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Frequency polygons for weakly dependent processes

Author

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  • Carbon, Michel
  • Garel, Bernard
  • Tran, Lanh Tat

Abstract

The purpose of this paper is to investigate the frequency polygon as a density estimator for stationary strong mixing processes. Optimal bin widths which asymptotically minimize integrated mean square errors (IMSE) are derived. Under weak conditions, frequency polygons achieve the same rate of convergence to zero of the IMSE as kernel estimators. They can also attain the optimal uniform rate of convergence ((n-1logn)1/3 under general conditions. Frequency polygons thus appear to be very good density estimators with respect to both criteria of IMSE and uniform convergence.

Suggested Citation

  • Carbon, Michel & Garel, Bernard & Tran, Lanh Tat, 1997. "Frequency polygons for weakly dependent processes," Statistics & Probability Letters, Elsevier, vol. 33(1), pages 1-13, April.
    • Handle: RePEc:eee:stapro:v:33:y:1997:i:1:p:1-13
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    References listed on IDEAS

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    1. Andrew Harvey (ed.), 1994. "Time Series," Books, Edward Elgar Publishing, volume 0, number 599.
    2. Ioannides, D. & Roussas, G. G., 1987. "Note on the uniform convergence of density estimates for mixing random variables," Statistics & Probability Letters, Elsevier, vol. 5(4), pages 279-285, June.
    3. Masry, Elias & Györfi, László, 1987. "Strong consistency and rates for recursive probability density estimators of stationary processes," Journal of Multivariate Analysis, Elsevier, vol. 22(1), pages 79-93, June.
    4. P. M. Robinson, 1983. "Nonparametric Estimators For Time Series," Journal of Time Series Analysis, Wiley Blackwell, vol. 4(3), pages 185-207, May.
    5. P. M. Robinson, 1987. "Time Series Residuals With Application To Probability Density Estimation," Journal of Time Series Analysis, Wiley Blackwell, vol. 8(3), pages 329-344, May.
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    Cited by:

    1. Michel Harel & Jean-François Lenain & Joseph Ngatchou-Wandji, 2016. "Asymptotic behaviour of binned kernel density estimators for locally non-stationary random fields," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(2), pages 296-321, June.
    2. Ahmad Younso, 2023. "On the consistency of mode estimate for spatially dependent data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 86(3), pages 343-372, April.
    3. Livasoa Andriamampionona & Victor Harison & Michel Harel, 2024. "Non-Parametric Estimation of the Renewal Function for Multidimensional Random Fields," Mathematics, MDPI, vol. 12(12), pages 1-22, June.
    4. Nadia Bensaïd & Sophie Dabo-Niang, 2010. "Frequency polygons for continuous random fields," Statistical Inference for Stochastic Processes, Springer, vol. 13(1), pages 55-80, April.
    5. Mohamed El Machkouri, 2013. "On the asymptotic normality of frequency polygons for strongly mixing spatial processes," Statistical Inference for Stochastic Processes, Springer, vol. 16(3), pages 193-206, October.
    6. Lardjane, Salim, 2022. "Strong uniform consistency of the Frequency Polygon density estimator for stable non-anticipative stochastic processes," Statistics & Probability Letters, Elsevier, vol. 189(C).

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