## The Laplace Distribution and Generalizations: A Revisit With Applications to Communications, Exonomics, Engineering, and FinanceThe aim of this monograph is quite modest: It attempts to be a systematic exposition of all that appeared in the literature and was known to us by the end of the 20th century about the Laplace distribution and its numerous generalizations and extensions. We have tried to cover both theoretical developments and applications. There were two main reasons for writing this book. The first was our conviction that the areas and situations where the Laplace distribution naturally occurs is so extensive that tracking the original sources is unfeasible. The second was our observation of the growing demand for statistical distributions having properties tangent to those exhibited by the Laplace laws. These two "necessary" conditions justified our efforts that led to this book. Many details are arranged primarily for reference, such as inclusion of the most commonly used terminology and notation. In several cases, we have proposed unification to overcome the ambiguity of notions so often present in this area. Personal taste may have done some injustice to the subject matter by omitting or emphasizing certain topics due to space limitations. We trust that this feature does not constitute a serious drawback-in our literature search we tried to leave no stone unturned (we collected over 400 references). |

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### Contents

Historical Background | 3 |

Classical Symmetric Laplace Distribution | 15 |

21 Definition and basic properties | 16 |

22 Representations and characterizations | 22 |

23 Functions of Laplace random variables | 35 |

24 Further properties | 46 |

25 Order statistics | 53 |

26 Statistical inference | 64 |

68 Linear transformations | 254 |

69 Infinite divisibility properties | 256 |

610 Stability properties | 258 |

611 Linear regression with Laplace errors | 261 |

612 Exercises | 268 |

Applications | 273 |

Introduction | 275 |

Engineering Sciences | 277 |

27 Exercises | 112 |

Asymmetric Laplace Distributions | 133 |

31 Definition and basic properties | 136 |

32 Representations | 144 |

33 Simulation | 149 |

34 Characterizations and further properties | 150 |

35 Estimation | 158 |

36 Exercises | 174 |

Related Distributions | 179 |

42 Laplace motion | 193 |

43 Linnik distribution | 199 |

44 Other cases | 219 |

45 Exercises | 222 |

Multivariate Distributions | 227 |

Introduction | 229 |

Symmetric Multivariate Laplace Distribution | 231 |

52 General symmetric multivariate case | 234 |

53 Exercises | 236 |

Asymmetric Multivariate Laplace Distribution | 239 |

Definition and basic properties | 240 |

62 General multivariate asymmetric case | 243 |

63 Representations | 246 |

64 Simulation algorithm | 248 |

65 Moments and densities | 249 |

66 Unimodality | 251 |

67 Conditional distributions | 253 |

72 Encoding and decoding of analog signals | 280 |

73 Optimal quantizer in image and speech compression | 281 |

74 Fracture problems | 284 |

75 Wind shear data | 285 |

76 Error distributions in navigation | 286 |

Financial Data | 289 |

82 Interest rate data | 290 |

83 Currency exchange rates | 292 |

84 Share market return models | 294 |

85 Option pricing | 296 |

86 Stochastic variance ValueatRisk models | 297 |

87 A jump diffusion model for asset pricing with Laplace distributed jumpsizes | 300 |

88 Price changes modeled by Laplace Weibull mixtures | 302 |

Inventory Management and Quality Control | 303 |

92 Acceptance sampling for Laplace distributed quality characteristics | 304 |

93 Steam generator inspection | 306 |

95 Duplicate checksampling of the metallic content | 308 |

Astronomy and the Biological and Environmental Sciences | 309 |

102 Pulses in long bright gammaray bursts | 310 |

103 Random fluctuations of response rate | 311 |

104 Modeling low dose responses | 312 |

106 ARMA models with Laplace noise in the environmental time series | 313 |

Bessel Functions | 315 |

319 | |

343 | |

### Common terms and phrases

ACd(m asymptotically normal Bessel function Bessel function distribution bivariate Brownian motion ch.f characteristic function characterization classical Laplace distribution classical Laplace r.v. coefficient conditional consider converges in distribution corresponding Cov(Y covariance matrix defined denote derived deviation discussed distribution function distribution with density distribution with mean entropy equal estimator Exercise exponential distribution finite variance gamma distribution geometric stable laws hyperbolic distributions i.i.d. standard infinitely divisible integral interval Kozubowski Laplace density Laplace laws Laplace motion Laplace random variables likelihood function limit linear location parameter log-Laplace distribution maximized mean zero mixture MLE of 9 MLE's multivariate Laplace normal distribution Note obtain order statistics probability Proof properties Proposition quantile r.v. with mean random sample relation Remark sample median scale parameter Section Show skewness standard classical Laplace standard Laplace standard normal symmetric Laplace distribution Theorem unbiased unimodal univariate Xn:n zero and variance