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Multivariate Slice Sampling Multivariate Slice Sampling A Thesis Submitted to the Faculty of Drexel University by Jingjing Lu in partial ful¯llment of the requirements for the degree of Doctor of Philosophy June 2008 °c Copyright June 2008 Jingjing Lu . All Rights Reserved. Dedications To My beloved family Acknowledgements I would like to express my gratitude to my entire thesis committee: Professor Avijit Banerjee, Professor Thomas Hindelang, Professor Seung-Lae Kim, Pro- fessor Merrill Liechty, Professor Maria Rieders, who were always available and always willing to help. This dissertation could not have been ¯nished without them. Especially, I would like to give my special thanks to my advisor and friend, Professor Liechty, who o®ered me stimulating suggestions and encouragement during my graduate studies. I would like to thank Decision Sciences department and all the faculty and sta® at LeBow College of Business for their help and caring. I thank my family for their support and encouragement all the time. Thank you for always being there for me. Many thanks also go to all my friends, who share a lot of happiness and dreams with me. i Table of Contents List of Figures ................................................................ v List of Tables ................................................................. vii Abstract ....................................................................... viii 1. Introduction ............................................................... 1 1.1 Bayesian Computations ............................................. 4 1.2 Monte Carlo Sampling............................................... 7 1.3 Markov chain Monte Carlo Methods ............................... 9 1.4 Markov Chain Sampling Methods .................................. 11 1.5 The Metropolis-Hastings Algorithm ................................ 12 1.6 The Gibbs Sampler .................................................. 14 1.7 The General Auxiliary Variable Method ........................... 16 1.8 The Simple Slice Sampler ........................................... 18 1.9 Contribution of This Thesis......................................... 21 2. Literature Review ......................................................... 24 3. Introduction to Multivariate Slice Samplers ............................ 36 3.1 Univariate Truncated Normal Slice Sampler ....................... 36 3.2 Bivariate Normal Slice Sampler..................................... 42 4. Multivariate Normal Slice Sampler ...................................... 48 ii 4.1 Slice Sampler with One Auxiliary Variable ........................ 48 4.1.1 Sampling xi Values One at a Time ......................... 50 4.1.2 Sampling All xi Values at Once ............................ 51 4.2 Slice Sampler with n + 1 Auxiliary Variables...................... 57 4.3 Slice Sampler with n(n + 1)=2 Auxiliary Variables................ 63 4.4 Conditionally Sampling Each Dimension .......................... 71 5. Generalized Multivariate Slice Samplers ................................ 74 5.1 The General Methodology of Introducing Auxiliary Variables ... 74 5.2 Generalized Multivariate Slice Sampler for Multivariate-T Dis- tribution .............................................................. 79 5.3 Generalized Multivariate Slice Sampler for Elliptical Distributions 84 5.4 Extension to Multivariate Skew Distributions ..................... 89 5.4.1 Multivariate Skew Normal Distributions................... 89 5.4.2 A General Class of Multivariate Skew Normal and Mul- tivariate Skew t Distributions............................... 90 5.4.3 Multivariate Skew Elliptical Distributions ................. 96 6. Discussion of Implementation Issues and Computational Complexity 99 6.1 Sampling One Dimensional Random Variable ..................... 99 6.2 Sampling Multivariate Random Variables ......................... 101 iii 6.3 Discussions of the Multivariate Normal Slice Sampling ........... 104 7. Simulation Study ......................................................... 111 7.1 The E®ect of Burn-in Period........................................ 112 7.2 The E®ect of Starting Point ........................................ 114 7.3 Number of Rejections to Achieve A Certain Number of Samples 116 7.4 Number of Rejections and Samples to Achieve A Certain Level of Accuracy........................................................... 118 7.5 Picking Out Quadrants Method .................................... 119 8. Applications ............................................................... 121 8.1 Applications in Productions and Operations Management ....... 121 8.1.1 The Original Problem ....................................... 122 8.1.2 Solving the Problem by Multivariate Slice Sampler ....... 127 8.2 Applications in Finance ............................................. 133 8.2.1 Introduction to the Portfolio Optimization Problem...... 134 8.2.2 New Bayesian Higher Moments Portfolio Optimization with A General Class of Multivariate Skew t Distribution 136 8.3 Applications in the Other Areas .................................... 141 9. Conclusions and Future Research ........................................ 144 Bibliography .................................................................. 149 iv Appendix A: Figures ......................................................... 152 Appendix B: Portfolio Parameter Summary ................................ 164 Vita ............................................................................ 166 v List of Figures 1 simple slice sampler ..................................................... 152 2 Region to sample from when ½ < 0, and u3 < 1 ...................... 153 3 Region to sample from when ½ < 0, and u3 > 1 ...................... 154 4 Region to sample from when ½ > 0, and u3 < 1 ...................... 155 5 Region to sample from when ½ > 0, and u3 > 1 ...................... 156 6 Truncated bivariate normal distribution 1 ............................ 157 7 Truncated bivariate normal distribution 2 ............................ 158 8 Truncated bivariate normal distribution 3 ............................ 159 9 Truncated bivariate normal distribution 4 ............................ 160 10 Two dimensional truncated sampling results ......................... 161 11 three dimensional of x1; x2; x3 .......................................... 162 12 three dimensional of x1; x2; x4 .......................................... 162 13 three dimensional of x2; x3; x4 .......................................... 162 14 three dimensional of x1; x3; x4 .......................................... 162 15 three dimensional quadrant 1 ........................................... 163 16 three dimensional quadrant 2 ........................................... 163 17 three dimensional quadrant 3 ........................................... 163 18 three dimensional quadrant 4 ........................................... 163 19 three dimensional quadrant 5 ........................................... 163 vi 20 three dimensional quadrant 6 ........................................... 163 vii List of Tables 1 Computational complexity of slice samplers ........................... 109 2 Computational complexity of slice samplers (continued) ............. 110 3 The e®ect of burn-in periods. ........................................... 114 4 The e®ect of starting point.............................................. 115 5 Number of rejections to achieve a certain number of samples ........ 117 6 Number of samples & rejections to achieve a certain level of accuracy118 7 The e®ect of picking out quadrants method. .......................... 120 8 Mean CPU time for generating 1000 samples in portfolio optimization139 9 Optimization results for portfolio selection ............................ 140 viii Abstract Multivariate Slice Sampling Jingjing Lu Advisor: Merrill Liechty In Bayesian decision theory, stochastic simulation techniques have been widely used by decision makers. The Markov chain Monte Carlo method (MCMC) plays a key role in stochastic simulation within the framework of Bayesian statistics. The notion of \slice sampling", or employing auxiliary variables in sampling, has been recently suggested in the literature for improv- ing the e±ciency of the traditional MCMC methods. In the existing literature, the one dimensional slice sampler has been extensively studied, yet the litera- ture on the multidimensional case is sparse. In this study, we utilize multiple auxiliary variables in our sampling algorithms for multivariate normal distri- butions, which adapt better to the local properties of the target probability distributions. We show that these methods are flexible enough to allow for truncation to rectangular regions and/or the exclusion of any n dimensional hyper-quadrant. We compare these algorithms for both e±ciency and accu- racy via simulation experiments. Our results show that our new sampling techniques are accurate and more e®ective than the traditional single auxil- ix iary variable slice sampler, especially for truncated multivariate distributions. Further, we extend our methods to general multivariate distributions includ- ing multivariate student t distributions, multivariate elliptical distributions, multivariate skew normal distributions, multivariate skew t distributions and a general class of multivariate skew elliptical distributions. We also discuss and outline some applications of our algorithms in the real business world, especially in the production and operations management and the ¯nance areas. With regard
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