Samuel Kotz

TomaszJ. Kozubowski

Krzysztof Podgorski

The Laplace Distribution and Generalizations

A Revisit with Applications to Communications,

Economics, Engineering, and Finance

Birkhäuser

Boston • Basel • Berlin

Contents

Preface xi

Abbreviations xiii

Notation xv

I Univariate Distributions 1

1 Historical Background 3

2 Classical Symmetrie Laplace Distribution 15 2.1 Definition and basic properties 16

2.1.1 Density and distribution functions 16 2.1.2 Characteristic and moment generating functions 19 2.1.3 Moments and related parameters 19

2.1.3.1 Cumulants 19 2.1.3.2 Moments 20 2.1.3.3 Mean deviation .1 20 2.1.3.4 Coefficients of skewness and kurtosis 21 2.1.3.5 Entropy 21 2.1.3.6 Quartiles and quantiles 21

2.2 Representations and characterizations 22 2.2.1 Mixture of normal distributions 22 2.2.2 Relation to exponential distribution 23 2.2.3 Relation to the Pareto distribution 24 2.2.4 Relation to 2 x 2 unit normal determinants 25 2.2.5 An orthogonal representation 25 2.2.6 Stability with respect to geometric summation 27

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2.2.7 Distributional limits of geometric sums 30 2.2.8 Stability with respect to the ordinary summation 32 2.2.9 Distributional limits of deterministic sums 35

2.3 Functions of Laplace random variables 35 2.3.1 The distribution of the sum of independent Laplace variates 35 2.3.2 The distribution of the product of two independent Laplace variates . . . . 40 2.3.3 The distribution of the ratio of two independent Laplace variates 41 2.3.4 The f-statistic for a double exponential (Laplace) distribution 43

2.4 Further properties 46 2.4.1 Infinite divisibility 46 2.4.2 Geometric infinite divisibility 48 2.4.3 Self-decomposability 49 2.4.4 Complete monotonicity 49 2.4.5 Maximum entropy property 51

2.5 Order statistics 53 2.5.1 Distribution of a Single order statistic 53

2.5.1.1 Theminimum 54 2.5.1.2 Themaximum 55 2.5.1.3 The median 55

2.5.2 Joint distributions of order statistics 55 2.5.2.1 Range, midrange, sample median 56

2.5.3 Moments of order statistics 60 2.5.4 Representation of order statistics via sums of exponentials 63

2.6 Statistical inference 64 2.6.1 Point estimation 65

2.6.1.1 Maximum likelihood estimation 66 2.6.1.2 Maximum likelihood estimation under censoring 77 2.6.1.3 Maximum likelihood estimation of monotone location parameters 78 2.6.1.4 Themethodofmoments 79 2.6.1.5 Linear estimation 84

2.6.2 Interval estimation 91 2.6.2.1 Confidence bands for the Laplace distribution function 93 2.6.2.2 Conditional inference 94

2.6.3 Tolerance intervals 99 2.6.4 Testing hypothesis 103

2.6.4.1 Testing the normal versus the Laplace 103 2.6.4.2 Goodness-of-fit tests 105 2.6.4.3 Neyman-Pearson test for location 106 2.6.4.4 Asymptotic optimality of the Kolmogorov-Smirnov test 110 2.6.4.5 Comparison of nonparametric tests of location 110

2.7 Exercises 112

3 Asymmetrie Laplace Distributions 133 3.1 Definition and basic properties 136

3.1.1 An alternative parametrization and special cases 136 3.1.2 Standardization 137 3.1.3 Densities and their properties 137 3.1.4 Moment and cumulant generating funetions 140 3.1.5 Moments and related parameters 141

Contents vii

3.1.5.1 Cumulants 141 3.1.5.2 Moments 142 3.1.5.3 Absolute moments 142 3.1.5.4 Mean deviation 142 3.1.5.5 Coefficient of Variation 143 3.1.5.6 Coefficients of skewness and kurtosis 143 3.1.5.7 Quantiles 143

3.2 Representations 144 3.2.1 Mixture of normal distributions 144 3.2.2 Convolution of exponential distributions 146 3.2.3 Self-decomposability 147 3.2.4 Relation to 2 x 2 normal determinants 148

3.3 Simulation 149 3.4 Characterizations and further properties 150

3.4.1 Infinite divisibility 150 3.4.2 Geometric infinite divisibility 151 3.4.3 Distributional limits of geometric sums 152 3.4.4 Stability with respect to geometric summation 155 3.4.5 Maximum entropy property 155

3.5 Estimation 158 3.5.1 Maximum likelihood estimation 158

3.5.1.1 Case 1: The values of/c and CT are known 159 3.5.1.2 Case 2: The values of 9 and K are known 161 3.5.1.3 Case 3: The values ofö and CT are known 162 3.5.1.4 Case 4: The value of K is known 166 3.5.1.5 Case 5: The value ofö is known 167 3.5.1.6 Case 6: The value of CT is known 170 3.5.1.7 Case 7: The values of all three parameters are unknown 172

3.6 Exercises 174

4 Related Distributions 179 4.1 Bessel function distribution 179

4.1.1 Definition and parametrizations 180 4.1.2 Representations 181

4.1.2.1 Mixture of normal distributions 181 4.1.2.2 Relation to gamma distribution 183

4.1.3 Self-decomposability 184 4.1.3.1 Relation to sample covariance 186

4.1.4 Densities 188 4.1.4.1 Asymmetrie Laplace laws 189 4.1.4.2 Symmetrie case 189 4.1.4.3 An integer value of T 191

4.1.5 Moments 192 4.2 Laplace motion 193

4.2.1 Symmetrie Laplace motion 193 4.2.2 Representations 194 4.2.3 Asymmetrie Laplace motion 197

4.2.3.1 Subordinated Brownian motion 198 4.2.3.2 Difference of gamma processes 198

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4.2.3.3 Compound Poisson approximation 198 4.3 Linnik distribution 199

4.3.1 Characterizations 200 4.3.1.1 Stability with respect to geometric summation 200 4.3.1.2 Distributional limits of geometric sums 202 4.3.1.3 Stability with respect to deterministic summation 203

4.3.2 Representations 204 4.3.3 Densities and distribution functions 206

4.3.3.1 Integral representations 206 4.3.3.2 Series expansions 208

4.3.4 Moments and tail behavior 212 4.3.5 Properties 213

4.3.5.1 Self-decomposability 213 4.3.5.2 Infinite divisibility 214

4.3.6 Simulation 215 4.3.7 Estimation 215

4.3.7.1 Method of moments type estimators 216 4.3.7.2 Least-squares estimators 216 4.3.7.3 Minimal distance method 217 4.3.7.4 Fractional moment estimation 218

4.3.8 Extensions 218 4.4 Othercases 219

4.4.1 Log-Laplace distribution 219 4.4.2 Generalized Laplace distribution 219 4.4.3 Sargan distribution 220 4.4.4 Geometric stable laws 220 4.4.5 v-stablelaws 222

4.5 Exercises 222

II Multivariate Distributions 227

Introduction 229

5 Symmetrie Multivariate Laplace Distribution 231 5.1 Bivariate case 231

5.1.1 Definition 231 5.1.2 Moments 232 5.1.3 Densities 232 5.1.4 Simulation of bivariate Laplace variates 233

5.2 General Symmetrie multivariate case 234 5.2.1 Definition 234 5.2.2 Moments and densities 235

5.3 Exercises 236

6 Asymmetrie Multivariate Laplace Distribution 239 6.1 Bivariate case: Definition and basic properties 240

6.1.1 Definition 240 6.1.2 Moments 241

Contents ix

6.1.3 Densities 241 6.1.4 Simulation of bivariate asymmetnc Laplace variates 242

6.2 General multivariate asymmetric case 243 6.2.1 Definition 243 6.2.2 Special cases 244

6.3 Representations 246 6.3.1 Basic representation 246 6.3.2 Polar representation 247 6.3.3 Subordinated Brownian motion 248

6.4 Simulation algorithm 248 6.5 Moments and densities 249

6.5.1 Mean vector and covariance matrix 249 6.5.2 Densities in the general case 249 6.5.3 Densities in the Symmetrie case 250 6.5.4 Densities in the one-dimensional case 250 6.5.5 Densities in the case of odd dimension 251

6.6 Unimodality 251 6.6.1 Unimodality 251 6.6.2 A related representation 252

6.7 Conditional distributions 253 6.7.1 Conditional distributions 253 6.7.2 Conditional mean and covariance matrix 254

6.8 Linear transformations 254 6.8.1 Linear combinations 254 6.8.2 Linear regression 255

6.9 Infinite divisibility properties 256 6.9.1 Infinite divisibility 256 6.9.2 Asymmetric Laplace motion 257 6.9.3 Geometrie infinite divisibility 258

6.10 Stability properties 258 6.10.1 Limits of random sums 258 6.10.2 Stability under random summation 259 6.10.3 Stability of deterministic sums 260

6.11 Linear regression with Laplace errors 261 6.11.1 Least-squares estimation 261 6.11.2 Estimation of er2 262 6.11.3 The distributions of Standard t and F statistics 263 6.11.4 Inference from the estimated regression function 264

6.11.4.1 Estimating the regression function at xo 264 6.11.4.2 Forecasting a new Observation at xo 264

6.11.5 Maximum likelihood estimation 265 6.11.6 Bayesian estimation 267

6.12 Exercises 268

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III Applications 273

Introduction 275

7 Engineering Sciences 277 7.1 Detection in the presence of Laplace noise 277 7.2 Encoding and decoding of analog signals 280 7.3 Optimal quantizer in image and speech compression 281 7.4 Fracture problems ' 284 7.5 Wind shear data 285 7.6 Error distributions in navigation 286

8 Financial Data 289 8.1 Underreported data 289 8.2 Interest rate data 290 8.3 Currency exchange rates 292 8.4 Share market return modeis 294

8.4.1 Introduction 294 8.4.2 Stock market returns 294

8.5 Option pricing 296 8.6 Stochastic variance Value-at-Risk modeis 297 8.7 A jump diffusion model for asset pricing with Laplace distributed jump-sizes . . . 300 8.8 Price changes modeled by Laplace-Weibull mixtures 302

9 Inventory Management and Quality Control 303 9.1 Demand during lead time 303 9.2 Acceptance sampling for Laplace distributed quality characteristics 304 9.3 Steam generator inspection 306 9.4 Adjustment of Statistical process control 306 9.5 Duplicate check-sampling of the metallic content 308

10 Astronomy and the Biological and Environmental Sciences 309 10.1 Sizes of sand particles, diamonds, and beans 309 10.2 Pulses in long bright gamma-ray bursts 310 10.3 Random fluctuations of response rate . 311 10.4 Modeling low dose responses 312 10.5 Multivariate elliptically contoured distributions for repeated measurements . . . . 312 10.6 ARMA modeis with Laplace noise in the environmental time series 313

Appendix: Bessel Functions 315

References 319

Index 343