UNIVERSITY OF CALIFORNIA Santa Barbara Parameter Estimation for Stable Distributions: Spacings-based and Indirect Inference A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Statistics and Applied Probability by Gaoyuan Tian Committee in Charge: Professor S.Rao Jammalamadaka, Chair Professor John Hsu Professor Tomoyuki Ichiba March 2016 The Dissertation of Gaoyuan Tian is approved: Professor John Hsu Professor Tomoyuki Ichiba Professor S.Rao Jammalamadaka, Committee Chairperson March 2016 Parameter Estimation for Stable Distributions: Spacings-based and Indirect Inference Copyright c 2016 by Gaoyuan Tian iii Acknowledgements First and foremost, I would like to express my deepest gratitude to my advisor, Professor S. Rao Jammalamadaka. Only his continuous guidance, support, and care made the research presented in this dissertation possible. I am very grateful to my committee members Professor Tomoyuki Ichiba and Professor John Hsu for their friendly advice and kindly help while serving on my committee. I thank Dr. St´ephaneGuerrier, who as a visiting faculty member in our department, intro- duced me to indirect inference and made many helpful suggestions. During the final phase of my work on indirect inference for stable distributions, I discovered this overlapped with some of the work of Professor David Veredas. I am grateful to him for pointing this out and for his support and suggestions. I would never have been able to finish my dissertation without the guidance of many faculty members, help from friends, as well as support from my family. I wish to thank Professors David Hinkley, Andrew Carter, Jean-Pierre Fouque, and Michael Ludkovski for their wonderful teaching and instruction during my studies here at UCSB. Also my time at UCSB has been made most enjoyable in large part due to the pleasure I had studying and working with many great graduate students, who have become my lifetime friends. I also want to thank them for their help and useful discussions in my study. Finally, I would like to thank my wonderful parents Jun Tian and Haifeng Yuan for all their love and support during all these years. iv Curriculum Vitæ Gaoyuan Tian Education 2015 Doctor of Philosophy in Statistics and Applied Probability, De- partment of Statistics and Applied Probability, University of Cal- ifornia, Santa Barbara 2011 Master of Art in Applied Statistics, University of California, Santa Barbara 2009 Master of Art in Economics,University of California, Santa Bar- bara 2008 Bachelor of Economics in Finance, Renmin University of China, China. Experience 2010-2015 Teaching Assistant, Department of Statistics and Applied Prob- ability, University of California, Santa Barbara 2009-2010 Readership Assistant, Department of Statistics and Applied Prob- ability, University of California, Santa Barbara. v Abstract Parameter Estimation for Stable Distributions: Spacings-based and Indirect Inference Gaoyuan Tian Stable distributions are important family of parametric distributions widely used in signal processing as well as in mathematical finance. Estimation of the parameters of this model, is not quite straightforward due to the fact that there is no closed-form expression for their probability density function. Besides the com- putationally intensive maximum likelihood method where the density has to be evaluated numerically, there are some existing adhoc methods such as the quantile method, and a regression based method. These are introduced in Chapter 2. In this thesis, we introduce two new approaches: One, a spacing based estimation method introduced in Chapter 3 and two, an indirect inference method considered in Chapter 4. Simulation studies show that both these methods are very robust and efficient and do as well or better than the existing methods in most cases. Fi- nally in Chapter 5, we use indirect inference approach to estimate the best fitting income distribution based on limited information that is often available. vi Contents Acknowledgements iv Curriculum Vitæ v Abstract vi List of Figures x List of Tables xi 1 Introduction to Stable Distributions 1 1.1 Definitions ::::::::::::::::::::::::::::::: 2 1.1.1 A Different Parametrization of Stable Laws :::::::: 3 1.2 Basic Properties :::::::::::::::::::::::::::: 5 1.2.1 Densities and Distribution Functions :::::::::::: 5 1.2.2 Tail Probabilities and Moments ::::::::::::::: 7 1.3 Simulation Methods :::::::::::::::::::::::::: 9 2 Existing Estimation Methods 10 2.1 Maximum Likelihood Estimation :::::::::::::::::: 11 2.1.1 The Integral Representations of Zolotarev ::::::::: 12 2.1.2 Fast Fourier Transformation ::::::::::::::::: 13 2.2 Quantile Based Estimation :::::::::::::::::::::: 15 2.3 Characteristic Function based Estimation :::::::::::::: 16 3 Spacing Based Estimation for Stable Distributions 18 3.1 Introduction of Spacings Based Estimation ::::::::::::: 19 3.1.1 Definition ::::::::::::::::::::::::::: 19 3.1.2 Properties ::::::::::::::::::::::::::: 20 3.1.3 Examples ::::::::::::::::::::::::::: 20 3.2 GSE Applied in Stable Distributions :::::::::::::::: 22 vii 3.2.1 Estimating Tail and Skewness Parameters ::::::::: 22 3.2.2 Monto Carlo Studies ::::::::::::::::::::: 25 3.3 Comparison Between Different Methods ::::::::::::::: 26 3.4 Conclusion ::::::::::::::::::::::::::::::: 27 4 Indirect Inference Method Applied to Stable Distributions 28 4.1 Introduction ::::::::::::::::::::::::::::: 28 4.2 Existing Estimation Methods :::::::::::::::::::: 30 4.2.1 Introduction of the Stable Distribution ::::::::::: 30 4.2.2 Available Estimation Methods :::::::::::::::: 31 4.3 Indirect Inference in Stable Distributions :::::::::::::: 33 4.3.1 Indirect Inference Method :::::::::::::::::: 33 4.3.2 Quanitle-based Indirect Inference :::::::::::::: 35 4.3.3 Theoretical Properties :::::::::::::::::::: 37 4.4 Simulation Study ::::::::::::::::::::::::::: 41 4.4.1 Choice of the number of quantiles, m:::::::::::: 41 4.4.2 Weight Matrix :::::::::::::::::::::::: 45 4.4.3 Comparison Between different Methods ::::::::::: 47 4.5 Case Study :::::::::::::::::::::::::::::: 47 4.6 Conclusion ::::::::::::::::::::::::::::::: 52 4.7 Appendix: Influence Function and Robust Property of Quantiles : 53 5 Indirect Inference Applied to Income Distributions 55 5.1 Introduction to Some Inequality Measures ::::::::::::: 56 5.1.1 Lorenz curve ::::::::::::::::::::::::: 56 5.1.2 Gini Index and Other Inequality Measures ::::::::: 57 5.1.3 Some Popular Parametric Income Distributions :::::: 59 5.2 Indirect Inference Method :::::::::::::::::::::: 60 5.2.1 Indirect inference framework ::::::::::::::::: 60 5.2.2 Theoretical Properties :::::::::::::::::::: 60 5.2.3 Goodness of Fit Analysis ::::::::::::::::::: 61 5.2.4 Data :::::::::::::::::::::::::::::: 61 5.3 Simulation Study ::::::::::::::::::::::::::: 63 5.3.1 Numerical Optimization ::::::::::::::::::: 63 5.3.2 Monte Carlo Study :::::::::::::::::::::: 65 5.4 Case Study :::::::::::::::::::::::::::::: 66 5.4.1 Data :::::::::::::::::::::::::::::: 68 5.4.2 Result ::::::::::::::::::::::::::::: 69 5.4.3 Conclusions :::::::::::::::::::::::::: 72 6 Conclusions and Discussion 73 viii A Code 75 A.1 R code of Estimating income distribution by indirect inference :: 75 A.2 Matlab code for spacing based estimation of stable distribution :: 77 Bibliography 80 ix List of Figures 1.1 Pdf of stable distribution of different parameters :::::::::: 6 1.2 Stable noise with different tail parameters ::::::::::::: 8 3.1 Greenwood statistic vs.α::::::::::::::::::::::: 23 3.2 Greenwood statistic vs.(α; β) ::::::::::::::::::::: 24 4.1 Estimation Algorithm ::::::::::::::::::::::::: 37 4.2 MSE ofα ^ by different q ::::::::::::::::::::::: 43 4.3 MSE of β^ by different q ::::::::::::::::::::::: 43 4.4 MSE of scaleσ ^ by different q :::::::::::::::::::: 44 4.5 MSE of locationu ^ by different q ::::::::::::::::::: 44 4.6 Sum of MSE :::::::::::::::::::::::::::::: 45 4.7 Plot of index and return ::::::::::::::::::::::: 49 4.8 histogram ::::::::::::::::::::::::::::::: 50 4.9 QQ plot :::::::::::::::::::::::::::::::: 50 5.1 Lorenz Curve of lognormal and exponential ::::::::::::: 58 5.2 Objective function vs (θ1; θ2) ::::::::::::::::::::: 64 5.3 Objective function vs θ2 ::::::::::::::::::::::: 64 5.4 Boxplot ofµ:::::::::::::::::::::::::::::: ^ 65 5.5 Boxplot ofσ:::::::::::::::::::::::::::::: ^ 66 5.6 GDP growth rate ::::::::::::::::::::::::::: 67 5.7 Lorenz curve 2010 USA vs China :::::::::::::::::: 69 5.8 Lorenz Curve of USA 1981 vs 2010 ::::::::::::::::: 70 5.9 Lorenz Curve of China 1981 vs 2010 ::::::::::::::::: 70 5.10 Income distribution of China 1981 vs 2010 ::::::::::::: 71 5.11 Income distribution of USA 1981 vs 2010 :::::::::::::: 71 x List of Tables 3.1 Mean Square Error: MLE vs GSE :::::::::::::::::: 22 3.2 Mean square error and bias ofα ^ with various sample size ::::: 25 3.3 Mean square error and bias of (^α; β^) with various sample size ::: 26 3.4 Monte Carlo mean and mean square error of (^α; β^) with different methods ::::::::::::::::::::::::::::::::::: 27 4.1 MSE comparison: Identity matrix vs Two-step weight matrix ::: 46 4.2 MSE of different methods :::::::::::::::::::::: 48 4.3 Stable vs Skewed-t :::::::::::::::::::::::::: 52 5.1 Lorenz Curve for some distributions ::::::::::::::::: 57 5.2 Original Data ::::::::::::::::::::::::::::: 62 5.3 Transformed Data ::::::::::::::::::::::::::: 62 5.4 Goodness of Fit Assessment ::::::::::::::::::::: 63 5.5 Income inequality of USA: 1981 v.s 2010 :::::::::::::: 68 5.6 Income inequality of China: 1981 v.s 2010 ::::::::::::: 68 xi