Non-uniform random variate generation:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
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New York u.a.
Springer
1986
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| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XVI, 843 S. |
| ISBN: | 0387963057 3540963057 |
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| 100 | 1 | |a Devroye, Luc |e Verfasser |0 (DE-588)170228444 |4 aut | |
| 245 | 1 | 0 | |a Non-uniform random variate generation |c Luc Devroye |
| 264 | 1 | |a New York u.a. |b Springer |c 1986 | |
| 300 | |a XVI, 843 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
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Datensatz im Suchindex
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| adam_text | TABLE OF CONTENTS
PREFACE
TABLE OF CONTENTS
I. INTRODUCTION 1
1. General outline. 1
2. About our notation. 5
2.1. Definitions. 5
2.2. A few Important unlvarlate densities. 7
3. Assessment of random varlate generators. 8
3.1. Distributions with no variable parameters. 9
3.2. Parametric families. 9
4. Operations on random variables. 11
4.1. Transformations. 11
4.2. Mixtures. 18
4.3. Order statistics. 17
4.4. Convolutions. Sums of Independent random variables. 19
4.5. Sums of Independent uniform random variables. 21
4.6. Exercises. 23
II. GENERAL PRINCIPLES IN RANDOM VARIATE
GENERATION 27
1. Introduction. 27
2. The Inversion method. 27
2.1. The Inversion principle. 27
2.2. Inversion by numerical solution of F {X )= U. 31
2.3. Explicit approximations. 35
2.4. Exercises. 36
3. The rejection method. 40
3.1. Definition. 40
3.2. Development of good rejection algorithms. 43
3.3. Generalizations of the rejection method. 47
3.4. Wald s equation. 50
3.5. Letac s lower bound. 52
viii CONTENTS
3.6. The squeeze principle. 53
3.7. Recycling random varlates. 58
3.8. Exercises. 60
4. Decomposition as discrete mixtures. 66
4.1. Definition. 66
4.2. Decomposition Into simple components. 66
4.3. Partitions Into Intervals. 67
4.4. The waiting time method for asymmetric mixtures. 71
4.5. Polynomial densities on [0,1]. 71
4.6. Mixtures with negative coefficients. 74
5. The acceptance complement method. 75
5.1. Definition. 75
5.2. Simple acceptance complement methods. 77
5.3. Acceleration by avoiding the ratio computation. 78
5.4. An example : nearly flat densities on [0,1]. 79
5.5. Exercises. 81
III. DISCRETE RANDOM VARIATES 83
1. Introduction. 83
2. The Inversion method. 85
2.1. Introduction. 85
2.2. Inversion by truncation of a continuous random varlate. 87
2.3. Comparison based Inversions. 88
2.4. The method of guide tables. 96
2.5. Inversion by correction. 98
2.6. Exercises. 101
3. Table look up methods. 102
3.1. The table look up principle. 102
3.2. Multiple table look ups. 104
4. The alias method. 107
4.1. Definition. 107
4.2. The alias urn method. 110
4.3. Geometrical puzzles. Ill
4.4. Exercises. 112
5. Other general principles. 113
5.1. The rejection method. 113
5.2. The composition and acceptance complement methods. 116
5.3. Exercises. 116
IV. SPECIALIZED ALGORITHMS 118
1. Introduction. 118
1.1. Motivation for the chapter. 118
1.2. Exercises. 118
2. The Forsythe von Neumann method. 121
2.1. Description of the method. 121
2.2. Von Neumann s exponential random varlate generator. 125
2.3. Monahan s generalization. 127
CONTENTS ix
2.4. An example : Vaduva s gamma generator. 130
2.5. Exercises. 132
3. Almost exact Inversion. 133
3.1. Definition. 133
3.2. Monotone densities on [0,oo). 134
3.3. Polya s approximation for the normal distribution. 135
3.4. Approximations by simple functions of normal random
varlates. 13g
3.5. Exercises. 143
4. Many to one transformations. 145
4.1. The principle. 145
4.2. The absolute value transformation. 147
4.3. The Inverse gausslan distribution. 148
4.4. Exercises. 150
5. The series method. 151
5.1. Description. 151
5.2. Analysis of the alternating series algorithm. 154
5.3. Analysis of the convergent series algorithm. 156
5.4. The exponential distribution. 157
5.5. The Raab Green distribution. 158
5.6. The Kolmogorov Smlrnov distribution. 161
5.7. Exercises. 168
6. Representations of densities as Integrals. 171
6.1. Introduction. 171
6.2. Khlnchlne s and related theorems. 171
6.3. The lnverse of / method for monotone densities. 178
6.4. Convex densities. 179
6.5. Recursive methods based upon representations. 180
6.6. A representation for the stable distribution. 183
6.7. Densities with Polya type characteristic functions. 186
6.8. Exercises. 191
7. The ratlo of unlforms method. 194
7.1. Introduction. 194
7.2. Several examples. 197
7.3. Exercises. 203
V. UNIFORM AND EXPONENTIAL SPACINGS 206
1. Motivation. 206
2. Uniform and exponential spaclngs. 207
2.1. Uniform spaclngs. 207
2.2. Exponential spaclngs. 211
2.3. Exercises. 213
3. Generating ordered samples. 213
3.1. Generating uniform [0,1] order statistics. 214
3.2. Bucket sorting. Bucket searching. 215
3.3. Generating exponential order statistics. 219
3.4. Generating order statistics with distribution function F. 220
3.5. Generating exponential random varlates In batches. 223
x CONTENTS
3.6. Exercises. 223
4. The polar method. 225
4.1. Radially symmetric distributions. 225
4.2. Generating random vectors uniformly distributed, on Crf . 230
4.3. Generating points uniformly In and on C2. 233
4.4. Generating normal random varlates in batches. 235
4.5. Generating radially symmetric random vectors. 236
4.6. The deconvolution method. 239
4.7. Exercises. 240
VI. THE POISSON PROCESS 246
1. The Polsson process. 246
1.1. Introduction. 246
1.2. Simulation of homogeneous Polsson processes. 248
1.3. Nonhomogeneous Polsson processes. 250
1.4. Global methods for nonhomogeneous Polsson
process simulation. 257
1.5. Exercises. 258
2. Generation of random varlates with a given hazard rate. 260
2.1. Hazard rate. Connection with Polsson processes. 260
2.2. The Inversion method. 261
2.3. The composition method. 262
2.4. The thinning method. 264
2.5. DHR distributions. Dynamic thinning. 267
2.6. Analysis of the dynamic thinning algorithm. 269
2.7. Exercises. 276
3. Generating random varlates with a given
discrete hazard rate. 278
3.1. Introduction. 278
3.2. The sequential test method. 279
3.3. Hazard rates bounded away from 1. 280
3.4. Discrete dynamic thinning. 283
3.5. Exercises. 284
VII. UNIVERSAL METHODS 286
1. Black box philosophy. 286
2. Log concave densities. 287
2.1. Definition. 287
2.2. Inequalities for log concave densities. 288
2.3. A black box algorithm. 290
2.4. The optimal rejection algorithm. 293
2.5. The mirror principle. 295
2.6. Non universal rejection methods. 298
2.7. Exercises. 308
3. Inequalities for families of densities. 310
3.1. Motivation. 310
3.2. Bounds for unlmodal densities. 310
CONTENTS xi
3.3. Densities satisfying a Llpschitz condition. 320
3.4. Normal scale mixtures. 325
3.5. Exercises. 328
4. The Inversion rejection method. 331
4.1. The principle. 331
4.2. Bounded densities. 332
4.3. Unlmodal and monotone densities. 334
4.4. Monotone densities on [0,1]. 335
4.5. Bounded monotone densities : Inversion rejection
based on Newton Raphson Iterations. 341
4.6. Bounded monotone densities : geometrically
Increasing Interval sizes. 344
4.7. Llpschltz densities on [0,00). 348
4.8. Exercises. 355
VIII. TABLE METHODS FOR CONTINUOUS RANDOM
VARIATES 358
1. Composition versus rejection. 358
2. Strip methods. 359
2.1. Definition. 359
2.2. Example 1 : monotone densities on [0,1]. 362
2.3. Other examples. 366
2.4. Exercises. 367
3. Grid methods. 368
3.1. Introduction. 368
3.2. Generating a point uniformly In a compact set A . 368
3.3. Avoidance problems. 372
3.4. Fast random varlate generators. 375
IX. CONTINUOUS UNIVARIATE DENSITIES 379
1. The normal density. 379
1.1. Definition. 379
1.2. The tall of the normal density. 380
1.3. Composition/rejection methods. 382
1.4. Exercises. 391
2. The exponential density. 392
2.1. Overview. 392
2.2. Marsaglla s exponential generator. 394
2.3. The rectangle wedge tail method. 397
2.4. Exercises. 401
3. The gamma density. 401
3.1. The gamma family. 401
3.2. Gamma varlate generators. 404
3.3. Uniformly fast rejection algorithms for a l. 407
3.4. The Welbull density. 414
3.5. Johnk s theorem and Its Implications. 416
3.6. Gamma variate generators when a i. 419
3.7. The tail of the gamma density. 420
xii CONTENTS
3.8. Stacy s generalized gamma distribution. 423
3.9. Exercises. 423
4. The beta density. 428
4.1. Properties of the beta density. 428
4.2. Overview of beta generators. 431
4.3. The symmetric beta density. 433
4.4. Uniformly fast rejection algorithms. 437
4.5. Generators when mln(a ,b ) l. 439
4.6. Exercises. 444
5. The t distribution. 445
5.1. Overview. 445
5.2. Ordinary rejection methods. 447
5.3. The Cauchy density. 450
5.4. Exercises. 451
6. The stable distribution. 454
6.1. Definition and properties. 454
6.2. Overview of generators. 458
6.3. The Bergstrom Feller series. 460
6.4. The series method for stable random varlates. 463
6.5. Exercises. 467
7. Nonstandard distributions. 468
7.1. Bessel function distributions. 468
7.2. The logistic and hyperbolic secant distributions. 471
7.3. The von Mlses distribution. 473
7.4. The Burr distribution. 476
7.5. The generalized Inverse gaussian distribution. 478
7.6. Exercises. 480
X. DISCRETE UNIVARIATE DISTRIBUTIONS 485
1. Introduction. 485
1.1. Goals of this chapter. 485
1.2. Generating functions. 486
1.3. Factorials. 489
1.4. A universal rejection method. 493
1.5. Exercises. 496
2. The geometric distribution. 498
2.1. Definition and genesis. 498
2.2. Generators. 499
2.3. Exercises. 500
3. The Polsson distribution. 501
3.1. Basic properties. 501
3.2. Overview of generators. 502
3.3. Simple generators. 502
3.4. Rejection methods. 506
3.5. Exercises. 518
4. The binomial distribution. 520
4.1. Properties. 520
CONTENTS xiii
4.2. Overview of generators. 523
4.3. Simple generators. 523
4.4. The rejection method. 526
4.5. Recursive methods. 536
4.6. Symmetric binomial random varlates. 538
4.7. The negative binomial distribution. 543
4.8. Exercises. 543
5. The logarithmic series distribution. 545
5.1. Introduction. 545
5.2. Generators. 546
5.3. Exercises. 54g
6. The Zlpf distribution. . 550
6.1. A simple generator. 550
6.2. The Planck distribution. 552
6.3. The Yule distribution. 553
6.4. Exercises. 553
XL MULTIVARIATE DISTRIBUTIONS 554
1. General principles. 554
1.1. Introduction. 554
1.2. The conditional distribution method. 555
1.3. The rejection method. 557
1.4. The composition method. 557
1.5. Discrete distributions. 559
1.6. Exercises. 562
2. Linear transformations. The multinomial distribution. 563
2.1. Linear transformations. 563
2.2. Generators of random vectors with a
given covarlance matrix. 564
2.3. The multlnormal distribution. 566
2.4. Points uniformly distributed In a hyperelllpsold. 567
2.5. Uniform polygonal random vectors. 568
{ 2.6. Time series. 571
2.7. Singular distributions. 571
2.8. Exercises. 572
3. Dependence. Blvarlate distributions. 573
3.1. Creating and measuring dependence. 573
3.2. Blvarlate uniform distributions. 576
3.3. Blvarlate exponential distributions. 583
3.4. A case study: blvariate gamma distributions. 586
3.5. Exercises. 588
4. The Dlrlchlet distribution. 593
4.1. Definitions and properties. 593
4.2. Liouvllle distributions. 596
4.3. Exercises. 599
5. Some useful multlvarlate families. 600
5.1. The Cook Johnson family. 600
xiv CONTENTS
5.2. Multivariate Khlnchlne mixtures. 603
5.3. Exercises. 604
6. Random matrices. 605
6.1. Random correlation matrices. 605
6.2. Random orthogonal matrices. 607
6.3. Random R X C tables. 608
6.4. Exercises. 610
XII. RANDOM SAMPLING 611
1. Introduction. 611
2. Classical sampling. 612
2.1. The swapping method. 612
2.2. Classical sampling with membership checking 613
2.3. Exercises. 619
3. Sequential sampling. 619
3.1. Standard sequential sampling. 619
3.2. The spacings method for sequential sampling. 621
3.3. The Inversion method for sequential sampling. 624
3.4. Inverslon with correctlon. 625
3.5. The ghost point method. 626
3.6. The rejection method. 631
3.7. Exercises. 635
4. Oversampllng. 635
4.1. Definition. 635
4.2. Exercises. 638
5. Reservoir sampling. 638
5.1. Definition. 638
5.2. The reservoir method with geometric jumps. 640
5.3. Exercises. 641
XIII. RANDOM COMBINATORIAL OBJECTS 642
1. General principles. 642
1.1. Introduction. 642
1.2. The decoding method. 643
1.3. Generation based upon recurrences. 645
2. Random permutations. 648
2.1. Simple generators. 648
2.2. Random binary search trees. 648
2.3. Exercises. 650
3. Random binary trees. 652
3.1. Representations of binary trees. 652
3.2. Generation by rejection. 655
3.3. Generation by sequential sampling. 656
3.4. The decoding method. 657
3.5. Exercises. 657
4. Random partitions. 657
4.1. Recurrences and codewords. 657
CONTENTS xv
4.2. Generation of random partitions. 660
4.3. Exercises. 661
5. Random free trees. 662
5.1. Prufer s construction. 662
5.2. KUngsberg s algorithm. 664
5.3. Free trees with a given number of leaves. 665
5.4. Exercises. 666
6. Random graphs. 667
6.1. Random graphs with simple properties. 667
6.2. Connected graphs. 668
6.3. Tlnhofer s graph generators. 669
6.4. Bipartite graphs. 671
6.5. Excerclses. 673
XIV. PROBABILISTIC SHORTCUTS AND ADDITIONAL
TOPICS 674
1. The maximum of Hd random variables. 674
1.1. Overview of methods. 674
1.2. The quick elimination principle. 675
1.3. The record time method. 679
1.4. Exercises. 681
2. Random varlates with given moments. 682
2.1. The moment problem. 682
2.2. Discrete distributions. 686
2.3. Unlmodal densities and scale mixtures. 687
2.4. Convex combinations. 689
2.5. Exercises. 693
3. Characteristic functions. 695
3.1. Problem statement. 695
3.2. The rejection method for characteristic functions. 696
3.3. A black box method. 700
3.4. Exercises. 715
4. The simulation of sums. 716
4.1. Problem statement. 716
4.2. A detour via characteristic functions. 718
4.3. Rejection based upon a local central limit theorem. 719
4.4. A local limit theorem. 720
4.5. The mixture method for simulating sums. 731
4.6. Sums of Independent uniform random variables. 732
4.7. Exercises. 734
5. Discrete event simulation. 735
5.1. Future event set algorithms. 735
5.2. Reeves s model. 738
5.3. Linear lists. 740
5.4. Tree structures. 747
5.5. Exercises. 748
6. Regenerative phenomena. 749
6.1. The principle. 749
xvi CONTENTS
6.2. Random walks. 749
6.3. Birth and death processes. 755
6.4. Phase type distributions. • 757
6.5. Exercises. 758
7. The generalization of a sample. 759
7.1. Problem statement. 759
7.2. Sample independence. 760
7.3. Consistency of density estimates. 762
7.4. Sample indistlngulshablllty. 763
7.5. Moment matching. 764
7.6. Generators for f n. 765
7.7. Exercises. 766
XV. THE RANDOM BIT MODEL 768
1. The random bit model. 768
1.1. Introduction. 768
1.2. Some examples. 769
2. The Knuth Yao lower bound. 771
2.1. DDG trees. 771
2.2. The lower bound. 771
2.3. Exercises. 775
3. Optimal and suboptimal DDG tree algorithms. 775
3.1. Suboptimal DDG tree algorithms. 775
3.2. Optimal DDG tree algorithms. 777
3.3. Distribution free Inequalities for the performance
of optimal DDG tree algorithms. 780
3.4. Exercises. 782
REFERENCES 784
INDEX 817
|
| any_adam_object | 1 |
| author | Devroye, Luc |
| author_GND | (DE-588)170228444 |
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| ctrlnum | (OCoLC)13269466 (DE-599)BVBBV000672093 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
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| id | DE-604.BV000672093 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:17:34Z |
| institution | BVB |
| isbn | 0387963057 3540963057 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000419402 |
| oclc_num | 13269466 |
| open_access_boolean | |
| owner | DE-12 DE-19 DE-BY-UBM DE-91 DE-BY-TUM DE-384 DE-739 DE-355 DE-BY-UBR DE-824 DE-N2 DE-29T DE-703 DE-706 DE-634 DE-11 DE-188 DE-83 |
| owner_facet | DE-12 DE-19 DE-BY-UBM DE-91 DE-BY-TUM DE-384 DE-739 DE-355 DE-BY-UBR DE-824 DE-N2 DE-29T DE-703 DE-706 DE-634 DE-11 DE-188 DE-83 |
| physical | XVI, 843 S. |
| publishDate | 1986 |
| publishDateSearch | 1986 |
| publishDateSort | 1986 |
| publisher | Springer |
| record_format | marc |
| spelling | Devroye, Luc Verfasser (DE-588)170228444 aut Non-uniform random variate generation Luc Devroye New York u.a. Springer 1986 XVI, 843 S. txt rdacontent n rdamedia nc rdacarrier Variables aléatoires Willekeurige variabelen gtt Random variables Zufallsgenerator (DE-588)4191097-7 gnd rswk-swf Stochastische Erzeugung (DE-588)4441717-2 gnd rswk-swf Zufallszahlen (DE-588)4124968-9 gnd rswk-swf Zufallsgenerator (DE-588)4191097-7 s DE-604 Zufallszahlen (DE-588)4124968-9 s Stochastische Erzeugung (DE-588)4441717-2 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000419402&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Devroye, Luc Non-uniform random variate generation Variables aléatoires Willekeurige variabelen gtt Random variables Zufallsgenerator (DE-588)4191097-7 gnd Stochastische Erzeugung (DE-588)4441717-2 gnd Zufallszahlen (DE-588)4124968-9 gnd |
| subject_GND | (DE-588)4191097-7 (DE-588)4441717-2 (DE-588)4124968-9 |
| title | Non-uniform random variate generation |
| title_auth | Non-uniform random variate generation |
| title_exact_search | Non-uniform random variate generation |
| title_full | Non-uniform random variate generation Luc Devroye |
| title_fullStr | Non-uniform random variate generation Luc Devroye |
| title_full_unstemmed | Non-uniform random variate generation Luc Devroye |
| title_short | Non-uniform random variate generation |
| title_sort | non uniform random variate generation |
| topic | Variables aléatoires Willekeurige variabelen gtt Random variables Zufallsgenerator (DE-588)4191097-7 gnd Stochastische Erzeugung (DE-588)4441717-2 gnd Zufallszahlen (DE-588)4124968-9 gnd |
| topic_facet | Variables aléatoires Willekeurige variabelen Random variables Zufallsgenerator Stochastische Erzeugung Zufallszahlen |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000419402&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT devroyeluc nonuniformrandomvariategeneration |