單峰穩定型分布之冪數的經驗分布及核密度函數估計法; Exponent Estimations for Unimodal Stable Distribution based on Empirical Distributions and Kernel Density Estimators

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题名:	單峰穩定型分布之冪數的經驗分布及核密度函數估計法;Exponent Estimations for Unimodal Stable Distribution based on Empirical Distributions and Kernel Density Estimators
作者:	周宗翰;Tsung-han Chou
贡献者:	數學研究所
关键词:	經驗分布;密度函數估計式;冪數;穩定型分布;stable distributions;empirical distributions;kernel density estimators;exponent
日期:	2007-07-05
上传时间:	2009-09-22 11:09:03 (UTC+8)
出版者:	國立中央大學圖書館
摘要:	穩定型分布之冪數因未出現於密度函數或分布函數，故不易估計，本文介紹一些估計冪數的方法。我們發現，單峰穩定型分布之冪數為密度函數或分布函數之泛函，故可由核密度函數估計式或經驗分布估計之。我們將討論這些估計式的性質及應用。 The collection of stable distributions is a particular class of distributions studied in probability and statistics. Let $X,X_1,ldots,X_k$ denote a sequence of i.i.d. random variables with a common distribution $R$. If for all positive integer $k$, $X$ and $frac{X_1+cdots+X_k}{k^alpha}$ have the same distribution for some constant $alpha$, then $R$ is a stable distribution with exponent $frac{1}{alpha}$. It is difficult to estimate exponent $alpha$ since $alpha$ does not appear in probability density function. The purpose of this paper is to study some estimators of $alpha$ and their applications. We find that under unimodal assumption $alpha$ is a functional of probability density function or distribution function. Consequently, $alpha$ can be estimated by kernel density estimators or empirical distributions.
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