu Q, Zhou D (2010) Learning gradients on manifolds. Bernoulli 16:181–207CrossRefMATHMathSciNetGoogle Scholar
  20. Müller H-G, Stadtmüller U, Schmitt T (1987) Bandwidth choice and confidence intervals for derivatives of noisy data. Biometrika 74:743–749CrossRefMATHMathSciNetGoogle Scholar
  21. Parzen E (1962) On estimation of a probability density function and mode. Ann Math Stat 33:1065–1076CrossRefMATHMathSciNetGoogle Scholar
  22. Rosenblatt M (1956) Remarks on some nonparametric estimates of a density function. Ann Math Stat 27:832–837CrossRefMATHMathSciNetGoogle Scholar
  23. Rosenblatt M (1971) Curve estimates. Ann Stat 42:1815–1842CrossRefMATHMathSciNetGoogle Scholar
  24. Rosenblatt M (1975) A quadratic measure of deviation of two-dimensional density estimates and a test of independence. Ann Stat 3:1–14CrossRefMATHMathSciNetGoogle Scholar
  25. Rosenblatt M (1976) On the maximal deviation of k-dimensional density estimates. Ann Probab 4:1009–1015CrossRefMATHMathSciNetGoogle Scholar
  26. Schweder T (1975) Window estimation of the asymptotic variance of rank estimators of location. Scand J Stat 2:113–126MATHMathSciNetGoogle Scholar
  27. Stone CJ (1980) Optimal rates of convergence for nonparametric estimators. Ann Stat 10:1040–1053CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of StatisticsUniversity of CaliforniaDavisUSA

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