The Laplace distribution and generalizations: a revisit with applications to communications, economics, engineering, and finance
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
| Published: |
New York
Springer
2001
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| Edition: | softcover repr. |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028213830&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Item Description: | Literaturverz. S. 319 - 342 |
| Physical Description: | XVIII, 349 S. graph. Darst. |
| ISBN: | 0817641661 3764341661 9781461266464 |
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| 245 | 1 | 0 | |a The Laplace distribution and generalizations |b a revisit with applications to communications, economics, engineering, and finance |c Samuel Kotz ; Tomasz J. Kozubowski ; Krzysztof Podgórski |
| 250 | |a softcover repr. | ||
| 264 | 1 | |a New York |b Springer |c 2001 | |
| 300 | |a XVIII, 349 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 319 - 342 | ||
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| 650 | 4 | |a Distribution (Probability theory) | |
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| adam_text | Contents
Preface xi
Abbreviations xiii
Notation xv
I Univariate Distributions 1
1 Historical Background 3
2 Classical Symmetric 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 .................................................... 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
vi Contents
2.3
2.4
2.5
2.6
2.7
2.2.7 Distributional limits of geometric sums · ■ · ..............................
2.2.8 Stability with respect to the ordinary summation............................
2.2.9 Distributional limits of deterministic sums.................................
Functions of Laplace random variables .............................................
2.3.1 The distribution of the sum of independent Laplace variates ................
2.3.2 The distribution of the product of two independent Laplace variates . . . .
2.3.3 The distribution of the ratio of two independent Laplace variates ......
2.3.4 The /-statistic for a double exponential (Laplace) distribution.............
Further properties.................................................................
2.4.1 Infinite divisibility.......................................................
2.4.2 Geometric infinite divisibility . . . ......................................
2.4.3 Self-decomposability........................................................
2.4.4 Complete monotonicity.......................................................
2.4.5 Maximum entropy property....................................................
Order statistics..................................................................
2.5.1 Distribution of a single order statistic ...................................
2.5.1.1 The minimum.......................................................
2.5.1.2 The maximum........................................................
2.5.1.3 The median........................................................
2.5.2 Joint distributions of order statistics.....................................
2.5.2.1 Range, midrange, sample median....................................
2.5.3 Moments of order statistics.................................................
2.5.4 Representation of order statistics via sums of exponentials............... .
Statistical inference.............................................................
2.6.1 Point estimation............................................................
2.6.1.1 Maximum likelihood estimation.....................................
2.6.1.2 Maximum likelihood estimation under censoring.....................
2.6.1.3 Maximum likelihood estimation of monotone location parameters
2.6.1.4 The method of moments.............................................
2.6.1.5 Linear estimádon..................................................
2.6.2 Interval estimation.........................................................
2.6.2.1 Confidence bands for the Laplace distribution function .
2.6.2.2 Conditional inference.............................................
2.6.3 Tolerance intervals.......................................................
2.6.4 Testing hypothesis......................................................
2.6.4.1 Testing the normal versus the Laplace ..........................
2.6.4.2 Goodness-of-fit tests.........................................
2.6.4.3 Neyman-Pearson test for location........................
2.6.4.4 Asymptotic optimality of the Kolmogorov-Smimov test
2.6.4.5 Comparison of nonparametric tests of location
Exercises..............................................
30
32
35
35
35
40
41
43
46
46
48
49
49
51
53
53
54
55
55
55
56
60
63
64
65
66
77
78
79
84
91
93
94
99
103
103
105
106
110
110
112
3 Asymmetric Laplace Distributions
3.1 Definition and basic properties.........................
3.1.1 An alternative parametrization and special cases
3.1.2 Standardization................................
3.1.3 Densities and their properties...............
3.1.4 Moment and cumulant generating functions . .
3.1.5 Moments and related parameters................
133
136
136
137
137
140
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 к and a are known........................ 159
3.5.1.2 Case 2: The values of Ѳ and к are known........................ 161
3.5.1.3 Case 3: The values of Ѳ and a are known........................ 162
3.5.1.4 Case 4: The value of к is known ............................... 166
3.5.1.5 Case 5: The value of Ѳ is known ............................... 167
3.5.1.6 Case 6: The value of a 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 Asymmetric Laplace laws........................................ 189
4.1.4.2 Symmetric case.................................................. 189
4.1.4.3 An integer value of r .......................................... 191
4.1.5 Moments................................................................. 192
4.2 Laplace motion .................................................... 193
4.2.1 Symmetric Laplace motion.................................................. 193
4.2.2 Representations........................................................... 194
4.2.3 Asymmetric Laplace motion................................................. 197
4.2.3.1 Subordinated Brownian motion.................................... 198
4.2.3.2 Difference of gamma processes................................... 198
viii Contents
4.3
4.4
4.5
4.2.3.3 Compound Poisson approximation....................
Linnik distribution.............................................
4.3.1 Characterizations.................................... ‘
4.3.1.1 Stability with respect to geometric summation .
4.3.1.2 Distributional limits of geometric sums .....
4.3.1.3 Stability with respect to deterministic summation
4.3.2 Representations...........................................
4.3.3 Densities and distribution functions......................
4.3.3.1 Integral representations........................
4.3.3.2 Series expansions................ . . . ........
4.3.4 Moments and tail behavior.................................
4.3.5 Properties...............................................
4.3.5.1 Self-decomposability.......................... ·
4.3.5.2 Infinite divisibility...........................
4.3.6 Simulation ..............................................
4.3.7 Estimation ..............................................
4.3.7.1 Method of moments type estimators ...............
4.3.7.2 Least-squares estimators.........................
4.3.7.3 Minimal distance method.........................
4.3.7.4 Fractional moment estimation....................
4.3.8 Extensions ................................................
Other cases ...................................................,
4.4.1 Log-Laplace distribution...................................
4.4.2 Generalized Laplace distribution...........................
4.4.3 Sargan distribution........................................
4.4.4 Geometric stable laws......................................
4.4.5 v-stable laws..............................................
Exercises........................................................
200
200
202
203
204
206
206
208
212
213
213
214
215
215
216
216
217
218
218
219
219
219
220
220
222
222
II Multivariate Distributions
227
Introduction
229
5 Symmetric Multivariate Laplace Distribution
5.1 Bivariate case..............................
5.1.1 Definition............................
5.1.2 Moments...............................
5.1.3 Densities.............................
5.1.4 Simulation of bivariate Laplace variates
5.2 General symmetric multivariate case.........
5.2.1 Definition............................
5.2.2 Moments and densities.................
5.3 Exercises.............................
231
231
231
232
232
233
234
234
235
236
6 Asymmetric Multivariate Laplace Distribution
6.1 Bivariate case: Definition and basic properties
6.1.1 Definition...........................
6.1.2 Moments...................
239
240
240
241
Contents ix
6.1.3 Densities................................................................. 241
6.1.4 Simulation of bivariate asymmetric 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 symmetric 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 Geometric 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 cr2......................................................... 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 xq............................. 264
6.11.5 Maximum likelihood estimation............................................. 265
6.11.6 Bayesian estimation....................................................... 267
6.12 Exercises....................................................................... 268
x Contents
III Applications
Introduction
7 Engineering Sciences
7.1 Detection in the presence of Laplace noise......
7.2 Encoding and decoding of analog signals ......
7.3 Optimal quantizer in image and speech compression
7.4 Fracture problems............................. · ·
7.5 Wind shear data.................................
7.6 Error distributions in navigation............
273
275
277
277
280
281
284
285
286
8 Financial Data
8.1 Underreported data................................................................289
8.2 Interest rate data................................................................290
8.3 Currency exchange rates ..........................................................292
8.4 Share market return models........................................................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 models ...................................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 models with Laplace noise in the environmental time series................313
Appendix: Bessel Functions
315
References
319
Index
343
|
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| id | DE-604.BV042783848 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T17:19:41Z |
| institution | BVB |
| isbn | 0817641661 3764341661 9781461266464 |
| language | English |
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| owner | DE-473 DE-BY-UBG |
| owner_facet | DE-473 DE-BY-UBG |
| physical | XVIII, 349 S. graph. Darst. |
| publishDate | 2001 |
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| publisher | Springer |
| record_format | marc |
| spellingShingle | Kotz, Samuel 1930-2010 Kozubowski, Tomasz J. 1962- Podgórski, Krzysztof 1963- The Laplace distribution and generalizations a revisit with applications to communications, economics, engineering, and finance Verdelingen (statistiek) gtt Distribution (Probability theory) Laplace-Verteilung (DE-588)4649789-4 gnd |
| subject_GND | (DE-588)4649789-4 |
| title | The Laplace distribution and generalizations a revisit with applications to communications, economics, engineering, and finance |
| title_auth | The Laplace distribution and generalizations a revisit with applications to communications, economics, engineering, and finance |
| title_exact_search | The Laplace distribution and generalizations a revisit with applications to communications, economics, engineering, and finance |
| title_full | The Laplace distribution and generalizations a revisit with applications to communications, economics, engineering, and finance Samuel Kotz ; Tomasz J. Kozubowski ; Krzysztof Podgórski |
| title_fullStr | The Laplace distribution and generalizations a revisit with applications to communications, economics, engineering, and finance Samuel Kotz ; Tomasz J. Kozubowski ; Krzysztof Podgórski |
| title_full_unstemmed | The Laplace distribution and generalizations a revisit with applications to communications, economics, engineering, and finance Samuel Kotz ; Tomasz J. Kozubowski ; Krzysztof Podgórski |
| title_short | The Laplace distribution and generalizations |
| title_sort | the laplace distribution and generalizations a revisit with applications to communications economics engineering and finance |
| title_sub | a revisit with applications to communications, economics, engineering, and finance |
| topic | Verdelingen (statistiek) gtt Distribution (Probability theory) Laplace-Verteilung (DE-588)4649789-4 gnd |
| topic_facet | Verdelingen (statistiek) Distribution (Probability theory) Laplace-Verteilung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028213830&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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