UNIVERSITY OF CALIFORNIA RIVERSIDE Estimation of the Parameters of Skew Normal Distribution by Approximating the Ratio of the Normal Density and Distribution Functions A Dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Applied Statistics by Debarshi Dey August, 2010 Dissertation Committee: Dr. Subir Ghosh, Chairperson Dr. Barry Arnold Dr. Aman Ullah Copyright by Debarshi Dey 2010 The Dissertation of Debarshi Dey is approved: _____________________________________________________ _____________________________________________________ _____________________________________________________ Committee Chairperson University of California, Riverside ACKNOWLEDGEMENTS I would take this opportunity to express my sincere gratitude and thanks to my advisor, Professor Subir Ghosh for his continuous and untiring guidance over the course of the last five years. His sincere interest in not only my academic progress, but also my personal well-being, has been a source of sustained motivation for me. I would also like to extend my sincere thanks to Professor Barry Arnold, of the Department of Statistics, UC Riverside, and Professor Aman Ullah, of the Department of Economics, UC Riverside, for graciously accepting to serve on my PhD Committee and for their valuable time and advice. I wish to thank the entire faculty of the Department of Statistics for enriching me with their vast knowledge in various fields of Statistics. I would like to thank the entire staff of the Department of Statistics who were always ready with their help. My friends here at UCR, have also been a source of major strength and support for the last five years. Your friendship has made my experience at Riverside all the more memorable and fulfilling. iv A very special thanks to my wife, Trupti, for being with me and for being my pillar of strength. Though she joined me only a few months ago, her boundless affection, and immense support are invaluable to me in this accomplishment. I would like to thank my parents, because it is for them and their hard work and sacrifice that made me achieve whatever little I have achieved. Without their constant support, and their immense faith in me, I might not have pursued and persevered. Together with them, my younger brother, Tukan, and my Grandmother, Danni, are equally committed to my success, and have taken immense pride in my accomplishments. Finally I would like to express my most humble gratitude to my Divine Master, Bhagawan Sri Sathya Sai Baba, without whose Grace and Benevolence I could not have achieved anything. v ABSTRACT OF THE DISSERTATION Estimation of the Parameters of Skew Normal Distribution Using Linear Approximations of the Ratio of the Normal Density and Distribution Functions by Debarshi Dey Doctor of Philosophy, Graduate Program in Applied Statistics University of California, Riverside, August 2010 Dr Subir Ghosh, Chairperson The normal distribution is symmetric and enjoys many important properties. That is why it is widely used in practice. Asymmetry in data is a situation where the normality assumption is not valid. Azzalini (1985) introduces the skew normal distribution reflecting varying degrees of skewness. The skew normal distribution is mathematically tractable and includes the normal distribution as a special case. It has three parameters: location, scale and shape. In this thesis we attempt to respond to the complexity and challenges in the maximum likelihood estimates of the three parameters of the skew normal distribution. The complexity is traced to the ratio of the normal density and distribution function in the likelihood equations in the presence of the skewness parameter. Solution to this problem is obtained by approximating this ratio by linear and non-linear functions. We observe that the linear approximation performs quite vi satisfactorily. In this thesis, we present a method of estimation of the parameters of the skew normal distribution based on this linear approximation. We define a performance measure to evaluate our approximation and estimation method based on it. We present the simulation studies to illustrate the methods and evaluate their performances. vii Contents 1. Introduction 1 1.1 Motivation and Historical Development. 2 1.2 Normal Distribution and Simple Linear Regression. .5 1.3 Skew Normal Distribution and Regression. .8 1.4 Thesis Description. .10 viii 2. The Univariate Skew Normal Distribution 11 2.1 The Univariate Skew Normal Distribution. 12 2.2 Moments of the Univariate Skew Normal Distribution. .15 2.3 Likelihood Function and Maximum Likelihood Estimates. .20 2.4 Challenges of the Maximum Likelihood Estimates of the Skew Normal Dsitribution. .30 2.5 Literature review on challenges. 33 3. Approximations of the ratio of the Standard Normal Density and Distribution Functions 35 3.1 Introduction. 36 3.2 Motivation. 38 3.3 Fitting the Linear Approximation to the Ratio. 41 3.4 Fitting the Non-linear Approximation to the Ratio. 46 4. Estimation of the Shape Parameter of the Standard Skew Normal Distribution 51 4.1 Introduction. .52 4.2 The Estimation Procedure using A (z) . 53 ix 4.2.1 Case I : Covariate X Present. 54 4.2.2 Case II : Covariate X Absent. 56 4.2.3 Measure of Goodness of Fit. 57 4.3 The Estimation Procedure using B (z) . 58 4.3.1 Case I : Covariate X Present. 59 4.3.2 Case II : Covariate X Absent. 61 4.3.3 Measure of Goodness of Fit. 63 4.4 A Simulated Data. 64 4.4.1 Case I : Covariate X Present. 65 4.4.2 Case II : Covariate X Absent. 70 4.5 Estimating Bias and Accuracy in Approximations using Simulations.. 75 5. Estimation of Location, Scale and Shape Parameter of a Skew Normal Distribution 80 5.1 Introduction. 81 5.2 Relations Among Estimated Parameters. 82 5.3 Estimation Procedure using . .91 5.4 A simulated Data. 99 5.5 Estimating Bias and Accuracy in Approximation using Simulations. .104 6. Conclusion 124 A (z) Bibliography 127 x List of Figures 2.1 The pdf of Z ~ SN(0,1,) , 1,2 and10 .. 14 2.2 Probability that Z < 0 for values of ranging from 0 to 30. .31 3.1 Plots of R (z) against z for 0.5, 1 and 2 .. .37 3.2 Plots of A (z) against z for , and .. 42 3.3 Plot of against z for 0.5 (continuous lines) and A (z) against z for (dotted lines). 43 3.4 Plot of against z for 1 (continuous lines) and against z for 1 (dotted lines). 44 xi 3.5 Plot of against z for 2 (continuous lines) and against z for 2 (dotted lines). 45 3.6 Plots of B (z) against z for , and .. 47 3.7 Plot of against z for (continuous lines) and B (z) against z for (dotted lines). 48 3.8 Plot of against z for (continuous lines) and against z for (dotted lines). .49 3.9 Plot of against z for (continuous lines) and against z for (dotted lines). .50 4.1 Plots of both R (z) and Aˆ (z) against z when covariate X is present. 67 1 1 4.2 Plots of both and Bˆ (z) against z when covariate X is present. 69 1 4.3 Plots of both and against z when covariate X is absent. 72 4.4 Plots of both and against z when covariate X is absent. .74 ˆ 5.1 Plot of the estimated log-likelihood l against the values of b [0,1) . 102 R (z) 0.5 1 2 5.2 Plot of the estimated log-likelihood against the values of b[0.35,0.50] . .103 0.5 A (z) 1 1 xii List of Tables ˆ ˆ2 ˆ 4.1. The values of Q1 ,Median, Mean, Q3, and SD for , ˆ , ˆ , and Ave A when 0 30, 1 5, 0.5, and 0.5 in approximating (3.1) by (3.2). .77 4.2. The values of Q1 ,Median, Mean, Q3, and SD for , , , and when , and 1.5 in approximating (3.1) by (3.2). .77 4.3. The values of Q1 ,Median, Mean, Q3, and SD for , , , and when 1.0, and in approximating (3.1) by (3.2). .77 4.4. The values of Q1 ,Median, Mean, Q3, and SD for , , , and when 1.0, and 1.5 in approximating (3.1) by (3.2). .78 xiii 2 4.5. The values of Q1 ,Median, Mean, Q3, and SD for , ˆ ˆ , ˆˆ , and Ave B when , and in approximating (3.1) by (3.3). .78 4.6. The values of Q1 ,Median, Mean, Q3, and SD for , , , and when , and 1.5 in approximating (3.1) by (3.3). 78 4.7. The values of Q1 ,Median, Mean, Q3, and SD for , , , and when 1.0, and in approximating (3.1) by (3.3). .79 ˆ 0 30, 1 5, 0.5 0.5 4.8. The values of Q1 ,Median, Mean, Q3, and SD for , , , and when 1.0, and 1.5 in approximating (3.1) by (3.3). 79 5.1 The true values of , the sample size n, and the proportion of times the sign of is correctly determined. .105 5.2 The values of Q1 ,Median, Mean, Q3, and SD for b ,ˆ0 , ˆ1 , ˆ , ,ˆ when n=20, 0 30, 1 5, 0.05, and 0.5in approximating (3.1) by (3.2). .108 5.3 The values of Q1 ,Median, Mean, Q3, and SD for , , , , , when n=50, , and in approximating (3.1) by (3.2). 108 xiv 5.4 The values of Q1 ,Median, Mean, Q3, and SD for , , , , , when n=100, , and in approximating (3.1) by (3.2). 109 5.5 The values of Q1 ,Median, Mean, Q3, and SD for , , , , , when n=500, , and in approximating (3.1) by (3.2).