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博碩士論文 etd-0624108-184139 詳細資訊
Title page for etd-0624108-184139
論文名稱
Title
關於分佈函數之探討
An Investigation of Distribution Functions
系所名稱
Department
畢業學年期
Year, semester
語文別
Language
學位類別
Degree
頁數
Number of pages
74
研究生
Author
指導教授
Advisor
召集委員
Convenor
口試委員
Advisory Committee
口試日期
Date of Exam
2008-06-07
繳交日期
Date of Submission
2008-06-24
關鍵字
Keywords
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
skew-normal distribution, nonhomogeneous Poisson process, conditional expectation, skew-Cauchy distribution, skew-t distribution., skew-symmetric distribution, order statistics, characterization, conditional distribution, record values, order statistics property
統計
Statistics
本論文已被瀏覽 5736 次,被下載 1930
The thesis/dissertation has been browsed 5736 times, has been downloaded 1930 times.
中文摘要
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。
Abstract
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.
目次 Table of Contents
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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