The extended Glivenko-Cantelli property for Kernel-Smoothed estimator of the cumulative distribution function in the length-biased sampling

Document Type : Research Paper

Authors

1 Department of Statistics, Faculty of Mathematical Sciences, Vali-e-Asr University of Rafsanjan, Rafsanjan, Iran

2 Department of Mathematics, Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran

3 Department of Statistics, University of Sistan and Baluchestan, Zahedan

Abstract

‎Let $\{Y_i; i = 1,\ldots,n \}$ be a length-biased sample from a population with cumulative distribution function $F(\cdot)$‎. If the probability of an item selected in the sample is proportional to its length‎, ‎then the distribution of the observed length is known as the length-biased distribution‎.‎We consider the kernel-type estimator $F_n^s(\cdot)$ of $F(\cdot)$‎. ‎Under suitable conditions‎, ‎the extended Glivenko-Cantelli theorem for $F_n^s(\cdot)$ is proved‎.

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References

[1] Blumenthal, S. (1967). Proportional sampling in life length studies. Technometrics, 9(2) 205{218. https://doi.org/10.1080/00401706.1967.10490456
[2] Cox, D. R. (1969). Some sampling problems in technology. In: Johnson NL, Smith H Jr New developments in survey sampling, Wiley, New York, 506{527.
[3] Degenhardt, HJA. (1993). Chung|Smirnov property for perturbed empirical distribution functions. Statistics and probability letters, 16(2), 97{101. https://doi.org/10.1016/0167-7152(93)90152-9
[4] Degenhardt, HJA. (1996). Strong limit theorems for the di erence of the perturbed empirical distribution function and the classical empirical distribution function. Scandi-navian journal of statistics, 23(3), 331{351.
[5] Fabian, V., & Hannan, J. (1985). Introduction to probability and mathematical statistics. John Wiley & Sons Incorporated.
[6] Falk, M. (1983). Relative eciency and de ciency of kernel type estima-tors of smooth distribution functions. Statistica Neerlandica, 37(2), 73{83. https://doi.org/10.1111/j.1467-9574.1983.tb00802.x
[7] Fernholz, L. T. (1991). Almost sure convergence of smoothed empirical distribution functions. Scandinavian Journal of Statistics, 22, 255{262.
[8] Horvath, L. (1985). Estimation from a length-biased distribution. Statistics & Risk Modeling, 3(1-2), 91{114. https://doi.org/10.1524/strm.1985.3.12.91
[9] Jahanshahi, SMA., Rad, A., & Fakoor, V. (2017). Goodness-of- t test under length-biased sampling. Communications in Statistics-Theory and Methods, 46(15), 7580{7592. https://doi.org/10.1080/03610926.2016.1157187
[10] McFadden, J. A. (1962). On the lengths of intervals in a stationary point process. Journal of the Royal Statistical Society. Series B (Methodological), 364{382. https://doi.org/10.1111/j.2517-6161.1962.tb00464.x
[11] Nadaraya, E. (1964). Some new estimates for distribution functions. Theory of Probability & Its Applications, 9(3), 497{500. https://doi.org/10.1137/1109069
[12] Reiss, R. D. (1981). Nonparametric estimation of smooth distribution functions. Scandinavian Journal of Statistics, 12, 116{119.
[13] Shorack, G. R, & Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. John Wiley & Sons Incorporated.
[14] Vardi, Y. (1982). Nonparametric estimation in renewal processes. The Annals of Statistics, 10(3), 772{785.
[15] Wicksell, S. D. (1925). The corpuscle problem: a mathematical study of a biometric problem. Biometrika, 17(1-2), 84{99. https://doi.org/10.2307/2332027
[16] Winter, BB. (1979). Convergence rate of perturbed empirical distribution functions. Journal of Applied Probability, 16(1), 163{173. https://doi.org/10.2307/3213384
[17] Yamato, H. (1973). Uniform convergence of an estimator of a distribution function. Bulletin of Mathematical Statistics, 15(3), 69{78. https://doi.org/10.5109/13073
[18] Yukich, J. E. (1992). Weak convergence of smoothed empirical processes. Scandinavian Journal of Statistics, 19(3), 271{279.
[19] Zamini, R., Ajami, M., & Fakoor, V. (2024). Berry-Esseen bound for smooth estimator of distribution function under length-biased data. Communications in Statistics-Theory and Methods, 53(5), 1800{1809. https://doi.org/10.1080/03610926.2022.2112695

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History

  • Receive Date: 03 December 2023
  • Revise Date: 22 May 2024
  • Accept Date: 12 July 2024

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