臺灣博碩士論文加值系統

The obtained confidence interval from using Central Limit
Theorem (CLT) is an approximate solution. On the other hand, the
confidence interval obtained by using Chebyshev's inequality is
strict but crude. In this work, we provide a new method that leads
us to a strict confidence interval which is close to the outcome
from using CLT. We primarily study interval estimation for
percentage. We apply Edgeworth expansion and Berry-Esseen
inequality to modify the results of using CLT for purpose of doing comparison.
We also apply the new method to obtain a strict solution instead of an approximate solution regarding population percentage p.

Contents
1 Introduction 4
2 The 2Kth moment method for strict confidence intervals 6
3 Use the 2Kth moment method in Bernoulli Distribution 12
4 Discussion on outcome of the new method on percentage
estimation 14
5 Comparison on the outcomes from using the new method,
CLT, and Chebyshev's inequality 17
6 Modified methods related to Edgeworth expansion and Berry-Esseen inequality 19
6.1 Method with use of Edgeworth expansion . . . . . . . . . . . . 19
6.2 Method with use of Berry-Esseen inequality . . . . . . . . . . 23
7 Appendix 25
7.1 Lyapunov's inequality . . . . . . . . . . . . . . . . . . . . . . . 25
7.2 R Code for Figure 1 . . . . . . . . . . . . . . . . . . . . . . . 25
7.3 R Code for Figure 2 . . . . . . . . . . . . . . . . . . . . . . . 26
7.4 R Code for Figure 3 . . . . . . . . . . . . . . . . . . . . . . . 28
8 Conclusion 30
Reference 31

References
Bickel, P., and Doksum, K. (2001) Mathematical Statistics, Prentice
Hall, Upper Saddle River, New Jersey.
Brown, L., Cai, T., and Dasgupta, A. (2001) Interval estimators for a
binomial proportion. Statistical Science 16, 101-133.
Maboukian, E. (1985) Modern Concepts and Theorems of Mathematical
Statistics, Springer-Verlag, New York Berlin Heidelberg Tokyo.