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 [A₁,B₁]=[0,1] And [A_n,B_n] 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_n,B_n]; And choose [A_n+1,B_n+1] to be [C_n,B_n], [A_n,D_n] or [C_n,D_n] with probabilities p, q, r respectively such that p+q+r=1. Kennedy showed that the limiting distribution of [A_n,B_n] 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.

本論文共一冊，約四萬五千餘字，分為六章。 第一章：緒論。除闡明問題的性質及其重要性外，並述明本研究的目的。 第二章：理論基礎與文獻探討。闡明β理論出現的背景及其要點。並簡要述明理論出 現後，前人對貝他（β）的研究或驗證，並扼要評估前人的研究。 第三章：研究方法。本研究的對象以在台灣證卷交易所上市的公司普通股為對象，選 取約五十餘家公司股票為樣本。並詳細述本研究所採用的研究方法、步驟及研究工具 等，並以SPSS套裝軟體為主要的資料處理工具。 第四章：研究結果。除對本研究之結果詳加述說外，並提出討論，且與前人的研究結 果相互比較。 第五章：摘要及結論。對本研究的結果摘要述明，並給後繼研究者進一步研究的建議 。最後對本研究下一總結論。

近來許多學者指出股票市場不如效率市場假說所述，反而存在低風險異常報酬之現象，而Frazzini and Pedersen (2014)提出貝他套利交易策略，藉由該策略可增加獲取之超額報酬。故本研究將探討低風險異常報酬是否仍存於近年的美國股票市場，另外，亦將加入投資人情緒及流動性指標，研究其是否能改良貝他套利交易策略。實證結果顯示，低風險異常報酬仍然存在於美國股票市場，且執行貝他套利交易策略可增加所獲取之超額報酬。此外，於該策略加入成交量變動率及成交量，分別作為情緒及流動性指標後，發現兩種方式皆能改良貝他套利交易策略之獲利，其中又以加入流動性指標能獲取較多報酬，然而上述兩種策略於2007年至2008年金融危機時，皆無法有效提高貝他套利交易策略的獲利。

本論文主要是介紹並評估一個R的軟體套件叫做"rBeta2009"。此套件是由Cheng et al. (2012) [8] 所設計，其目的是用來產生貝他分配(Beta Distribution)及狄氏分配(Dirichlet Distribution)的亂數。本論文特別針對此套件之(i)有效性(effiniency)、(ii)精確性(accuracy)及(iii)隨機性(randomness)進行評估，並與現有的R套件作比較。此外，本論文也介紹如何應用此套件來產生(i)反貝他分配(Inverted Beta Distribution)、(ii)反狄氏分配(Inverted Dirichlet Distribution)、(iii)Liouville分配及(iv)凸面區域上的均勻分配之亂數。 / A package in R called "rBeta2009", originally designed by Cheng et al. (2012) [6], was introduced and evaluated in this thesis. The purpose of the package is generating beta random numbers and Dirichlet random vectors. In this paper, we not only evaluated (i) the efficiency, (ii) the accuracy and (iii) the randomness, but also compare it with other R packages currently in use. In addition, it was also scrutinized in this thesis how to generate (i) inverted beta random numbers, (ii) inverted Dirichlet random vectors, (iii) Liouville random vectors, and (iv) uniform random vectors over convex polyhedron by using the same package.

本研究利用元大台灣50 ETF作為樣本資料，檢測2006年至2016年實證期間風險基礎指數和市值加權指數所分別建構的投資組合，其績效表現、風險表現、分散性表現的優劣性；其中Choueifaty, Froidure, and Reynier (2011) 所建構的最佳風險分散投資組合 (most diversified portfolio) 為近年來新起的風險基礎指數投資組合，我們將證實在獲得良好的投資組合分散性同時，如同其他的風險基礎指數投資組合的目標，同時也能獲得超越以追蹤市值加權指數為標的的投資組合績效。 本研究以夏普比率、信息比率、阿爾法作為衡量績效的指標；以標準差、貝他作為風險衡量的指摽；另以Choueifaty and Coignard (2008) 提出的分散性比率作為分散性衡量的指標。實證結果顯示，在整體實證期間，最佳風險分散投資組合在績效、風險、分散性的指標上皆有超越市值加權指數投資組合的能力，再以年為單位的個別期間，其績效衡量上大致優於市值加權指數投資組合，風險和分散性衡量上則優於市值加權指數投資組合的表現，但論以其整體表現，並非為本研究所提出的風險基礎指數投資組合中最佳者，因此投資人在選擇該類投資組合策略時，建議從該投資組合過去表現中判斷，選擇符合自己投資習慣者為之。 / This article examines the performance, risks and diversification of different types of portfolio strategies such as risk-based indexes and cap-weighted index during 2006- 2016. We introduce the recent most diversified portfolio (MDP), which was proposed by Choueifaty, Froidure, and Reynier (2011) and find the result that like the goal of other risk-based portfolios, which is to improve the risk-return profile of cap-weighted portfolio, MDP surpasses overall performance, risks and diversification compared to cap-weighted portfolio while achieving diversification. We use Sharpe ratio, information ratio and alpha as the performance indicators, use standard deviation, beta as the risk indicators, and adopt diversification ratio (DR), which was proposed by Choueifaty and Coignard (2008), as the diversification indicator in our analysis. The results of this study show that MDP surpasses overall performance, risks and diversification compared to cap-weighted portfolio in the full empirical period. In addition, MDP is generally superior to cap-weighted portfolios in terms of performance in many single years of the whole period, and completely beat cap-weighted portfolios in terms of risks and diversification in every single year of the whole period. Although the ability of exceeding cap-weighted portfolio, MDP do not win first place of mentioned risk-based portfolios in our research. As a result, we suggest investors choose their portfolio strategies refer to its past performance, risks and diversification, and select the best according to their investment preference.