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研究生： 李世懿
Shih-yi Li
論文名稱： 分布函數之反函數之核估計的模擬研究
A Simulation Study for Kernel Estimator of Inverse Distribution Function
指導教授： 許玉生
Yu-sheng Hsu
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系
Department of Mathematics
畢業學年度： 96
語文別： 中文
論文頁數： 34
中文關鍵詞： 分布函數之反函數 、核估計
外文關鍵詞： inverse distribution function, kernel estimator
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    • 分布函數為機率上重要的分析工具，其重要性不亞於機率密度函數及特徵函數。在統計上分布函數也有很多應用，令$F$表一分布函數，則$F^{-1}$可用於隨機變數之模擬及穩定型分布(stable distribution)之冪數(exponent)的估計。通常分布函數是未知的，必需用樣本估計。分布函數未知時，常用之分布函數的估計式為經驗分布(empirical distribution function)。本文之目的為研究$F^{-1}$的估計，但上述經驗分布卻因其反函數不存在，故不能直接運用。本文提出$F^{-1}(y)$之核估計式$widehat{F}^{-1}(y)$，因此式之機率性質非常複雜，故本文將以電腦模擬方式研究$widehat{F}^{-1}(y)$之漸近一致性(asymptotic consistency)及漸近常態性(asymptotic normality)。

    The inverse function of a distribution function has many applications in statistics. In practice, the inverse function is unknown and has to be estimated. The purpose of this paper is to discuss a kernel estimator $widehat{F}^{-1}(y)$ of the inverse function $F^{-1}(y)$ of a distribution function $F(x)$. Since the theoretical property of $widehat{F}^{-1}(y)$ is extremely complicated, we will investigate the asymptotic consistency and asymptotic normality of $widehat{F}^{-1}(y)$ via computer simulations.

    中文摘要...........................................i 英文摘要..........................................ii 致謝辭...........................................iii 目錄..............................................iv 1 簡介.............................................1 2 常態數據之模擬結果...............................5 3 柯西數據之模擬結果...............................8 4 羅吉斯數據之模擬結果............................11 5 模擬結果之比較..................................14 6 結論............................................15 參考文獻..........................................16 附錄一............................................18 附錄二............................................23 附錄三............................................28 附錄四............................................33

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