國立中山大學,National Sun Yat-sen University,學位論文,thesis/dissertation,利用泰勒展開式求分位點的近似值,Approximation for Quantile Using Taylor Expansion

論文名稱 Title	利用泰勒展開式求分位點的近似值 Approximation for Quantile Using Taylor Expansion
系所名稱 Department	應用數學系 Department of Applied Mathematics
畢業學年期 Year, semester	100 學年度第 2 學期 The spring semester of Academic Year 100	語文別 Language	英文 English
學位類別 Degree	碩士 Master	頁數 Number of pages	24
研究生 Author	邱昇昱 Sheng-Yu Chiou
指導教授 Advisor	張福春 Fu-Chuen Chang
召集委員 Convenor	羅夢娜 Mong-Na Lo Huang
口試委員 Advisory Committee	郭美惠, 張中, 陳美如 Mei-Hui Guo; Chung Chang; May-Ru Chen
口試日期 Date of Exam	2012-06-21	繳交日期 Date of Submission	2012-07-03
關鍵字 Keywords	泰勒展開式、分位點、中位數、牛頓法、連續型分配 Newton's method, Median, continuous distribution, Quantile, Taylor expansion
統計 Statistics	本論文已被瀏覽 5751 次，被下載 1587 次 The thesis/dissertation has been browsed 5751 times, has been downloaded 1587 times.

中文摘要
分位點是隨機變數中的一個基本且重要的統計量。對於某些分佈，它們的分位點可由公式所得。但是大部分分佈的分位點之公式並不存在。Yu and Zelterman (2011) 和 Chang (2004) 已經發表了有關求分位點的近似方法。而本論文我們提出一個改進的方法，此方法結合泰勒展開式與牛頓法。並舉了一些例子來比較Yu and Zelterman (2011) 和 Chang (2004) 的方法和我們所提出的方法之優劣。
Abstract
Quantile is a basic and an important quantity of a random variable. In some distributions, their quantiles have closed-form expressions. However, for many continuous distributions, the closed-form expressions of their quantiles do not exist. Yu and Zelterman (2011) and Chang (2004) have proposed an approximation of quantiles. In this paper, we propose an improved method which is combined the Taylor expansion with Newton’s method. Some examples are given to compare the computing time of the method we proposed with the methods in Yu and Zelterman (2011) and Chang (2004).

目次 Table of Contents
誌謝 i 摘要 ii Abstract iii 1 Introduction 1 2 General approximation for the quantile function 2 3 Examples 6 4 Conclusion 14 References 15 Appendix 16

參考文獻 References
Abramowitz, M. and Stegun, I.A. (1970). Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Washington, DC: National Bureau of Standards. Burden, R.L. and Faires, J.D. (2010). Numerical Analysis, 9th ed. Boston, MA: Brooks/Cole. Chang, F.-C. (2004). A recursive formula for computing Taylor polynomial of quantile. Unpublished technical report. Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, Taiwan 804, R.O.C. Dette, H. Melas, V.B. and Pepelyshev, A. (2004). Optimal designs for estimating individual coefficients in polynomial regression — a functional approach. J. Statist. Plann. Inference 118, pp. 201-209. Ruskeepaa, H. (2009). Mathematica Navigator: Mathematics, Statistics and Graphics, 3rd ed. Boston, MA: Elsevier/Academic Press. Yu, C. and Zelterman, D. (2011). A general approximation to quantiles. Unpublished technical report. Department of Bioststistics, Vanderbilt University Medical Center, Nashville, TN 37232, USA. Yu, X.-C., Yuan, Z.-Y., Yu, C. and Yang, M. (2005). Computer Implementation of Probability Distribution Quantile Estimation. IEEE. ICMLC, Guangzhou, August 2005, pp. 18-21.

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