Extended Glivenko—Cantelli theorem for simple random sampling without replacement from a finite population

References

• Bardenet, R., and O.-A. Maillard. 2015. Concentration inequalities for sampling without replacement. Bernoulli 21 (3):1361–85. doi: 10.3150/14-BEJ605.
   Web of Science ®Google Scholar
• Boistard, H., H. P. Lopuhaä, and A. Ruiz-Gazen. 2017. Functional central limit theorems for single-stage sampling designs. The Annals of Statistics 45 (4):1728–58. doi: 10.1214/16-AOS1507.
   Web of Science ®Google Scholar
• Bonnéry, D., F. J. Breidt, and F. Coquet. 2012. Uniform convergence of the empirical cumulative distribution function under informative selection from a finite population. Bernoulli 18 (4):1361–85. doi: 10.3150/11-BEJ369.
   Web of Science ®Google Scholar
• Breidt, F., J. D. Jay, and Opsomer. 2008. Endogenous post-stratification in surveys: Classifying with a sample-fitted model. The Annals of Statistics 36 (1):403–27. doi: 10.1214/009053607000000703.
   Web of Science ®Google Scholar
• Chatterjee, A. 2011. Asymptotic properties of sample quantiles from a finite population. Annals of the Institute of Statistical Mathematics 63 (1):157–79. doi: 10.1007/s10463-008-0210-4.
   Web of Science ®Google Scholar
• Cheng, F. 2008. Extended Glivenko–Cantelli theorem in ARCH(p)-time series. Statistics & Probability Letters 78 (12):1434–9. doi: 10.1016/j.spl.2007.12.009.
   Web of Science ®Google Scholar
• Cheng, F., J. Yan, and L. Yang. 2014. Extended Glivenko–Cantelli theorem in nonparametric regression. Communications in Statistics- Theory and Methods 43 (17):3720–5. doi: 10.1080/03610926.2012.700377.
   Web of Science ®Google Scholar
• Conti, P. L. 2014. On the estimation of the distribution function of a finite population under high entropy sampling designs, with applications. Sankhya B 76 (2):234–59. doi: 10.1007/s13571-014-0083-x.
   Google Scholar
• Conti, P. L., and D. Marella. 2015. Inference for quantiles of a finite population: Asymptotic versus resampling results. Scandinavian Journal of Statistics 42 (2):545–61. doi: 10.1111/sjos.12122.
   Web of Science ®Google Scholar
• Erdős, P., and A. Rényi. 1959. On the central limit theorem for samples from a finite population. Publications of the Mathematical Institute of the Hungarian Academy of Sciences 4:49–61.
   Google Scholar
• Fabian, V., and J. Hannan, 1985. Introduction to probability and mathematical statistics, In Probability and mathematical statistics: Probability and mathematical statistics, Wiley Series. New York: John Wiley & Sons, Inc.
   Google Scholar
• Francisco, C. A., and W. A. Fuller. 1991. Quantile estimation with a complex survey design. The Annals of Statistics 19 (1):454–69. doi: 10.1214/aos/1176347993.
   Web of Science ®Google Scholar
• Hájek, J. 1960. Limiting distributions in simple random sampling from a finite population. Publications of the Mathematical Institute of the Hungarian Academy of Sciences 5:361–74.
   Google Scholar
• Han, Q., and J. A. Wellner. 2021. Complex sampling designs: Uniform limit theorems and applications. The Annals of Statistics 49 (1):459–85. doi: 10.1214/20-AOS1964.
   Web of Science ®Google Scholar
• Horowitz, J. 1990. A uniform law of large numbers and empirical central limit theorem for limits of finite populations. Statistics & Probability Letters 10 (2):159–66. doi: 10.1016/0167-7152(90)90012-V.
   Web of Science ®Google Scholar
• Loève, M. 1977. Probability theory. I. In Graduate texts in mathematics, 4th ed., 45. New York-Heidelberg: Springer-Verlag.
   Google Scholar
• Motoyama, H., and H. Takahashi, 2006. On estimators for finite population distributon function. (Yugen boshuudan niokeru bunpukansu no suitei). In Kakei deta no keizaibunseki to toukeiteki shuhou, ed. Yoshizoe, Y. Tokyo: Tokei Kenkyukai, 117–31 (in Japanese).
   Google Scholar
• Motoyama, H., and H. Takahashi. 2008. Smoothed versions of statistical functionals from a finite population. Journal of the Japan Statistical Society 38 (3):475–504. doi: 10.14490/jjss.38.475.
   Google Scholar
• Rényi, A. 1970. Probability theory. In North-Holland series in applied mathematics and mechanics, ed. László Vekerdi, vol. 10. Amsterdam: North-Holland Pub. Co.
   Google Scholar
• Rosén, B. 1965. Limit theorems for sampling from finite populations. Arkiv För Matematik 5 (5):383–424. doi: 10.1007/BF02591138.
   Google Scholar
• Rubin-Bleuer, S. 2003. On the convergence of sample empirical processes. Working Paper 03-3. Canada: Statistics Canada. Methodology Branch. https://publications.gc.ca/site/eng/9.840393/publication.html.
   Google Scholar
• Saegusa, T. 2019. Large sample theory for merged data from multiple sources. The Annals of Statistics 47 (3):1585–615. doi: 10.1214/18-AOS1727.
   Web of Science ®Google Scholar
• Saegusa, T., and J. A. Wellner. 2013. Weighted likelihood estimation nder two-phase sampling. Annals of Statistics 41 (1):269–95. 24563559 doi: 10.1214/12-AOS1073.
   PubMed Web of Science ®Google Scholar
• Sen, P. K. 1972. A Hájek-Rényi type inequality for generalized U-Statistics. Calcutta Statistical Association Bulletin 21 (3-4):171–80. doi: 10.1177/0008068319720306.
   Google Scholar
• Serfling, R. J. 1980. Approximation theorems of mathematical statistics. In Wiley series in probability and mathematical statistics. New York: John Wiley & Sons, Inc.
   Google Scholar
• Shao. 1994. L-statistics in complex survey problems. The Annals of Statistics 22 (2):946–67. doi: 10.1214/aos/1176325505.
   Web of Science ®Google Scholar
• Shorack, G. R., and J. A. Wellner, 1986. Empirical processes with applications to statistics. In Wiley series in probability and mathematical statistics: Probability and mathematical statistics. New York: John Wiley & Sons, Inc.
   Google Scholar
• Sitter, R. R., and C. Wu. 2001. A note on Woodruff confidence intervals for quantiles. Statistics & Probability Letters 52 (4):207–15. doi: 10.1016/S0167-7152(00)00207-8.
   Web of Science ®Google Scholar
• Van De Geer, S. A. 2000. Applications of empirical process theory. In Cambridge series in statistical and probabilistic mathematics. Vol. 6. Cambridge: Cambridge University Press.
   Google Scholar
• Van Der Vaart, A. W. 1998. Cambridge series in statistical and probabilistic mathematics. Vol. 3. Cambridge: Cambridge University Press. doi: 10.1017/CBO9780511802256.
   Google Scholar
• Van Der Vaart, A. W., and J. A. Wellner, 1996. Weak convergence and empirical processes. In Springer series in statistics, With applications to statistics. New York: Springer-Verlag. doi: 10.1007/978-1-4757-2545-2.
   Google Scholar
• Zhong, C. 2022. Extended Glivenko-Cantelli theorem and L1 strong consistency of innovation density estimator for time-varying semiparametric ARCH model. Journal of Nonparametric Statistics 35 (2):373–96. doi: 10.1080/10485252.2022.2152813.
   Web of Science ®Google Scholar