Long-memory processes: probabilistic properties and statistical methods
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Heidelberg [u.a.]
Springer
2013
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| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVII, 884 S. graph. Darst. |
| ISBN: | 3642355110 9783642355110 |
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Record in the Search Index
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IMAGE 1
CONTENTS
1 DEFINITION O F LONG MEMORY 1
1.1 HISTORIC OVERVIEW 1
1.2 DATA EXAMPLES 3
1.3 DEFINITION O F DIFFERENT TYPES OF MEMORY 19
1.3.1 SECOND-ORDER DEFINITIONS FOR STATIONARY PROCESSES 19
1.3.2 VOLATILITY DEPENDENCE 30
1.3.3 SECOND-ORDER DEFINITIONS FOR NONSTATIONARY PROCESSES . . . 31
1.3.4 CONTINUOUS-TIME PROCESSES 32
1.3.5 SELF-SIMILAR PROCESSES: BEYOND SECOND-ORDER DEFINITIONS . 33 1.3.6
OTHER APPROACHES 38
2 ORIGINS AND GENERATION O F LONG MEMORY 4 3
2.1 GENERAL PROBABILISTIC MODELS 43
2.1.1 LINEAR PROCESSES WITH FINITE SECOND MOMENTS 4 3
2.1.2 LINEAR PROCESSES WITH INFINITE SECOND MOMENTS 52
2.1.3 NONLINEAR PROCESSES-VOLATILITY MODELS 55
2.1.4 COUNTING PROCESSES 76
2.2 PHYSICAL MODELS 81
2.2.1 TEMPORAL AGGREGATION 81
2.2.2 CROSS-SECTIONAL AGGREGATION 85
2.2.3 PARTICLE SYSTEMS, TURBULENCE, ECOLOGICAL SYSTEMS 90
2.2.4 NETWORK TRAFFIC MODELS 95
2.2.5 CONTINUOUS-TIME MODELS 101
2.2.6 FRACTALS 105
3 MATHEMATICAL CONCEPTS 107
3.1 ORTHOGONAL POLYNOMIALS AND EXPANSIONS 108
3.1.1 ORTHOGONAL POLYNOMIALS-GENERAL INTRODUCTION 108
3.1.2 HERMITE POLYNOMIALS AND HERMITE EXPANSION 110
3.1.3 LAGUERRE POLYNOMIALS 114
3.1.4 JACOBI POLYNOMIALS 116
XI
HTTP://D-NB.INFO/1027870716
IMAGE 2
XII C O N T E N T S
3.1.5 GEGENBAUER POLYNOMIALS 117
3.1.6 LEGENDRE POLYNOMIALS 118
3.2 MULTIVARIATE HERMITE EXPANSIONS 119
3.3 APPELL POLYNOMIALS 129
3.3.1 GENERAL MOTIVATION 129
3.3.2 DEFINITION 130
3.3.3 ORTHOGONALITY 133
3.3.4 COMPLETENESS AND UNIQUENESS 136
3.3.5 EXTENSION BEYOND ANALYTIC FUNCTIONS 150
3.4 MULTIVARIATE APPELL POLYNOMIALS AND WICK PRODUCTS 153
3.4.1 DEFINITION 153
3.4.2 CONNECTION TO CUMULANTS AND OTHER IMPORTANT PROPERTIES . 155 3.4.3
DIAGRAM FORMULAS 160
3.5 WAVELETS 166
3.5.1 THE CONTINUOUS WAVELET TRANSFORM (CWT) 166
3.5.2 THE DISCRETE WAVELET TRANSFORM (DWT) 170
3.5.3 COMPUTATIONAL ASPECTS AND THE TRANSITION FROM DISCRETE TO
CONTINUOUS TIME 176
3.6 FRACTALS 178
3.7 FRACTIONAL AND STABLE PROCESSES 188
3.7.1 FRACTIONAL BROWNIAN MOTION AND HERMITE-ROSENBLATT PROCESSES 188
3.7.2 LINEAR FRACTIONAL STABLE MOTION 201
3.7.3 FRACTIONAL CALCULUS 206
4 LIMIT THEOREMS 209
4.1 TOOLS 209
4.1.1 INTRODUCTION 209
4.1.2 HOW TO DERIVE LIMIT THEOREMS? 209
4.1.3 SPECTRAL REPRESENTATION O F STATIONARY SEQUENCES 214
4.2 LIMIT THEOREMS FOR SUMS WITH FINITE MOMENTS 218
4.2.1 INTRODUCTION 218
4.2.2 NORMALIZING CONSTANTS FOR STATIONARY PROCESSES 219
4.2.3 SUBORDINATED GAUSSIAN PROCESSES 221
4.2.4 LINEAR PROCESSES 231
4.2.5 SUBORDINATED LINEAR PROCESSES 239
4.2.6 STOCHASTIC VOLATILITY MODELS AND THEIR MODIFICATIONS . . . 257
4.2.7 ARCH(OO) MODELS 260
4.2.8 LARCH MODELS 262
4.2.9 SUMMARY OF LIMIT THEOREMS FOR PARTIAL SUMS 264
4.3 LIMIT THEOREMS FOR SUMS WITH INFINITE MOMENTS 265
4.3.1 INTRODUCTION 265
4.3.2 GENERAL TOOLS: REGULAR VARIATION, STABLE LAWS AND POINT PROCESSES
266
4.3.3 SUMS O F LINEAR AND SUBORDINATED LINEAR PROCESSES 274
IMAGE 3
C O N T E N T S XIII
4.3.4 STOCHASTIC VOLATILITY MODELS 286
4.3.5 SUBORDINATED GAUSSIAN PROCESSES WITH INFINITE VARIANCE . . 294
4.3.6 QUADRATIC LARCH MODELS 298
4.3.7 SUMMARY O F LIMIT THEOREMS FOR PARTIAL SUMS 298
4.4 LIMIT THEOREMS FOR SAMPLE COVARIANCES 299
4.4.1 GAUSSIAN SEQUENCES 300
4.4.2 LINEAR PROCESSES WITH FINITE MOMENTS 307
4.4.3 LINEAR PROCESSES WITH INFINITE MOMENTS 310
4.4.4 STOCHASTIC VOLATILITY MODELS 312
4.4.5 SUMMARY O F LIMIT THEOREMS FOR SAMPLE COVARIANCES . . . 313 4.5
LIMIT THEOREMS FOR QUADRATIC FORMS 314
4.5.1 GAUSSIAN SEQUENCES 315
4.5.2 LINEAR PROCESSES 321
4.5.3 SUMMARY OF LIMIT THEOREMS FOR QUADRATIC FORMS 324
4.6 LIMIT THEOREMS FOR FOURIER TRANSFORMS AND THE PERIODOGRAM . . . 325
4.6.1 PERIODOGRAM AND DISCRETE FOURIER TRANSFORM (DFT) . . . . 325 4.6.2
SECOND-ORDER PROPERTIES OF THE FOURIER TRANSFORM AND THE PERIODOGRAM 326
4.6.3 LIMITING DISTRIBUTION 333
4.7 LIMIT THEOREMS FOR WAVELETS 334
4.7.1 INTRODUCTION 334
4.7.2 DISCRETE WAVELET TRANSFORM OF STOCHASTIC PROCESSES . . . . 334
4.7.3 SECOND-ORDER PROPERTIES OF WAVELET COEFFICIENTS 336
4.8 LIMIT THEOREMS FOR EMPIRICAL AND QUANTILE PROCESSES 340
4.8.1 LINEAR PROCESSES WITH FINITE MOMENTS 340
4.8.2 APPLICATIONS AND EXTENSIONS 345
4.8.3 EMPIRICAL PROCESSES WITH ESTIMATED PARAMETERS 347
4.8.4 LINEAR PROCESSES WITH INFINITE MOMENTS 349
4.8.5 TAIL EMPIRICAL PROCESSES 350
4.9 LIMIT THEOREMS FOR COUNTING PROCESSES AND TRAFFIC MODELS . . . . 356
4.9.1 . COUNTING PROCESSES 356
4.9.2 SUPERPOSITION OF COUNTING PROCESSES 358
4.9.3 TRAFFIC MODELS 360
4.9.4 RENEWAL REWARD PROCESSES 362
4.9.5 SUPERPOSITION OF O N - O F F PROCESSES 368
4.9.6 SIMULTANEOUS LIMITS AND FURTHER EXTENSIONS 372
4.10 LIMIT THEOREMS FOR EXTREMES 374
4.10.1 GUMBEL DOMAIN O F ATTRACTION 376
4.10.2 FRECHET DOMAIN O F ATTRACTION 380
4.10.3 STATIONARY STABLE PROCESSES 383
5 STATISTICAL INFERENCE FOR STATIONARY PROCESSES 385
5.1 INTRODUCTION 385
5.2 LOCATION ESTIMATION 389
5.2.1 TESTS AND CONFIDENCE INTERVALS BASED ON THE SAMPLE MEAN 389
IMAGE 4
X I V C O N T E N T S
5.2.2 EFFICIENCY OF THE SAMPLE MEAN 393
5.2.3 M-ESTIMATION 397
5.3 SCALE ESTIMATION 405
5.4 HEURISTIC ESTIMATION O F LONG MEMORY 409
5.4.1 VARIANCE PLOT 409
5.4.2 RESCALED RANGE METHOD 410
5.4.3 KPSS STATISTIC 4 1 3
5.4.4 RESCALED VARIANCE METHOD 414
5.4.5 DETRENDED FLUCTUATION ANALYSIS (DFA) 414
5.4.6 TEMPORAL AGGREGATION 415
5.4.7 COMMENTS 416
5.5 GAUSSIAN MAXIMUM LIKELIHOOD AND WHITTLE ESTIMATION 416
5.5.1 EXACT GAUSSIAN OR QUASI-MAXIMUM LIKELIHOOD ESTIMATION 416 5.5.2
WHITTLE ESTIMATION 420
5.5.3 FURTHER COMMENTS ON THE WHITTLE ESTIMATOR 426
5.5.4 SOME TECHNICAL DETAILS FOR THE WHITTLE ESTIMATOR 430
5.5.5 FURTHER APPROXIMATION METHODS FOR THE MLE 431
5.5.6 MODEL CHOICE 435
5.5.7 COMMENTS ON FINITE SAMPLE PROPERTIES AND FURTHER EXTENSIONS 439
5.6 SEMIPARAMETRIC NARROWBAND METHODS IN THE FOURIER DOMAIN . . . 440
5.6.1 INTRODUCTION 440
5.6.2 LOG-PERIODOGRAM REGRESSION-NARROWBAND LSE 441
5.6.3 LOCAL WHITTLE ESTIMATION-NARROWBAND WHITTLE ESTIMATION 445 5.6.4
TECHNICAL DETAILS FOR SEMIPARAMETRIC ESTIMATORS IN THE FOURIER DOMAIN
448
5.6.5 COMPARISON AND MODIFICATIONS OF SEMIPARAMETRIC ESTIMATORS IN THE
FOURIER DOMAIN 457
5.7 SEMIPARAMETRIC NARROWBAND METHODS IN THE WAVELET DOMAIN . . . 461
5.7.1 LOG WAVELET REGRESSION 461
5.7.2 TECHNICAL DETAILS FOR WAVELET ESTIMATORS 465
5.8 OPTIMAL RATE FOR NARROWBAND METHODS 470
5.9 BROADBAND METHODS 476
5.9.1 BROADBAND LSE FOR FEXP(OO) MODELS 476
5.9.2 BROADBAND WHITTLE ESTIMATION FOR FEXP(OO) MODELS . . . 484 5.9.3
ADAPTIVE FRACTIONAL AUTOREGRESSIVE FITTING 486
5.9.4 GENERAL CONCLUSIONS ON BROADBAND ESTIMATORS 489
5.10 PARAMETRIC AND SEMIPARAMETRIC ESTIMATORS-SUMMARY 491
5.11 ESTIMATION FOR PANEL DATA 491
5.12 ESTIMATING PERIODICITIES 494
5.12.1 IDENTIFYING LOCAL MAXIMA 494
5.12.2 IDENTIFYING STRONG STOCHASTIC PERIODICITIES 496
5.13 QUANTILE ESTIMATION 499
5.14 DENSITY ESTIMATION 501
5.14.1 I N T R O D U C T I O N 5 0 1
IMAGE 5
C O N T E N T S X V
5.14.2 NONPARAMETRIC KERNEL DENSITY ESTIMATION UNDER LRD . . . 504
5.14.3 DENSITY ESTIMATION BASED ON THE CUMULANT GENERATING FUNCTION 513
5.15 TAIL INDEX ESTIMATION 516
5.16 GOODNESS-OF-FIT TESTS 523
6 STATISTICAL INFERENCE FOR NONLINEAR PROCESSES 529
6.1 STATISTICAL INFERENCE FOR STOCHASTIC VOLATILITY MODELS 531
6.1.1 LOCATION ESTIMATION 532
6.1.2 ESTIMATION OF DEPENDENCE PARAMETERS 534
6.1.3 TAIL INDEX ESTIMATION 537
6.2 STATISTICAL INFERENCE FOR LARCH PROCESSES 538
6.2.1 LOCATION ESTIMATION 539
6.2.2 ESTIMATION OF DEPENDENCE PARAMETERS 540
6.3 STATISTICAL INFERENCE FOR ARCH(OO) PROCESSES 551
6.3.1 LOCATION ESTIMATION 552
6.3.2 ESTIMATION OF DEPENDENCE PARAMETERS 552
7 STATISTICAL INFERENCE FOR NONSTATIONARY PROCESSES 555
7.1 PARAMETRIC LINEAR FIXED-DESIGN REGRESSION 556
7.1.1 ASYMPTOTIC DISTRIBUTION O F THE LSE 556
7.1.2 THE REGRESSION SPECTRUM AND EFFICIENCY OF THE LSE . . . . 561
7.1.3 ROBUST LINEAR REGRESSION 577
7.1.4 OPTIMAL DETERMINISTIC DESIGNS 578
7.2 PARAMETRIC LINEAR RANDOM-DESIGN REGRESSION 580
7.2.1 SOME EXAMPLES, ESTIMATION O F CONTRASTS 581
7.2.2 SOME GENERAL RESULTS AND THE HETEROSKEDASTIC CASE . . . . 587
7.2.3 RANDOMLY WEIGHTED PARTIAL SUMS 593
7.2.4 SPURIOUS CORRELATIONS 597
7.2.5 FRACTIONAL COINTEGRATION 604
7.3 PIECEWISE POLYNOMIAL AND SPLINE REGRESSION 612
7.4 NONPARAMETRIC REGRESSION WITH LRD ERRORS-KERNEL AND LOCAL POLYNOMIAL
SMOOTHING 616
7.4.1 INTRODUCTION 618
7.4.2 FIXED-DESIGN REGRESSION WITH HOMOSCEDASTIC LRD ERRORS 637 7.4.3
FIXED-DESIGN REGRESSION WITH HETEROSKEDASTIC LRD ERRORS 648 7.4.4
BANDWIDTH CHOICE FOR FIXED DESIGN NONPARAMETRIC REGRESSION-PART I 648
7.4.5 THE SEMIFAR MODEL 649
7.4.6 BANDWIDTH CHOICE FOR FIXED DESIGN NONPARAMETRIC REGRESSION-PART
II: DATA-DRIVEN SEMIFAR ALGORITHMS . 652 7.4.7 TREND ESTIMATION FROM
REPLICATES 659
7.4.8 RANDOM-DESIGN REGRESSION UNDER LRD 664
7.4.9 CONDITIONAL VARIANCE ESTIMATION 670
7.4.10 ESTIMATION OF TREND FUNCTIONS FOR LARCH PROCESSES . . . 673
7.4.11 FURTHER BIBLIOGRAPHIC COMMENTS 674
IMAGE 6
X V I C O N T E N T S
7.5 TREND ESTIMATION BASED ON WAVELETS 675
7.5.1 INTRODUCTION 675
7.5.2 FIXED DESIGN 675
7.5.3 RANDOM DESIGN 683
7.6 ESTIMATION OF TIME DEPENDENT DISTRIBUTION FUNCTIONS AND QUANTILES
685
7.7 PARTIAL LINEAR MODELS 689
7.8 INFERENCE FOR LOCALLY STATIONARY PROCESSES 692
7.8.1 INTRODUCTION 692
7.8.2 OPTIMAL ESTIMATION FOR LOCALLY STATIONARY PROCESSES . . . . 694
7.8.3 COMPUTATIONAL ISSUES 699
7.9 ESTIMATION AND TESTING FOR CHANGE POINTS, TRENDS AND RELATED
ALTERNATIVES 700
7.9.1 INTRODUCTION 700
7.9.2 CHANGES IN THE MEAN UNDER LONG MEMORY 701
7.9.3 CHANGES IN THE MARGINAL DISTRIBUTION 707
7.9.4 CHANGES IN THE LINEAR DEPENDENCE STRUCTURE 711
7.9.5 CHANGES IN THE MEAN VS. LONG-RANGE DEPENDENCE 717
7.10 ESTIMATION O F RAPID CHANGE POINTS IN THE TREND FUNCTION 724
8 FORECASTING 733
8.1 FORECASTING FOR LINEAR PROCESSES 733
8.1.1 INTRODUCTION 733
8.1.2 FORECASTING FOR FARIMA PROCESSES 738
8.1.3 FORECASTING FOR FEXP PROCESSES 741
8.2 FORECASTING FOR NONSTATIONARY PROCESSES 743
8.3 FORECASTING FOR NONLINEAR PROCESSES 745
8.4 NONPARAMETRIC PREDICTION OF EXCEEDANCE PROBABILITIES 746
9 SPATIAL AND SPACE-TIME PROCESSES 753
9.1 SPATIAL MODELS ON Z* 753
9.2 SPATIAL FARIMA PROCESSES 755
9.3 MAXIMUM LIKELIHOOD ESTIMATION 756
9.4 LATENT SPATIAL PROCESSES: AN EXAMPLE FROM ECOLOGY 762
10 RESAMPLING 771
10.1 GENERAL INTRODUCTION 771
10.2 SOME BASICS ON BOOTSTRAP FOR I.I.D. DATA 775
10.3 SELF-NORMALIZATION 776
10.4 THE MOVING BLOCK BOOTSTRAP (MBB) 778
10.5 THE SAMPLING WINDOW BOOTSTRAP (SWB) 782
10.6 SOME PRACTICAL ISSUES 787
10.7 MORE COMPLEX MODELS 790
10.7.1 B O O T S T R A P F O R T H E H E A V Y - T A I L E D S V M O D E
L 7 9 0
10.7.2 T E S T I N G F O R J U M P S IN A T R E N D F U N C T I O N 7 9
2
IMAGE 7
C O N T E N T S XVII
APPENDIX A FUNCTION SPACES 797
A. 1 CONVERGENCE O F FUNCTIONS AND BASIC DEFINITIONS 797
A.2 L SPACES 797
A.3 THE SPACES C AND D 798
APPENDIX B REGULARLY VARYING FUNCTIONS 799
APPENDIX C VAGUE CONVERGENCE 801
APPENDIX D SOME USEFUL INTEGRALS 803
GLOSSARY 805
REFERENCES 807
AUTHOR INDEX 855
SUBJECT INDEX 867 |
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| dewey-ones | 519 - Probabilities and applied mathematics |
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| dewey-tens | 510 - Mathematics |
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| isbn | 3642355110 9783642355110 |
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| publisher | Springer |
| record_format | marc |
| spelling | Long-memory processes probabilistic properties and statistical methods Jan Beran ... Heidelberg [u.a.] Springer 2013 XVII, 884 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Long-memory-Prozess (DE-588)4345167-6 gnd rswk-swf Long-memory-Prozess (DE-588)4345167-6 s DE-604 Beran, Jan 1959- Sonstige (DE-588)141540060 oth Erscheint auch als Online-Ausgabe 978-3-642-35512-7 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4179702&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026069600&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Long-memory processes probabilistic properties and statistical methods Long-memory-Prozess (DE-588)4345167-6 gnd |
| subject_GND | (DE-588)4345167-6 |
| title | Long-memory processes probabilistic properties and statistical methods |
| title_auth | Long-memory processes probabilistic properties and statistical methods |
| title_exact_search | Long-memory processes probabilistic properties and statistical methods |
| title_full | Long-memory processes probabilistic properties and statistical methods Jan Beran ... |
| title_fullStr | Long-memory processes probabilistic properties and statistical methods Jan Beran ... |
| title_full_unstemmed | Long-memory processes probabilistic properties and statistical methods Jan Beran ... |
| title_short | Long-memory processes |
| title_sort | long memory processes probabilistic properties and statistical methods |
| title_sub | probabilistic properties and statistical methods |
| topic | Long-memory-Prozess (DE-588)4345167-6 gnd |
| topic_facet | Long-memory-Prozess |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4179702&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026069600&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT beranjan longmemoryprocessesprobabilisticpropertiesandstatisticalmethods |