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題名 產生貝他分配的演算法研究
A Study on an Algorithm for Generating Beta Distribution
作者 洪英超
Hung, Ying Chau
貢獻者 江振東
Jiang, Jen Dung
洪英超
Hung, Ying Chau
關鍵詞 貝他分配
Beta Distribution
日期 1994
上傳時間 29-Apr-2016 09:21:07 (UTC+8)
摘要 　　在眾多產生貝他分配的方法中，我們研究Kennedy的演算法。在本文中，我們探討在小樣本下，不同參數組合(k,p,q,r) 產生同一貝他分配的情形。
　　There are mAny methods for generating a beta distribution. In this study, we focus on the method proposed by Kennedy (1988). Let [A<sub>1</sub>,B<sub>1</sub>]=[0,1] And [A<sub>n</sub>,B<sub>n</sub>] be rAndom subinterval of [0,1] defined recursively as follows. Take C , D to be the minimum And maximum of k i.i.d rAndom points uniformly distributed on [A<sub>n</sub>,B<sub>n</sub>]; And choose [A<sub>n+1</sub>,B<sub>n+1</sub>] to be [C<sub>n</sub>,B<sub>n</sub>], [A<sub>n</sub>,D<sub>n</sub>] or [C<sub>n</sub>,D<sub>n</sub>] with probabilities p, q, r respectively such that p+q+r=1. Kennedy showed that the limiting distribution of [A<sub>n</sub>,B<sub>n</sub>] has a beta distribution on [0,1] with parameters k(p+r) And k(q+r).
描述 碩士
國立政治大學
統計學系
82354014
資料來源 http://thesis.lib.nccu.edu.tw/record/#B2002003389
資料類型 thesis
dc.contributor.advisor 江振東zh_TW
dc.contributor.advisor Jiang, Jen Dungen_US
dc.contributor.author (Authors) 洪英超zh_TW
dc.contributor.author (Authors) Hung, Ying Chauen_US
dc.creator (作者) 洪英超zh_TW
dc.creator (作者) Hung, Ying Chauen_US
dc.date (日期) 1994en_US
dc.date.accessioned 29-Apr-2016 09:21:07 (UTC+8)-
dc.date.available 29-Apr-2016 09:21:07 (UTC+8)-
dc.date.issued (上傳時間) 29-Apr-2016 09:21:07 (UTC+8)-
dc.identifier (Other Identifiers) B2002003389en_US
dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/87867-
dc.description (描述) 碩士zh_TW
dc.description (描述) 國立政治大學zh_TW
dc.description (描述) 統計學系zh_TW
dc.description (描述) 82354014zh_TW
dc.description.abstract (摘要) 　　在眾多產生貝他分配的方法中，我們研究Kennedy的演算法。在本文中，我們探討在小樣本下，不同參數組合(k,p,q,r) 產生同一貝他分配的情形。zh_TW
dc.description.abstract (摘要) 　　There are mAny methods for generating a beta distribution. In this study, we focus on the method proposed by Kennedy (1988). Let [A<sub>1</sub>,B<sub>1</sub>]=[0,1] And [A<sub>n</sub>,B<sub>n</sub>] be rAndom subinterval of [0,1] defined recursively as follows. Take C , D to be the minimum And maximum of k i.i.d rAndom points uniformly distributed on [A<sub>n</sub>,B<sub>n</sub>]; And choose [A<sub>n+1</sub>,B<sub>n+1</sub>] to be [C<sub>n</sub>,B<sub>n</sub>], [A<sub>n</sub>,D<sub>n</sub>] or [C<sub>n</sub>,D<sub>n</sub>] with probabilities p, q, r respectively such that p+q+r=1. Kennedy showed that the limiting distribution of [A<sub>n</sub>,B<sub>n</sub>] has a beta distribution on [0,1] with parameters k(p+r) And k(q+r).en_US
dc.description.tableofcontents 謝辭
ABSTRACT
Contents
Figures
Tables
Chapter 1 Introduction-----1
Chapter 2 Literature Review-----3
　　2.1 Stochastic Search Methods for Global Optimization-----3
　　2.2 Kennedy`s Algorithm-----4
　　2.3 The RAnge of Parameters-----8
Chapter 3 Tools for Use in Simulations-----10
　　3.1 Introduction-----10
　　3.2 RAndom Number Generator-----10
　　3.3 Chi-Square Goodness-of-Fit Test-----11
　　3.4 The Up-And-Down Test-----12
　　3.5 Flow Diagrams for the Simulation-----13
Chapter 4 Simulation Results-----17
　　4.1 Uniformity And RAndomness of the Congruential Generator-----17
　　4.2 The AcceptAnce Probability of the Goodness-of-Fit Test-----18
　　4.3 The "Best" Combination of k, p, q, r-----19
　　　　4.3.1 Symmetric Cases-----20
　　　　4.3.2 Right-Skewed Cases-----26
　　　　4.3.3 Left-Skewed Cases-----37
　　4.4 The Speed of Convergence-----44
Chapter 5 Conclusions-----48
Appendix-----50
References-----59

Figures
Figure 2.1　Diagram for choosing the first subinterval from [0,1].-----5
Figure 3.5.1　Flow diagram for testing "randomness" and Uniform(0,1) of a random number generator.-----14
Figure 3.5.2　Flow diagram for comparing all possible combinations of k, p, q, r which generate Beta(k(p+r),k(q+r)).-----15
Figure 3.5.3　Flow diagram for calculating the mean frequencies T of the interval which converges with the final length < 0.001 for all possible combinations of k, p, q, r.-----16
Figure 4.1　100 sample points are grouped into 10 categories where each category has length 0.1.-----18
Figure 4.2　The acceptance probability of the test hypothesis H0：F(x) = Beta(4,4) distribution for different sample sizes given k = 5, p = 1/5, q =1/5, r =3/5.-----19
Figure 4.3　The pdf of Beta(m,m) distribution.-----20
Figure 4.4　The pdf of a Beta(m,n) distribution with mFigure 4.5.1　The probability that the goodness-of-fit test accepts vs k when a Beta(2,5) distribution is generated.-----27
Figure 4.5.2　The probability that the goodness-of-fit test accepts vs k when a Beta(3,4) distribution is generated.-----27
Figure 4.5.3　The probability that the goodness-of-fit test accepts vs k when a Beta(3,5) distribution is generated.-----27
Figure 4.5.4　The probability that the goodness-of-fit test accepts vs k when a Beta(2,7) distribution is generated.-----27
Figure 4.6　The pdf of a Beta(m,n) distribution with m>n.-----37

Tables
Table 4.1　100 sample points generated by generator (4.1.1)-----17
Table 4.2　The values of P(goodness-of-fit test accept ∣k, p, q, r) and standard deviations when a Beta(1,1) distribution is generated-----22
Table 4.3　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(2,2) distribution is generated-----22
Table 4.4　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(3,3) distribution is generated-----23
Table 4.5　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(4,4) distribution is generated-----24
Table 4.6　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(5,5) distribution is generated-----25
Table 4.7　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(2/3,5/3) distribution is generated-----28
Table 4.8　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(1.2) distribution is generated-----28
Table 4.9　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(1,3) distribution is generated-----29
Table 4.10　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(1,4) distribution is generated-----29
Table 4.11　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(2,3) distribution is generated-----30
Table 4.12　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(1,5) distribution is generated-----30
Table 4.13　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(2,4) distribution is generated-----31
Table 4.14　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(1,6) distribution is generated-----31
Table 4.15　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(3,4) distribution is generated-----32
Table 4.16　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(2,5) distribution is generated-----33
Table 4.17　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(1,7) distribution is generated-----33
Table 4.18　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(3,5) distribution is generated-----34
Table 4.19　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(2,6) distribution is generated-----35
Table 4.20　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(2,7) distribution is generated-----36
Table 4.21　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(4,1) distribution is generated-----38
Table 4.22　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(4,2) distribution is generated-----38
Table 4.23　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(5,1) distribution is generated-----39
Table 4.24　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(5,2) distribution is generated-----39
Table 4.25　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(4,3) distribution is generated-----40
Table 4.26　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(6,1) distribution is generated-----41
Table 4.27　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(7,1) distribution is generated-----41
Table 4.28　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(5,3) distribution is generated-----42
Table 4.29　The values of P(goodness-of-fit test accept │k, p, q, r) and standard deviations when a Beta(6,1) distribution is generated-----43
Table 4.30　The mean frequencies And variances for the convergence using the criterion that the final interval length <0.001-----46		zh_TW
dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#B2002003389en_US
dc.subject (關鍵詞) 貝他分配zh_TW
dc.subject (關鍵詞) Beta Distributionen_US
dc.title (題名) 產生貝他分配的演算法研究zh_TW
dc.title (題名) A Study on an Algorithm for Generating Beta Distributionen_US
dc.type (資料類型) thesisen_US