95
Views
1
CrossRef citations to date
0
Altmetric
Original Articles

On Wavelet Estimation of the Derivatives of a Density Based on Biased Data

&
Pages 4491-4506 | Received 24 Sep 2012, Accepted 26 Sep 2013, Published online: 11 Nov 2015

References

  • Ahmad, I.A. (1995). On multivariate kernel estimation for samples from weighted distributions. Statist. Probab. Lett. 22:121–129.
  • Antoniadis, A., Grégoire, G., Nason, G. (1999). Density and hazard rate estimation for right-censored data by using wavelet methods. J. Roy. Statist. Soc. 61:63–84.
  • Bhattacharya, P.K. (1967). Estimation of a probability density function and its derivatives. Sankhyā A29:373–382.
  • Buckheit, J., Donoho, D.L. (1995). Wavelab and reproducible research. In: Antoniadis, A., ed., Wavelets and Statistics. New York: Springer, pp. 55–81.
  • Cai, T. (1999). Adaptive wavelet estimation: A block thresholding and oracle inequality approach. Ann. Statist. 27:898–924.
  • Cai, T. (2002). On block thresholding in wavelet regression: adaptivity, block size, and threshold level. Statistica Sinica 12:1241–1273.
  • Chaubey, Y.P., Chesneau, C., Doosti, H. (2011). On linear wavelet density estimation: Some recent developments. J. Ind. Statist. Assoc. 65:169–179.
  • Chaubey, Y.P., Laïb, N., Li, J. (2012). Generalized kernel regression estimator for dependent size-biased data. J. Statist. Plann. Infere. 142:708–727.
  • Chaubey, Y.P., Laïb, N., Sen, A. (2010). Generalised kernel smoothing for non-negative stationary ergodic processes. J. Nonparametr. Statist. 22:973–997.
  • Chesneau, C. (2008). Wavelet estimation via block thresholding: a minimax study under risk. Statistica Sinica 18(3):1007–1024.
  • Chesneau, C. (2010). Wavelet block thresholding for density estimation in the presence of bias. J. Kor. Statist. Soc. 39:43–53.
  • Chesneau, C. (2011). Adaptive wavelet estimation of a biased density for strongly mixing sequences. Int. J. Math. Mathemat. Sci. Article ID 604150, 21 pages.
  • Cox, D.R. (1969). Some sampling problems in technology. In: Johnson, N. L., Smith, H. eds., New Developments in Survey Sampling, 506–527. New York: John Wiley.
  • Cristóbal, J.A., Alcalá, J.T. (2000). Nonparametric regression estimators for length biased data. J. Statist. Plann. Infere. 89:145–168.
  • Dabrowska, D.M. (1995). Nonparametric regression with censored covariates. J. Multivariate Anal. 54:253–283.
  • Daubechies, I. (1988). Orthogonal bases of compactly supported wavelets. Statistica Sinica 41:909–996.
  • Daubechies, I. (1992). Ten lectures on wavelets. CBMS-NSF Regional Conferences Series in Applied Mathematics. SIAM, Philadelphia.
  • de Ũna-Álarez, J., Rodríguez, A. (2007). Nonparametric estimation from length-biased data under competing risks. Computat. Statist. Data Anal. 51:2653–2669.
  • Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D. (1995). Wavelet shrinkage: asymptopia? (with discussion). J. Roy. Statist. Soc. B57:301–369.
  • Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D. (1996). Density estimation by wavelet tresholding. Ann. Statist. 24:508–539.
  • Doosti, H., Fakoor, V., Chaubey, Y.P. (2007). Wavelet linear density estimation for negatively associated sequences. J. Ind. Statist. Assoc. 44:127–135.
  • Efromovich, S. (2004). Density estimation for biased data. Ann. Statist. 32:1137–1161.
  • Fan, J., Marron, J.S. (1994). Fast implementations of nonparametric curve estimators. J. Computat. Graph. Statist. 3:35–56.
  • Fix, E., Hodges, J.L. (1951). Discriminatory analysis. Nonparametric discrimination; consistency properties. Report Number 4, Project Number 21-49-004, USAF School of Aviation Medicine, Randolph Field, Texas. (Reprinted as pp. 261–279 of Agrawala, 1977).
  • Guillamón, A., Navarro, J., Ruiz, J.M. (1998). Kernel density estimation using weighted data. Commun. Statist. Theor. Meth. 27:2123–2135.
  • Hall, P., Patil, P. (1995). Formulae for mean integated squared error of non-linear wavelet-based density estimators. Ann. Statist. 23:905–928.
  • Hall, P., Kerkyacharian, G., Picard, D. (1999). On the minimax optimality of block thresholded wavelet estimators. Ann. Statist. 9:33–50.
  • Härdle, W., Kerkycharian, Picard, D., Tsybakov, T. (1998). Wavelets, approximations, and statistical applications. Lecture Notes Statist. 129: New York: Springer.
  • Härdle, W., Marron, J.S., Wand, M.P. (1990). Bandwidth choice for density derivatives. J. Roy. Statist. Soc. B52:223–232.
  • Hildenbrand, K., Hildenbrand, W. (1986). On the mean income effect: A data analysis of the U.K. family expenditure survey. In: Hildenbrand, K., Mas-Colell, A., eds., Contributions to Mathematical Economics, in Honor of Gerard Debreu. Amsterdam: North Holland, pp. 247–268.
  • Jones, M.C. (1991). Kernel density estimation for length biased data. Biometrika 78:511–519.
  • Jones, M.C., Marron, J.S., Sheather, S.J. (1996). A brief survey of bandwidth selection for density estimation. J. Amer. Statist. Assoc. 91:401–407.
  • Kerkyacharian, G., Picard, C. (1992). Density estimation in Besov spaces. Statist. Probab. Lett. 13:15–24.
  • Kerkyacharian, G., Picard, D. (2000). Thresholding algorithms, maxisets and well concentrated bases. Test. 9(2):283–345.
  • Leblanc, F. (1996). Wavelet linear density estimator for a discrete-time stochastic process: Lp − losses. Statist. Probab. Lett. 27:71–84.
  • Marron, J.S., de Ũna-Álvarez, J. (2004). SiZer for length biased, censored density and hazard estimation. J. Statist. Plann. Infere. 121:149–161.
  • Masry, E. (2005). Nonparametric regression estimation for dependent functional data: asymptotic normality. Stoch. Process. Applic. 115:155–177.
  • Meyer, Y. (1992). Wavelets and Operators. Cambridge: Cambridge University Press.
  • Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Statist. 33:1065–1076.
  • Petrov, V.V. (1995). Limit Theorems of Probability Theory: Sequences of Independent Random Variables. Oxford: Clarendon Press.
  • Rao, B. L.S. (1996). Nonparametric estimation of the derivatives of a density by the method of wavelets. Bull. Inform. Cybernat. 28:91–100.
  • Rao, B. L.S. (1999). Estimation of the integrated squared density derivative by wavelets. Bull. Inform. Cyb. 31:47–65.
  • Ramirez, P., Vidakovic, B. (2010). Wavelet density estimation for stratified size-biased sample. J. Statist. Plann. Infere. 140:419–432.
  • Raykar, V.C., Duraiswami, R., Zhao, L.H. (2010). Fast computation of kernel estimators. J. Computat. Graph. Statist. 19:205–220.
  • Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Ann. Mathemat. Statist. 27:832–837.
  • Rosenthal, H.P. (1970). On the subspaces of Lp (p ⩾ 2) spanned by sequences of independent random variables. Israel J. Math. 8:273–303.
  • Silverman, B.W. (1986). Density Estimation for Statistics and Data Analysis. New York: Chapman and Hall.
  • Silverman, B.W., Jones, M.C. (1989). Fix, E., Hodges, J. L. (1951): An important contribution to nonparametric discriminant analysis and density estimation: Commentary on Fix and Hodges (1951). Int. Statist. Rev. 57:233–247.
  • Singh, R. S. (1977). Applications of estimators of a density and its derivatives to certain statistical problems. J. Roy. Statist. Soc. B39:357–363.
  • Sköld, M. (1999). Kernel regression in the presence of size-bias. J. Nonparametr. Statist. 12:41–51.
  • Triebel, H. (1992). Theory of Function Space II. Berlin: Birkhaüser Verlag.
  • Tribouley, K. (1995). Adaptative density estimation. In: Antoniadis, A., Oppenheim, G. eds., Lecture Notes in Statistics. New York: Springer-Verlag, 103, pp. 385–395.
  • Vardi, Y. (1982). Nonparametric estimation in the presence of length bias. Ann. Statist. 10:616–620.
  • Vardi, Y. (1985). Empirical distributions in selection bias models (with discussions). Ann. Statist. 13:178–205.
  • Vidakovic, B. (1999). Statistical Modeling by Wavelets. New York: John Wiley and Sons.
  • Walter, G., Ghorai, J. (1992). Advantages and disadvantages of density estimation with wavelets. Proc. 24th Symp. Interface, Newton, H. J. ed., Interface FNA, VA 24: 234–343.
  • Wand, M.P. (1994). Fast computation of multivariate kernel estimators. J. Computat. Graph. Statist. 3:433–445.
  • Wu, C.O. (1997a). The effects of kernel choices in density estimation with biased data. Statist. Probab. Lett. 34:373–383.
  • Wu, C.O. (1997b). A cross-validation bandwidth choice for kernel density estimates with selection biased data. J. Multivariate Anal. 61:38–60.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.