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t分布的代表点及其在统计模拟中的应用
Representative Points of Student's tn Distribution and Their Applications in Statistical Simulation
本文首先讨论了在均方误最小意义下学生氏tn 分布代表点， 利用方开泰、贺曙东的算法找出代表点， 证明了当n≥3时， tn 分布总体下算法的收敛性， 代表点的存在性及其唯一性， 并从tn 分布角度研究了代表点在统计模拟方面的应用。传统的蒙特卡罗方法和自助法及重抽样方法对随机样本进行抽样， 是统计模拟方法的基础. Fang， Zhou， Wang讨论了一元正态分布的代表点在统计模拟中的应用并且首次提出用代表点代替独立同分布的随机样本， 构造一个离散的近似总体， 通过对近似总体重复抽样来进行统计推断. 这是一个新思想. 本文继续探讨这个问题， 文中统计推断主要有两部分： 经典估计和稳健估计. 经典估计主要集中考虑参数点估计 （均值、方差、偏度和峰度）； 稳健估计主要考虑简单的位置参数 （中位数和均值）和尺度参数 （中位数绝对偏差和四分位距）. 我们的结果再次验证， 代表点方法可以明显地提高统计估计量的精确度以及收敛速度.
The paper considers the representative points (RP) of student's tn distribution for minimizing the mean of square error. Fang and He's algorithm is applied to find its RP. We show the existence and uniqueness of RP of tn distribution when n≥3 and discuss applications of RP in statistical simulation. Traditional Monte Carlo, Bootstrap and Resampling are the basic methods in statistical simulation based on a random sample. Fang, Zhou and Wang illustrated the application of representative points of univariate normal distribution in statistical simulation, and firstly proposed to use RP instead of i.i.d. random samples, to construct an approximate distribution and then resample from the approximation for statistical inference. This is a new idea. In this article, we continue to talk about this issue and focus on classical estimation and robust estimation of t-distribution's parameters. Mean, variance, skewness and kurtosis are discussed in classical estimation of parameters; and robust estimation involves the estimation of the location parameters (mean and median) and the scale parameters (median absolute deviation and interquartile range). Our results once again indicate that the new method can significantly improve the accuracy of the estimator of the statistics, and accelerates the converging speed of the statistics.
伪蒙特卡罗方法 / 代表点 / 统计模拟 / tn分布 / 重抽样 / 稳健估计 {{custom_keyword}} /
quasi-Monte Carlo methds / representative points / statistical simulation / tn distribution / resampling / robust estimation {{custom_keyword}} /
[1] Cox D R. Note on Grouping. J. American Statistical Association, 1957, 52: 543-547
[2] 方开泰, 贺曙东. 在正态总体中如何选择给定数目的代表点的问题. 应用数学学报, 1984, 7(3): 293-306 (Fang K T, He S D. The problem of selecting a given number of representative points in a normal population and a generlized Mill's ration. Acta Mathematica Applicatae Sinica, 1984, 7(3): 293-306)
[3] Flury B A. Principal points. Biometrika Trust, 1990, 77(1): 33-41
[4] Max J. Quantizing for minimum distortion. Information Theory, IRE Transactions On, 1960, l6(1): 7-12
[5] Bofinger E. Maximizing the Correlation of Grouped Observations. J. American Statistical Association, 1970, 65: 1632-1638
[6] 傅洪海. 在 Γ分布的总体中如何选取给定数目的代表点问题. 中国矿业学院学报, 1985, 4: 107-117 (Fu H H. The problem of selecting a given number of representative points in a Γ distribution. Jornual of China University of Mining and Technology, 1985, 4: 107-117)
[7] 傅洪海. 威布尔分布总体的最有代表点问题. 中国矿业大学学报, 1993, 22(3): 78-83 (Fu H H. The problem of the optimal representative points about Weibull distribution. Journal of China University of Mining and Technology, 1993, 22(3): 78-83)
[8] 费荣昌. 在 Pearson 分布族总体中选取代表点的问题. 无锡轻工业学院学报, 1990, 9(1): 71-78 (Fei R C. The problem of selecting the representative points in the family of Pearson distribution. Journal of Wuxi Institute of Light Industry, 1990, 9(1): 71-78)
[9] 费荣昌. 代表点与总体之间的统计关系. 无锡轻工业学院学报, 1991, 10(1): 123-134 (Fei R C. The statistical relationship between the representative points and population distribution. Journal of Wuxi Institute of Light Industry, 1991, 10(1): 123-134)
[10] Yamamoto W, Shinozaki N. On uniqueness of two principal points for univariate location mixtures. Statist. Probab. Lett., 2000, 46: 33-42
[11] Fang K T, Wang Y. Number-Theoretic Methods in Statistics. London: Chapman & Hall, 1994
[12] Fang K T, Zhou M, Wang W J. Applications of the representative points in statistical simulations. Science China Mathematics (A), 2014, 57(12): 2609-2620
[13] 刘智敏. 统计 t 分布的新计算与应用. 中国计量学院学报, 2004, 15(1): 1-12 (Liu Z M. New statistical calculation and Application of t distribution. Journal of China University of Metrology., 2004, 15(1): 1-12)
[14] Ross S M. A Course in Simulation. New York: Macmillan Publishing Company, 1990
[15] Efron B. Bootstrap methods: another look at the jacknife. The Annals of Statistics, 1979, 7: 1-26
[16] Efron B, Tibshirani R J. An Introduction to the Bootstrap. New York: Chapman & Hall, 1993
[17] Niederreiter H. Random number generaion and quasi-Mente Carlo methods. Philadelphia: SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, 1992
[18] Li L, Flury B D. Uniqueness of principal points for univariate distributions. Statist. Probab. Lett., 1995, 25: 323-327
[19] Peter J Huber, Elvezio M. Ronchetti. Robust statistics (Second Edition). New Jersey: John Wiley & Sons, Inc., 2009
国家自然科学基金（11401146，11261016），中国科学院自然科学奖的研究基金（11261016），海南省自然科学基金（20156233，20151006），海南师范大学青年基金（QN1429）以及北京师范大学-香港浸会大学联合国际学院研究基金（R201409）资助项目.
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