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A note on uniform consistency of monotone function estimators

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  • Neumeyer, Natalie

Abstract

Recently, Dette, Neumeyer and Pilz (2005a) proposed a new monotone estimator for strictly increasing nonparametric regression functions and proved asymptotic normality. We explain two modifications of their method that can be used to obtain monotone versions of any nonparametric function estimators, for instance estimators of densities, variance functions or hazard rates. The method is appealing to practitioners because they can use their favorite method of function estimation (kernel smoothing, wavelets, orthogonal series,. . . ) and obtain a monotone estimator that inherits desirable properties of the original estimator. In particular, we show that both monotone estimators share the same rates of uniform convergence (almost sure or in probability) as the original estimator.

Suggested Citation

  • Neumeyer, Natalie, 2005. "A note on uniform consistency of monotone function estimators," Technical Reports 2005,35, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    • Handle: RePEc:zbw:sfb475:200535
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    References listed on IDEAS

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    1. P. Groeneboom & H. P. Lopuhaa, 1993. "Isotonic estimators of monotone densities and distribution functions: basic facts," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 47(3), pages 175-183, September.
    2. Dette, Holger & Birke, Melanie, 2005. "A note on estimating a monotone regression by combining kernel and density estimates," Technical Reports 2005,24, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    3. de Jong, Robert M., 2002. "A note on "Convergence rates and asymptotic normality for series estimators": uniform convergence rates," Journal of Econometrics, Elsevier, vol. 111(1), pages 1-9, November.
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    5. Masry, Elias, 1997. "Multivariate probability density estimation by wavelet methods: Strong consistency and rates for stationary time series," Stochastic Processes and their Applications, Elsevier, vol. 67(2), pages 177-193, May.
    6. Michael G. Akritas & Ingrid Van Keilegom, 2001. "Non‐parametric Estimation of the Residual Distribution," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 28(3), pages 549-567, September.
    7. Dette, Holger & Pilz, Kay F., 2004. "On the estimation of a monotone conditional variance in nonparametric regression," Technical Reports 2004,42, Technische Universität Dortmund, Sonderforschungsbereich 475: Komplexitätsreduktion in multivariaten Datenstrukturen.
    8. Polonik, W., 1995. "Density Estimation under Qualitative Assumptions in Higher Dimensions," Journal of Multivariate Analysis, Elsevier, vol. 55(1), pages 61-81, October.
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