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

周宗翰(Tsung-han Chou)

指導教授 ： 許玉生

國立中央大學/理學院/數學系/碩士(2007年)

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。

關鍵字

經驗分布 ； 密度函數估計式 ； 冪數 ； 穩定型分布

並列摘要

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.

並列關鍵字

stable distributions ； empirical distributions ； kernel density estimators ； exponent

參考文獻

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