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| 研究生： |
洪英超 Hung, Ying Chau |
|---|---|
| 論文名稱： |
產生貝他分配的演算法研究 A Study on an Algorithm for Generating Beta Distribution |
| 指導教授： |
江振東
Jiang, Jen Dung |
| 學位類別： |
碩士
Master |
| 系所名稱： |
商學院 - 統計學系 Department of Statistics |
| 論文出版年： | 2013 |
| 畢業學年度： | 83 |
| 語文別： | 英文 |
| 論文頁數： | 72 |
| 中文關鍵詞： | 貝他分配 |
| 外文關鍵詞： | Beta Distribution |
| 相關次數： | 點閱：88 下載：0 |
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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).
Based upon the known asymptotic result, we study the small-sample behaviors among those combinations of k, p, q, r that have the same Beta(m, n) distribution, where m = k(p+r), n = k(q+r), through simulations. We conclude that smaller k's basically have better performAnces.
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 H<sub>0</sub>：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 m<n.-----26
Figure 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
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