當母體分配為小樣本的常態分配時，通常使用T分配統計量建立信賴區間；然而當母體分配不為常態而是偏態分配時，若在偏態系數愈大且樣本數愈小的情況下，使用T分配統計量會造成誤差，為解決這項問題，許多學者提出將T分配統計量進行修正，期望透過修改後的T分配統計量以降低母體偏態造成的誤差，Zhou（2005）根據Hall（1992）的方法，提出修正後的T分配統計量，並證明其方法可以得到近似的覆蓋精度且信賴區間長度較短的結果，但其衡量與比較方法皆是以傳統的衡量指標進行，在衡量方法的優劣上容易發生相互矛盾的情形，假如追求高覆蓋機率，則會得到較長的信賴區間，反之若追求較短的信賴區間則會犧牲覆蓋機率，因此本文中將採用Yeh and Schemeiser（2015）提出的VAMP1RE 標準對Zhou（2005）、Hall（1992）提出的方法進行比較，該標準以Schruben （1980）提出的覆蓋值（Coverage value）為基礎，該標準將CIP的衡量分成偏離有效性以及無效模仿兩項指標，計算CIP與理想CIP的覆蓋值，取其平均平方誤差，並在不同樣本大小下對其結果進行分析。

When we are trying to construct a confidence interval with small sample size, we can use t-statistic construct the(1-α)100%confidence interval to evaluate the population mean if the data is independent and identical from normal distribution. If data comes from a skewed distribution, the coverage accuracy of t-statistic is poor. In order to solve this problem, Hall (1992) proposed the modified t-statistic to remove the effect of skewness and Zhou (2005) proposed a new t-statistic which he claim the new statistic can get shorter confidence interval length than Hall’s t-statistic. Zhou’s comparison is based on traditional criterion; it has some contradiction when we use it to measure the CIP is good or not (i.e. when we want a tighter confidence width, we will get less coverage probability, vice versa. We will use a new criterion to evaluate the CIP build by these authors, which proposed by Yeh and Schemeiser (2015) called VAMP1RE Criterion. The new Criterion is based on the coverage value which is proposed by Schruben (1980). VAMP1RE Criterion can be decomposing into two causes, Departure form Validity and Inability to mimic. VAMP1RE Criterion can be find by the coverage value of Ideal CIP and proposed CIP then calculate the mean-squared error of the two coverage value. We will use VAMP1RE Criterion to compare the t-statistic proposed by Zhou (2005)、Hall (1992) and Johnson (1978).

1. Davison, A.C., & Hinkley, D.V., Bootstrap Methods and their Application., Cambridge University Press., New York ,1997.
2. Mundform, D. J, et al., Number of Replications Required in Monte Carlo Simulation Studies: A Synthesis of Four Studies., Journal of Modern Applied Statistical Methods., New York ,2001.
3. Efron, B., Bootstrap Methods: Another Look at Jackknife., The Annals of Statistics., New York ,1979.
4. Efron B., & Tibshirani R.J., An Introduction to the Bootstrap., Chapman and Hall., London, 1993.
5. Efron, B., Better Bootstrap Confidence Intervals., Journal of the American Statistical Association., 1987.
6. Hall, P., On the removal of skewness by transformation., Journal of the Royal., 1992.
7. 許力升，中央極限定理應用於偏斜分布時樣本大小之探討，淡江大學數學系碩士班，新北市，民國99年。
8. Hutchinson, S. R., & Bandalos, D. L. A guide to Monte Carlo simulation research for applied researchers., Journal of Vocational Education Research., 1997, 22(4): 233-45.
9. Johnson, N. J., Modified t tests and confidence intervals for asymmetrical populations., Journal of the American Statistical Association., 1978, 73(363): 536-544.
10. Kang, K. & Schmeiser, B., Graphical methods for evaluating and comparing
confidence-interval procedures., Operations Research., 1990, 38(3): 546-553.
11. Kao, L, et al., Good Luck or Good Strategy? : Bootstrapped Mutual Funds Performance, Journal of Management & Systems., 2007, 14(3): 341-358.
12. Kroese, D. P, et al. Why the Monte Carlo method is so important today., Computational Statistics., 2014, 6(6): 386-392.　
13. Law, A. M., & Kelton, W. D., Confidence interval procedures for steady-state simulations, II: A survey of sequential procedures., Management Science., 1982, 28(5): 550-562.
14. Neyman, J., On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection., Journal of the Royal Statistical Society., 1934, 97(4): 558-625.
15. Carpenter, J. & Bithell, J., Bootstrap confidence intervals: when, which, what? A practical guide for medical statisticians., Statistics in medicine., 2000, 19(9): 1141-1164.
16. Shi, W. & Kibria, B. M. G., On some confidence intervals for estimating the mean of a skewed population., International Journal of Mathematical Education in Science and Technology., 2006, 38(3): 412-421.
17. Schruben, L.W., A coverage function for interval estimators of simulation response.,
Management Science., 1980, 26(1): 18-27.
18. Schmeiser, B. W., Some myths and common errors in simulation experiments., Proceedings of the 2001 Winter Simulation Conference., 2001, 1: 39-46.
19. Schmeiser, B.W. & Scott, M. D., SERVO: Simulation experiments with random-vector output., Proceedings of the 1991 Winter Simulation Conference., New Jersey, 1991,
927-936.
20. Schmeiser, B.W. & Yeh, Y., On choosing a single criterion for confidence-interval procedures., Proceedings of the 2002 Winter Simulation Conference., California, 2002, 345-352.
21. Scholz, F.W., The Bootstrap Small Sample Properties., Tech. Rep., 2007.
22. Zhou, X.U ., Nonparametric confidence intervals for the one- and two-sample problems. Biostatistics., 2005, 6(2): 187-200.
23. Yeh, Y. & Schmeiser, B.W., VAMP1RE: a single criterion for rating and
ranking confidence-interval procedures., IIE Transactions., 2015, 47(11): 1203-1216.
24. 周心怡，拔靴法(Bootstrap)之探討及其應用，國立中央大學統計研究所，桃園市，2004，民國93年。
25. Nelson, B.L., Stochastic Simulation Research in Management Science., Management Science., 2004, 50(7): 855-868..