Parameter Estimation and Prediction Interval Construction for Location{Scale Models with Nuclear Applications By Xingli Wei A Dissertation Submitted to the School of Graduate Studies in Partial Fulfilment of the Requirements for the Degree of Doctor of Philosophy McMaster University @Copyright by Xingli Wei, July, 2014 Ph.D Math(Stats) McMaster University Hamilton, Ontario TITLE: Parameter Estimation and Prediction Interval Construction for Location{Scale Models with Nuclear Applications AUTHOR: Xingli Wei SUPERVISOR: Dr. Fred M. Hoppe ii Contents List of Figures vi Acknowledgement viii Abstract ix 1 Introduction 1 1.1 Motivation . .1 1.2 Literature Review and Preliminaries . .4 1.3 Thesis Outline and Contributions . 11 2 A Simple Method for Estimating Parameters of Location-Scale Fam- ilies 13 2.1 Introduction . 13 2.2 Estimation Based on the Sample Mean and Sample Standard Deviation 16 2.2.1 Equivariance Principle . 16 2.2.2 Unbiasedness Principle . 18 2.2.3 Minimum Risk Principle . 23 2.3 Estimation Based on the Sample Mean and Gini's Mean Difference . 31 2.4 An Adaptive Unbiased Estimator for Location-Scale Families . 33 3 A Mixed Method for Estimating Location, Scale and Shape Param- eters of a 3-Parameter Distribution 41 3.1 Mixed Method . 41 3.2 Example 1: Generalized Extreme Value (GEV) Distribution . 43 3.2.1 Introduction . 43 3.2.2 Methods of Estimation for the Generalized Extreme Value Dis- tribution . 46 3.2.3 Simulation Results and Conclusions . 52 3.3 Example 2: Power Gumbel Distribution . 56 3.3.1 Maximum Likelihood Method . 56 3.3.2 Mixed Method . 58 3.3.3 Simulation Results and Conclusions . 61 4 Bayesian Estimation for the Gumbel Minimum Distribution 69 4.1 Bayesian Method . 69 4.2 Full Bayesian Estimation for Gumbel Minimum Distribution . 72 4.2.1 An Approximation Theorem for an Exponential Sum of Gumbel Random Variables . 72 4.2.2 Priors for the Gumbel Minimum Distribution . 77 iii 4.2.3 Bayesian Estimators Based on Jeffrey’s Non-informative Prior 80 4.2.4 Bayesian Estimators Based on an Asymptotically Conjugate Prior 83 4.3 Bayesian-Maximum Likelihood Estimation for the Gumbel Minimum Distribution . 86 4.3.1 Estimation for the Gumbel Minimum Distribution Using a Non- informative Prior . 86 4.3.2 Estimation for the Gumbel Maximum Distribution Using a Noninformative Prior . 89 4.4 Another Example: Estimation for the Normal Distribution . 93 5 Frequentist Prediction Interval for a Location{Scale Family 97 5.1 Introduction . 97 5.2 Construction of a Conditional Prediction Interval . 98 5.3 Construction of an Unconditional Prediction Interval . 102 5.4 Construction of Prediction Intervals Using a Monte Carlo Method . 104 6 Bayesian Prediction Intervals for the Gumbel Minimum Distribu- tion 107 6.1 Construction of a Bayesian Prediction Interval Using a Noninformative Prior on a when b is Known . 108 6.1.1 Prediction Interval for Ymax ................... 108 6.1.2 Prediction Intervals for Y(j) ................... 111 6.2 Construction of a Bayesian Prediction Interval Using a Conjugate Prior for a with Known b ........................... 114 6.2.1 Prediction Interval for Ymax ................... 116 6.2.2 Prediction Interval for Y(j) .................... 119 6.3 Constructing Bayesian Prediction Intervals Using Gibbs Sampling . 121 6.3.1 Case 1: a Unknown and b Known . 124 6.3.2 Case 2: Both a and b Unknown . 128 7 Comparative Study of Parameter Estimation and Prediction Inter- vals for the Gumbel Distribution 135 7.1 Introduction . 135 7.2 Methods of Parameter Estimation for the Gumbel Distribution . 137 7.3 Simulation Results and Conclusions . 148 7.4 Methods of Prediction Interval Construction for the Gumbel Minimum Distribution . 153 7.4.1 The Approximation for the Distribution . 153 7.4.2 Conditional and Unconditional Probability Method . 156 7.4.3 Monte Carlo Estimation of the Distribution Percentiles and Method 1 . 158 7.4.4 Markov Chain Monte Carlo Estimation of the Distribution Per- centiles and Method 2 . 158 iv 7.5 Numerical Examples . 159 8 Resampling and Prediction Interval Construction 173 8.1 Introduction . 173 8.2 Bootstrap Resampling . 174 8.3 A New Resampling Method . 176 8.4 Another Look at the Bootstrap Method . 179 8.5 Smoothed Resampling . 180 8.6 A Simple Version of the Generalized Bootstrap . 183 8.7 Numerical Examples . 188 8.8 Wilks' Formula for Symmetric Distributions in the GRS Method . 205 8.8.1 Symmetrization . 207 8.8.2 Comparison with the Generalized Bootstrap . 212 A Useful Formulas 219 B Bibliography 221 v List of Figures 1.1 Pickering A Feeder Pipes . .2 3.1 Location estimator of GEV with a = 12; b = 5; κ = 0:25; 10000 trials . 53 3.2 Scale estimator of GEV with a = 12; b = 5; κ = 0:25; 10000 trials . 53 3.3 Shape estimator of GEV with a = 12; b = 5; κ = 0:25; 10000 trials . 54 3.4 Location estimator of GEV with a = 12; b = 5; κ = −0:25; 10000 trials 54 3.5 Scale estimator of GEV with a = 12; b = 5; κ = −0:25; 10000 trials . 55 3.6 Shape estimator of GEV with a = 12; b = 5; κ = −0:25; 10000 trials . 55 3.7 Distribution function and density of power Gumbel . 60 3.8 Location estimator of power Gumbel with a = 5; b = 12; r = 5 . 67 3.9 Scale estimator of power Gumbel with a = 5; b = 12; r = 5 . 67 3.10 Shape estimator of power Gumbel with a = 5; b = 12; r = 5 . 67 3.11 Location estimator of power Gumbel with a = 1; b = 5; r = 5 . 68 3.12 Scale estimator of power Gumbel with a = 12; b = 5; r = 5 . 68 3.13 Shape estimator of power Gumbel with a = 12; b = 5; r = 5 . 68 7.1 Bias of location estimator of Gumbel . 151 7.2 MSE of location estimator of Gumbel . 151 7.3 Bias of scale estimator of Gumbel . 152 7.4 MSE of scale estimator of Gumbel . 152 7.5 Difference between two quantiles for n = 23; m = 8; j = 1 with 2000000 simulations . 160 7.6 Difference between two quantiles for n = 23; m = 8; j = 3 with 2000000 simulations . 161 7.7 Difference between two quantiles for n = 23; m = 8; j = 8 with 2000000 simulations . 161 7.8 Predictive distribution of Gumbel . 167 8.1 Scatter plot of standard normal distribution . 210 8.2 Scatter plot of standard Cauchy distribution . 210 8.3 Scatter plot of Laplace distribution . 210 8.4 Scatter plot of Student distribution with df=5 . 211 8.5 Scatter plot of Student distribution with df=10 . 211 8.6 Scatter plot of logistic distribution . 211 8.7 Scatter plot of uniform distribution . 212 8.8 Estimated Probability of P (W > χγ) with P (X1 ≤ χγ) = 0:95 and X1 ∼ Uniform(−1; 1) .......................... 214 8.9 Estimated Probability of P (W > χγ) with P (X1 ≤ χγ) = 0:95 and X1 ∼ logistic distribution with a = 0 b =5 .............. 214 vi 8.10 Estimated Probability of P (W > χγ) with P (X1 ≤ χγ) = 0:95 and X1 ∼ Norm(0; 1)............................. 215 8.11 Estimated Probability of P (W > χγ) with P (X1 ≤ χγ) = 0:95 and X1 ∼ Laplace(0; 1) ............................ 216 8.12 Estimated Probability of P (W > χγ) with P (X1 ≤ χγ) = 0:95 and X1 ∼ t − distribution with df =1.................... 216 8.13 Estimated Probability of P (W > χγ) with P (X1 ≤ χγ) = 0:95 and X1 ∼ t − distribution with df =5.................... 217 8.14 Estimated Probability of P (W > χγ) with P (X1 ≤ χγ) = 0:95 and X1 ∼ t − distribution with df =10................... 217 vii Acknowledgement This thesis would not be finished in time and shape without the help and support of my supervisor Dr. Fred M. Hoppe. My greatest thanks go to him for his supporting, guiding, and working very closely with me for the last few years. I feel tremendously lucky to have had the opportunity to work with him. I am very grateful for his patience, encouragement, criticism, persistence, and friendly relationship with me. A very special thanks goes to the other members of my committee: Dr. Alexan- der Rosa and Dr Paul Sermer. They were always available for my questions, and their suggestions and comments helped me a lot to stay on my path and to do the right thing. I would like to thank them for their comments, suggestions and review that helped to produce this document. Finally, I would like to thank my family, Hua Zheng, and Cynthia Wei, for their love and support. This dissertation is dedicated to them. viii Abstract This thesis presents simple efficient algorithms to estimate distribution parameters and to construct prediction intervals for location{scale families. Specifically, we study two scenarios: one is a frequentist method for a general location{scale family and then extend to a 3{parameter distribution, another is a Bayesian method for the Gumbel distribution. At the end of the thesis, a generalized bootstrap resampling scheme is proposed to construct prediction intervals for data with an unknown distribution. Our estimator construction begins with the equivariance principle, and then makes use of unbiasedness principle. These two estimates have closed form and are functions of the sample mean, sample standard deviation, sample size, as well as the mean and variance of a corresponding standard distribution. Next, we extend the previous result to estimate a 3-parameter distribution which we call a mixed method. A central idea of the mixed method is to estimate the location and scale parameters as functions of the shape parameter. The sample mean is a popular estimator for the population mean.