Abstract
A monotone estimate of the conditional variance function in a heteroscedastic, nonparametric regression model is proposed. The method is based on the application of a kernel density estimate to an unconstrained estimate of the variance function and yields an estimate of the inverse variance function. The final monotone estimate of the variance function is obtained by an inversion of this function. The method is applicable to a broad class of nonparametric estimates of the conditional variance and particularly attractive to users of conventional kernel methods, because it does not require constrained optimization techniques. The approach is also illustrated by means of a simulation study.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akritas M., Van Keilegom I. (2001). Nonparametric estimation of the residual distribution. Scandinavian Journal of Statistics 28, 549–567
Box G.E.P. (1988). Signal to noise ratios, performance criteria and transformation. Technometrics (with discussions) 30, 1–40
Brunk H.D. (1955). Maximum likelihood estimates of monotone parameters. The Annals of Mathematical Statistics 26, 607–616
Carroll R.J. (1982). Adapting for heteroscedasticity in linear models. The Annals of Statistics 10, 1224–1233
Carroll R.J. (1987). The effect of variance function estimation on prediction-intervals. In: Berger J.O., Gupta S.S.(eds) Proceedings of the 4th purdue symposium statistical decision theory and related topics, Vol. II. Heidelberg, Springer
Dette H., Munk A. (1998). Testing heteroscedasticity in nonparametric regression. Journal of the Royal Statistical Society, Series B, 60, 693–708
Dette H., Munk A., Wagner T. (1998). Estimating the variance in nonparametric regression—what is a reasonable choice?. Journal of the Royal Statistical Society, Series B 60, 751–764
Dette H., Neumeyer N., Pilz K.F. (2006). A simple nonparametric estimator of a monotone regression function. Bernoulli 12, 469–490
Fan J., Gijbels I. (1995). Data driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaption. Journal of the Royal Statistical Society, Series B 57, 371–394
Fan J., Gijbels I. (1996). Local polynomial modelling and its applications. London, Chapman and Hall
Fan J., Yao Q. (1998). Efficient estimation of conditional variance functions in stochastic regression. Biometrika 85, 645–660
Gasser T., Sroka L., Jennen-Steinmetz G. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 73, 626–633
Hall P., Carroll R.J. (1989). Variance estimation in regression: the effect of estimating the mean. Journal of the Royal Statistical Society, Series B 51, 3–14
Hall P., Huang L.S. (2001). Nonparametric kernel regression subject to monotonicity constraints. The Annals of Statistics 29, 624–647
Hall P., Kay J.W., Titterington D.M. (1990). Asymptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77, 521–528
Hall P., Marron J.S. (1990). On variance estimation in nonparametric regression. Biometrika77, 415–419
Mack Y.P., Silverman B.W. (1982). Weak and strong uniform consistency of kernel regression estimates. Zeitschrift Wahrscheinlichkeitstheorie verwandte Gebiete 61, 405–415
Mammen E. (1991). Estimating a smooth monotone regression function. The Annals of Statistics 19, 724–740
Mukerjee R. (1988). Monotone nonparametric regression. The Annals of Statistics 16, 741–750
Müller H.G., Stadtmüller U. (1987). Estimation of heteroscedasticity in regression analysis. The Annals of Statistics 15, 610–625
Müller H.G., Stadtmüller U. (1993). On variance function estimation with quadratic forms. Journal of Statistical Planning and Inference 35, 213–231
Nash, W. J., Sellers, T. L., Talbot, S. R., Cawthorn, A. J., Ford, W. B. (1994). The population biology of abalone (Haliotis) in Tasmania. I. Blacklip abalone (H.rubra) from the north coast and islands of Bass Strait. Sea Fisheries Division, Technical Report No. 48.
Orey S. (1958). A central limit theorem for m-dependent random variables. Duke Mathematical Journal 25, 543–546
Rice J. (1984). Bandwidth choice for nonparametric regression. The Annals of Statistics 12, 1215–1230
Ruppert D., Wand M.P., Holst U., Hössjer O. (1997). Local polynomial variance-function estimation. Technometrics 39, 262–273
Sacks J., Ylvisaker D. (1970). Designs for regression problems for correlated errors. The Annals of Mathematical Statistics 41, 2057–2074
Yu K., Jones M.C. (2004). Likelihood-based local linear estimation of the conditional variance function. Journal of the American Statistical Association 99, 139–155
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Dette, H., Pilz, K. On the estimation of a monotone conditional variance in nonparametric regression. Ann Inst Stat Math 61, 111–141 (2009). https://doi.org/10.1007/s10463-007-0126-4
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10463-007-0126-4