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Kernel based estimation of the distribution function for length biased data

Author

Listed:
  • Arup Bose

    (Indian Statistical Unit)

  • Santanu Dutta

    (Tezpur University)

Abstract

Empirical and kernel estimators are considered for the distribution of positive length biased data. Their asymptotic bias, variance and limiting distribution are obtained. For the kernel estimator, the asymptotically optimal bandwidth is calculated and rule of thumb bandwidths are proposed. At any point below the median, the asymptotic mean squared error of the kernel estimator is smaller than that of the empirical estimator. A suitably truncated kernel estimator is positive and we prove the strong uniform, and $$L_2$$ L 2 consistency of this estimator. Simulations reveal the improved performance of the truncated kernel estimator in estimating tail probabilities based on length biased data.

Suggested Citation

  • Arup Bose & Santanu Dutta, 2022. "Kernel based estimation of the distribution function for length biased data," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 85(3), pages 269-287, April.
    • Handle: RePEc:spr:metrik:v:85:y:2022:i:3:d:10.1007_s00184-021-00824-3
    DOI: 10.1007/s00184-021-00824-3
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    References listed on IDEAS

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    3. Chiung-Yu Huang & Jing Qin, 2011. "Nonparametric estimation for length-biased and right-censored data," Biometrika, Biometrika Trust, vol. 98(1), pages 177-186.
    4. Jan W. H. Swanepoel & Francois C. Van Graan, 2005. "A New Kernel Distribution Function Estimator Based on a Non‐parametric Transformation of the Data," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 32(4), pages 551-562, December.
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