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Asymptotics for function derivatives estimators based on stationary and ergodic discrete time processes

Author

Listed:
  • Salim Bouzebda

    (Sorbonne Universités)

  • Mohamed Chaouch

    (Qatar University)

  • Sultana Didi Biha

    (Qassim University)

Abstract

The main purpose of the present work is to investigate kernel-type estimate of a class of function derivatives including parameters such as the density, the conditional cumulative distribution function and the regression function. The uniform strong convergence rate is obtained for the proposed estimates and the central limit theorem is established under mild conditions. Moreover, we study the asymptotic mean integrated square error of kernel derivative estimator which plays a fundamental role in the characterization of the optimal bandwidth. The obtained results in this paper are established under a general setting of discrete time stationary and ergodic processes. A simulation study is performed to assess the performance of the estimate of the derivatives of the density function as well as the regression function under the framework of a discretized stochastic processes. An application to financial asset prices is also considered for illustration.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:aistmt:v:74:y:2022:i:4:d:10.1007_s10463-021-00814-2
    DOI: 10.1007/s10463-021-00814-2
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    1. Wu, Wei Biao & Huang, Yinxiao & Huang, Yibi, 2010. "Kernel estimation for time series: An asymptotic theory," Stochastic Processes and their Applications, Elsevier, vol. 120(12), pages 2412-2431, December.
    2. Karunamuni, R. J. & Mehra, K. L., 1990. "Improvements on strong uniform consistency of some known kernel estimates of a density and its derivatives," Statistics & Probability Letters, Elsevier, vol. 9(2), pages 133-140, February.
    3. Henderson, Daniel J. & Parmeter, Christopher F., 2012. "Canonical higher-order kernels for density derivative estimation," Statistics & Probability Letters, Elsevier, vol. 82(7), pages 1383-1387.
    4. Jeffrey S. Racine, 2016. "Local Polynomial Derivative Estimation: Analytic or Taylor?," Advances in Econometrics, in: Essays in Honor of Aman Ullah, volume 36, pages 617-633, Emerald Group Publishing Limited.
    5. Salim Bouzebda & Sultana Didi, 2017. "Multivariate wavelet density and regression estimators for stationary and ergodic discrete time processes: Asymptotic results," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(3), pages 1367-1406, February.
    6. Abdous, Belkacem & Germain, Stéphane & Ghazzali, Nadia, 2002. "A unified treatment of direct and indirect estimation of a probability density and its derivatives," Statistics & Probability Letters, Elsevier, vol. 56(3), pages 239-250, February.
    7. Henderson, Daniel J. & Parmeter, Christopher F., 2012. "Normal reference bandwidths for the general order, multivariate kernel density derivative estimator," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2198-2205.
    8. Tiee-Jian Wu & Chih-Yuan Hsu & Huang-Yu Chen & Hui-Chun Yu, 2014. "Root $$n$$ n estimates of vectors of integrated density partial derivative functionals," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 66(5), pages 865-895, October.
    9. Anne Leucht & Michael Neumann, 2013. "Degenerate $$U$$ - and $$V$$ -statistics under ergodicity: asymptotics, bootstrap and applications in statistics," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 65(2), pages 349-386, April.
    10. 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.
    11. Salim Bouzebda & Sultana Didi, 2017. "Additive regression model for stationary and ergodic continuous time processes," Communications in Statistics - Theory and Methods, Taylor & Francis Journals, vol. 46(5), pages 2454-2493, March.
    12. Paul Deheuvels & David Mason, 2004. "General Asymptotic Confidence Bands Based on Kernel-type Function Estimators," Statistical Inference for Stochastic Processes, Springer, vol. 7(3), pages 225-277, October.
    13. Park, Cheolwoo & Kang, Kee-Hoon, 2008. "SiZer analysis for the comparison of regression curves," Computational Statistics & Data Analysis, Elsevier, vol. 52(8), pages 3954-3970, April.
    14. Singh, R. S., 1976. "Nonparametric estimation of mixed partial derivatives of a multivariate density," Journal of Multivariate Analysis, Elsevier, vol. 6(1), pages 111-122, March.
    15. Rice, John A., 1986. "Bandwidth choice for differentiation," Journal of Multivariate Analysis, Elsevier, vol. 19(2), pages 251-264, August.
    16. D. Blanke & B. Pumo, 2003. "Optimal sampling for density estimation in continuous time," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(1), pages 1-23, January.
    17. Hardle, W. & Marron, J.S. & Wand, Mp., 1990. "Bandwith choice for density derivatives," LIDAM Reprints CORE 945, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    18. Henderson, Daniel J. & Li, Qi & Parmeter, Christopher F. & Yao, Shuang, 2015. "Gradient-based smoothing parameter selection for nonparametric regression estimation," Journal of Econometrics, Elsevier, vol. 184(2), pages 233-241.
    19. Chaouch, Mohamed & Laïb, Naâmane, 2019. "Optimal asymptotic MSE of kernel regression estimate for continuous time processes with missing at random response," Statistics & Probability Letters, Elsevier, vol. 154(C), pages 1-1.
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