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

Abstract

In this study, we consider the rate of uniform almost sure convergence of the empirical distribution function for simple random sampling without replacement from a finite population. Utilizing Hoeffding’s inequality for simple random sampling without replacement, this study extends the classical Glivenko—Cantelli theorem for the empirical distribution function for samples from a finite population. Our numerical simulation results are consistent with theoretical results.

KEYWORDS:

• Rate of convergence
• Glivenko—Cantelli theorem
• Hoeffding’s inequality
• empirical distribution function
• finite population
• simple random sampling without replacement

Acknowledgment

The author thanks two anonymous referees for their comments and suggestions which greatly helped to improve the article.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by a Grant-in-Aid for Scientific Research (Japan Society for the Promotion of Science) Grant Numbers 18H00837, 21K01429.