分位數平均:統計推論與應用

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(1)國立交通大學統計學研究所碩士論文. 分位數平均:統計推論與應用. Quantile Mean : Statistical Inferences and Applications. 研究生：卓韋婷指導教授：陳鄰安. 教授. 中華民國九十三年六月.

(2) 分位數平均:統計推論與應用 Quantile Mean :Statistical Inferences and Applications 研究生：卓韋婷. Student：Wei-Ting Jwo. 指導教授：陳鄰安. Advisor：Lin-An Chen. 國立交通大學統計學研究所碩士論文. A Thesis Submitted to Institute of Statistics College of Science National Chiao Tung University in partial Fulfillment of the Requirements for the Degree of Master in Statistics June 2004 Hsinchu, Taiwan, Republic of China. 中華民國九十三年六月.

(3) 分位數平均:統計推論與應用學生：卓韋婷. 指導教授：陳鄰安教授. 國立交通大學統計學研究所碩士班. 摘. 要. 常用的分位數平均是第 α 及 1 − α 這兩個分位數所取的平均數，此一平均數具有穩健性的特質。我們把此分位數平均擴展到由相同覆蓋機率但長度最短的兩個分位數來取的平均數，討論此一新的分位數平均，我們考慮了有母數及無母數兩種方法並做了分析，最後我們也把此一分位數平均擴展到一般的 L-估計上。.

(4) Quantile Mean : Statistical Inferences and Applications. Student: Wei-Ting Jwo. Advisor : Dr. Lin-An Chen. Institute of Statistics National Chiao Tung University. ABSTRACT. The quantile mean being the average of a pair of symmetric type quantiles, F −1 (α ) and F −1 (1 − α ) , is a robust type location parameter playing an alternative to the population mean. We extend this quantile mean to a pair of quantiles where this corresponding quantile interval is the one with smallest width among all choices of 1 − 2α quantile intervals. Parametric statistical inferences and nonparametric estimation techniques are all addressed. Moreover, an extension of the quantile mean to a new general L-estimation has also been provided..

(5) 目中文提要英文提要目錄表目錄 1.. 2. 3. 3.1 3.2 3.3 4 4.1 4.2 4.3 4.4 5 5.1 5.2 6 7 參考文獻. 錄. ………………………………………………… ………………………………………………… ………………………………………………… ………………………………………………… Introduction……………………………………… Mode Type Quantile Mean…………………… Estimation for Mode Type Quantile Mean…… Normal Distribution…………………………… Exponential Distribution……………………… Uniform Distribution…………………………… Confidence Interval and Hypothesis Testing for Mode Type Quantile Mean…………………… C.I. for Normal Distribution…………………… C.I. for Exponential Distribution……………… C.I. for Uniform Distribution………………… C.I. for Gamma distribution…………………… Testing Hypothesis for Mode Type Quantile Mean…………………………………………… Test for Normal Distribution…………………… Test for Exponential Distribution……………… Asymptotic Analysis and Monte Carlo Study for Nonparametric Estimation……………………… Extension of Mode Type Quantile Mean……… …………………………………………………. i ii iii iv 1. 2 7 8 9 11 11 11 12 13 13 14 14 14 14 17 19.

(6) 表目錄 Table1. Mode type quantile interval and mean for a discrete distribution ………………………………………………5 Table 2. Powers control charts based on median and mode intervals …………………………………………………7 Table 3. Estimates for γ = 1 − 2α median and mode types quantile means for revenue data …………………………………9 Table 4. Estimates for γ median and mode type quantile means for examination score data …………………………………10 Table 5. Tests for. H 0 : τ mod = t0. for Normal distribution …………14. Table 6. Tests for. H 0 : τ mod = t. for Exponential distribution ……14. Table 7. Asymptotic variance analyses for distribution Gamma (3.5, β ) with β = 0.3 and 1.0 ………………………………………15 Table 8. MSE’s for right skewed Exponential distribution ………16 Table 9. MSE’s for Gamma (α , β ) distribution ……………………17 Table 10. MSE’s for Gamma (2, 2) distribution ……………………18.

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