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研究生:蘇南誠
研究生(外文):Nan-cheng Su
論文名稱:關於分佈函數之探討
論文名稱(外文):An Investigation of Distribution Functions
指導教授:黃文璋黃文璋引用關係
指導教授(外文):Wen-Jang Huang
學位類別:博士
校院名稱:國立中山大學
系所名稱:應用數學系研究所
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2008
畢業學年度:96
語文別:英文
論文頁數:74
中文關鍵詞:非齊性Poisson過程順序統計量條件分佈刻劃條件期望值偏斜t分佈偏斜對稱分佈偏斜常態分佈偏斜柯西分佈記錄值順序統計量性質
外文關鍵詞:skew-normal distributionnonhomogeneous Poisson processconditional expectationskew-Cauchy distributionskew-t distribution.skew-symmetric distributionorder statisticscharacterizationconditional distributionrecord valuesorder statistics property
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研究機率分佈的性質一直以來都是統計和應用機率領域的主要課題。本論文主要經由下列兩個主題來探討分佈函數:(i) 基於記錄值和順序統計量的分佈刻劃 (ii) 關於偏斜t分佈的性質。
在刻劃文獻中,有很多結果隱含關於記錄值和順序統計量的性質。雖然在基於記錄值和順序統計量的刻劃中,已經有許多大家熟悉的結果,但是發掘新的刻劃結果仍相當吸引人的。本論文的第一部分,將給出在給定最大順序統計量之下,任何一個記錄值的條件分佈。接著探討基於記錄值和最大順序統計量的分佈刻劃。此外,也會在順序統計量的點過程當中,藉由到達時刻或現在壽命的條件動差之間的某些關係,來刻畫點過程的期望函數。這些結果可應用在順序統計量中對均勻分佈的刻劃,和記錄值中對指數分佈的刻劃。
Azzalini (1985, 1986)提出的偏斜常態分佈,不僅涵蓋常態分佈,且具有一些與常態分佈相同的性質。此類分佈有助於穩健性的研究和偏斜性的建模。此後,便有許\多人投入基於對稱分佈的偏斜分佈之研究。本論文的第二部分,將定義和研究所謂的廣義偏斜t分佈,並給出一些例子。這些例子是由兩個獨立的偏斜對稱隨機變數相除所產生的。最後我們也會對偏斜對稱分佈的性質作一些探討。
The study of properties of probability distributions has always been a persistent theme of statistics and of applied probability. This thesis deals with an investigation of distribution functions under the following two topics: (i) characterization of distributions based on record values and order statistics, (ii) properties of the skew-t distribution.
Within the extensive characterization literature there are several results involving properties of record values and order statistics. Although there have been many well known results already developed, it is still of great interest to find new characterization of distributions based on record values and order statistics. In the first part, we provide the conditional distribution of any record value given the maximum order statistics and study characterizations of distributions based on record values and the maximum order statistics. We also give some characterizations of the mean value function within the class of order statistics point processes, by using certain relations between the conditional moments of the jump times or current lives. These results can be applied to characterize the uniform distribution using the sequence of order statistics, and the exponential distribution using the sequence of record values, respectively.
Azzalini (1985, 1986) introduced the skew-normal distribution which includes the normal distribution and has some properties like the normal and yet is skew. This class of distributions is useful in studying robustness and for modeling skewness. Since then, skew-symmetric distributions have been proposed by many authors. In the second part, the so-called generalized skew-t distribution is defined and studied. Examples of distributions in this class, generated by the ratio of two independent skew-symmetric distributions, are given. We also investigate properties of the skew-symmetric distribution.
1 Introduction 1
2 Characterizations based on record values and order statistics 7
2.1 Introduction 7
2.2 The conditional distribution of record values given Xn:n 9
2.3 Characterizations based on conditional expectations of record values given Xn:n 12
2.4 Further characterizations based on record values and Xn:n 19
3 Characterizations of the order statistics point process by the relations between its conditional moments 25
3.1 Introduction 25
3.2 Characterizations by using conditional moments of jump times of the process 28
3.3 Characterizations based on relationship of conditional moments of jump time and current life 35
3.4 Some characterizations related to Abu-Youssef (2003) 37
4 A study of generalized skew-t distribution 41
4.1 Introduction 41
4.2 Preliminaries 42
4.3 Generalized skew-t distribution 44
4.4 Skew-symmetric distribution 54
4.5 Appendix 57
References 61
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