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Rates of consistency for nonparametric estimation of the mode in absence of smoothness assumptions

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  • Herrmann, Eva
  • Ziegler, Klaus

Abstract

Nonparametric estimation of the mode of a density or regression function via kernel methods is considered. It is shown that the rate of consistency of the mode estimator can be determined without the typical smoothness conditions. Only the uniform rate of the so-called stochastic part of the problem together with some mild conditions characterizing the shape or "acuteness" of the mode influence the rate of the mode estimator. In particular, outside the location of the mode, our assumptions do not even imply continuity. Overall, it turns out that the location of the mode can be estimated at a rate that is the better the "peakier" (and hence nonsmooth) the mode is, while the contrary holds with estimation of the size of the mode.

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  • Herrmann, Eva & Ziegler, Klaus, 2004. "Rates of consistency for nonparametric estimation of the mode in absence of smoothness assumptions," Statistics & Probability Letters, Elsevier, vol. 68(4), pages 359-368, July.
    • Handle: RePEc:eee:stapro:v:68:y:2004:i:4:p:359-368
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    References listed on IDEAS

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    1. Vieu, Philippe, 1996. "A note on density mode estimation," Statistics & Probability Letters, Elsevier, vol. 26(4), pages 297-307, March.
    2. Liebscher E., 2001. "Estimation Of The Density And The Regression Function Under Mixing Conditions," Statistics & Risk Modeling, De Gruyter, vol. 19(1), pages 9-26, January.
    3. Joseph Romano, 1988. "Bootstrapping the mode," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 40(3), pages 565-586, September.
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    Cited by:

    1. Ouafae Benrabah & Elias Ould Saïd & Abdelkader Tatachak, 2015. "A kernel mode estimate under random left truncation and time series model: asymptotic normality," Statistical Papers, Springer, vol. 56(3), pages 887-910, August.
    2. Jan Beran & Klaus Telkmann, 2021. "On inference for modes under long memory," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 48(2), pages 429-455, June.
    3. Shi, Xiaoping & Wu, Yuehua & Miao, Baiqi, 2009. "A note on the convergence rate of the kernel density estimator of the mode," Statistics & Probability Letters, Elsevier, vol. 79(17), pages 1866-1871, September.
    4. Salim Bouzebda & Mohamed Chaouch & Sultana Didi Biha, 2022. "Asymptotics for function derivatives estimators based on stationary and ergodic discrete time processes," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 74(4), pages 737-771, August.
    5. Eunju Hwang & Dong Shin, 2016. "Kernel estimators of mode under $$\psi $$ ψ -weak dependence," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 68(2), pages 301-327, April.
    6. Obereder, Andreas & Scherzer, Otmar & Kovac, Arne, 2007. "Bivariate density estimation using BV regularisation," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5622-5634, August.
    7. Salah Khardani & Mohamed Lemdani & Elias Ould Saïd, 2012. "On the strong uniform consistency of the mode estimator for censored time series," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 75(2), pages 229-241, February.

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