Basics of modern mathematical statistics:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2015
|
| Series: | Springer texts in statistics
|
| Subjects: | |
| Links: | http://deposit.dnb.de/cgi-bin/dokserv?id=4421219&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=027058260&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XVIII, 296 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783642399084 |
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Record in the Search Index
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| adam_text | 1 BASIC NOTIONS 1
1.1 EXAMPLE OF A BERNOULLI EXPERIMENT 1
1.2 LEAST SQUARES ESTIMATION IN A LINEAR MODEL 4
1.3 GENERAL PARAMETRIC MODEL 7
1.4 STATISTICAL DECISION PROBLEM. LOSS AND RISK 8
1.5 EFFICIENCY 9
2 PARAMETER ESTIMATION FOR AN I.I.D. MODEL 11
2.1 EMPIRICAL DISTRIBUTION: GLIVENKO-CANTELLI THEOREM 11
2.2 SUBSTITUTION PRINCIPLE: METHOD OF MOMENTS 16
2.2.1 METHOD OF MOMENTS: UNIVARIATE PARAMETER 16
2.2.2 METHOD OF MOMENTS: MULTIVARIATE PARAMETER 17
2.2.3 METHOD OF MOMENTS: EXAMPLES 17
2.3 UNBIASED ESTIMATES, BIAS, AND QUADRATIC RISK 22
2.3.1 UNIVARIATE PARAMETER 23
2.3.2 MULTIVARIATE CASE 23
2.4 ASYMPTOTIC PROPERTIES 24
2.4.1 ROOT- NORMALITY: UNIVARIATE PARAMETER 25
2.4.2 ROOT-N NORMALITY: MULTIVARIATE PARAMETER 27
2.5 SOME GEOMETRIC PROPERTIES OF A PARAMETRIC FAMILY 30
2.5.1 KULLBACK-LEIBLER DIVERGENCE 30
2.5.2 HELLINGER DISTANCE 32
2.5.3 REGULARITY AND THE FISHER INFORMATION:
UNIVARIATE PARAMETER 33
2.5.4 LOCAL PROPERTIES OF THE KULLBACK-LEIBLER
DIVERGENCE AND HELLINGER DISTANCE 36
2.6 CRAMER-RAO INEQUALITY 37
2.6.1 UNIVARIATE PARAMETER 37
2.6.2 EXPONENTIAL FAMILIES AND R-EFFICIENCY 39
XIII
HTTP://D-NB.INFO/1038377005
XIV CONTENTS
2.7 CRAMDR-RAO INEQUALITY: MULTIVARIATE PARAMETER 41
2.7.1 REGULARITY AND FISHER INFORMATION: MULTIVARIATE
PARAMETER 41
2.7.2 LOCAL PROPERTIES OF THE KULLBACK-LEIBLER
DIVERGENCE AND HELLINGER DISTANCE 42
2.7.3 MULTIVARIATE CRAMER-RAO INEQUALITY 43
2.7.4 EXPONENTIAL FAMILIES AND R-EFFICIENCY 45
2.8 MAXIMUM LIKELIHOOD AND OTHER ESTIMATION METHODS 47
2.8.1 MINIMUM DISTANCE ESTIMATION 47
2.8.2 M-ESTIMATION AND MAXIMUM LIKELIHOOD ESTIMATION 47
2.9 MAXIMUM LIKELIHOOD FOR SOME PARAMETRIC FAMILIES 51
2.9.1 GAUSSIAN SHIFT 51
2.9.2 VARIANCE ESTIMATION FOR THE NORMAL LAW 53
2.9.3 UNIVARIATE NORMAL DISTRIBUTION 54
2.9.4 UNIFORM DISTRIBUTION ON [0,0] 54
2.9.5 BERNOULLI OR BINOMIAL MODEL 55
2.9.6 MULTINOMIAL MODEL 55
2.9.7 EXPONENTIAL MODEL 56
2.9.8 POISSON MODEL 56
2.9.9 SHIFT OF A LAPLACE (DOUBLE EXPONENTIAL) LAW 57
2.10 QUASI MAXIMUM LIKELIHOOD APPROACH 58
2.10.1 LSE AS QUASI LIKELIHOOD ESTIMATION 58
2.10.2 LAD AND ROBUST ESTIMATION AS QUASI
LIKELIHOOD ESTIMATION 59
2.11 UNIVARIATE EXPONENTIAL FAMILIES 60
2.11.1 NATURAL PARAMCTRIZATION 61
2.11.2 CANONICAL PARAMETRIZATION 64
2.11.3 DEVIATION PROBABILITIES FOR THE MAXIMUM LIKELIHOOD 67
2.12 HISTORICAL REMARKS AND FURTHER READING 72
3 REGRESSION ESTIMATION 75
3.1 REGRESSION MODEL 75
3.1.1 OBSERVATIONS 75
3.1.2 DESIGN 76
3.1.3 ERRORS 77
3.1.4 REGRESSION FUNCTION 78
3.2 METHOD OF SUBSTITUTION AND M-ESTIMATION 79
3.2.1 MEAN REGRESSION: LEAST SQUARES ESTIMATE 79
3.2.2 MEDIAN REGRESSION: LAD ESTIMATE 80
3.2.3 MAXIMUM LIKELIHOOD REGRESSION ESTIMATION 81
3.2.4 QUASI MAXIMUM LIKELIHOOD APPROACH 82
3.3 LINEAR REGRESSION 83
3.3.1 PROJECTION ESTIMATION 84
3.3.2 POLYNOMIAL APPROXIMATION 86
3.3.3 ORTHOGONAL POLYNOMIALS 87
CONTENTS
XV
3.3.4 CHEBYSHEV POLYNOMIALS 90
3.3.5 LEGENDRE POLYNOMIALS 94
3.3.6 LAGRANGE POLYNOMIALS 96
3.3.7 HERMITE POLYNOMIALS 98
3.3.8 TRIGONOMETRIC SERIES EXPANSION 101
3.4 PIECEWISE METHODS AND SPLINES 102
3.4.1 PIECEWISE CONSTANT ESTIMATION 102
3.4.2 PIECEWISE LINEAR UNIVARIATE ESTIMATION 104
3.4.3 PIECEWISE POLYNOMIAL ESTIMATION 106
3.4.4 SPLINE ESTIMATION 106
3.5 GENERALIZED REGRESSION 110
3.5.1 GENERALIZED LINEAR MODELS 112
3.5.2 LOGIT REGRESSION FOR BINARY DATA 113
3.5.3 PARAMETRIC POISSON REGRESSION 114
3.5.4 PIECEWISE CONSTANT METHODS IN GENERALIZED
REGRESSION 115
3.5.5 SMOOTHING SPLINES FOR GENERALIZED REGRESSION 117
3.6 HISTORICAL REMARKS AND FURTHER READING 118
4 ESTIMATION IN LINEAR MODELS 119
4.1 MODELING ASSUMPTIONS 119
4.2 QUASI MAXIMUM LIKELIHOOD ESTIMATION 120
4.2.1 ESTIMATION UNDER THE HOMOGENEOUS NOISE
ASSUMPTION 122
4.2.2 LINEAR BASIS TRANSFORMATION 122
4.2.3 ORTHOGONAL AND ORTHONORMAL DESIGN 124
4.2.4 SPECTRAL REPRESENTATION 125
4.3 PROPERTIES OF THE RESPONSE ESTIMATE F 127
4.3.1 DECOMPOSITION INTO A DETERMINISTIC
AND A STOCHASTIC COMPONENT 127
4.3.2 PROPERTIES OF THE OPERATOR N 127
4.3.3 QUADRATIC LOSS AND RISK OF THE RESPONSE ESTIMATION 129
4.3.4 MISSPECIFIED COLORED NOISE 130
4.4 PROPERTIES OF THE MLE 0 131
4.4.1 PROPERTIES OF THE STOCHASTIC COMPONENT 131
4.4.2 PROPERTIES OF THE DETERMINISTIC COMPONENT 132
4.4.3 RISK OF ESTIMATION: R-EFFICIENCY 134
4.4.4 THE CASE OF A MISSPECIFIED NOISE 136
4.5 LINEAR MODELS AND QUADRATIC LOG-LIKELIHOOD 137
4.6 INFERENCE BASED ON THE MAXIMUM LIKELIHOOD 139
4.6.1 A MISSPECIFIED LPA 143
4.6.2 A MISSPECIFIED NOISE STRUCTURE 143
4.7 RIDGE REGRESSION, PROJECTION, AND SHRINKAGE 145
4.7.1 REGULARIZATION AND RIDGE REGRESSION 146
4.7.2 PENALIZED LIKELIHOOD: BIAS AND VARIANCE 146
XVI CONTENTS
4.7.3 INFERENCE FOR THE PENALIZED MLE 149
4.7.4 PROJECTION AND SHRINKAGE ESTIMATES 150
4.7.5 SMOOTHNESS CONSTRAINTS AND ROUGHNESS PENALTY
APPROACH 153
4.8 SHRINKAGE IN A LINEAR INVERSE PROBLEM 154
4.8.1 SPECTRAL CUT-OFF AND SPECTRAL PENALIZATION:
DIAGONAL ESTIMATES 154
4.8.2 GALERKIN METHOD 156
4.9 SCMIPARAMETRIC ESTIMATION 157
4.9.1 (0, N)- AND W-SETUP 157
4.9.2 ORTHOGONALITY AND PRODUCT STRUCTURE 158
4.9.3 PARTIAL ESTIMATION 161
4.9.4 PROFILE ESTIMATION 162
4.9.5 SEMIPARAMETRIC EFFICIENCY BOUND 165
4.9.6 INFERENCE FOR THE PROFILE LIKELIHOOD APPROACH 166
4.9.7 PLUG-IN METHOD 168
4.9.8 TWO-STEP PROCEDURE 169
4.9.9 ALTERNATING METHOD 170
4.10 HISTORICAL REMARKS AND FURTHER READING 172
5 BAYES ESTIMATION 173
5.1 BAYES FORMULA 174
5.2 CONJUGATED PRIORS 176
5.3 LINEAR GAUSSIAN MODEL AND GAUSSIAN PRIORS 177
5.3.1 UNIVARIATE CASE 178
5.3.2 LINEAR GAUSSIAN MODEL AND GAUSSIAN PRIOR 179
5.3.3 HOMOGENEOUS ERRORS, ORTHOGONAL DESIGN 182
5.4 NON-INFORMATIVE PRIORS 183
5.5 BAYES ESTIMATE AND POSTERIOR MEAN 184
5.6 POSTERIOR MEAN AND RIDGE REGRESSION 186
5.7 BAYES AND MINIMAX RISKS 187
5.8 VAN TREES INEQUALITY 188
5.9 HISTORICAL REMARKS AND FURTHER READING 193
6 TESTING A STATISTICAL HYPOTHESIS 195
6.1 TESTI
NG PROBLEM 195
6.1.1 SIMPLE HYPOTHESIS 195
6.1.2 COMPOSITE HYPOTHESIS 196
6.1.3 STATISTICAL TESTS 196
6.1.4 ERRORS OF THE FIRST KIND, TEST LEVEL 197
6.1.5 RANDOMIZED TESTS 198
6.1.6 ALTERNATIVE HYPOTHESES, ERROR OF THE SECOND
KIND, POWER OF A TEST 199
6.2 NEYMAN-PEARSON TEST FOR TWO SIMPLE HYPOTHESES 201
6.2.1 NEYMAN-PEARSON TEST FOR AN I.I.D. SAMPLE 203
CONTENTS XVN
6.3 LIKELIHOOD RATIO TEST 204
6.4 LIKELIHOOD RATIO TESTS FOR PARAMETERS OF A NORMAL DISTRIBUTION....
206
6.4.1 DISTRIBUTIONS RELATED TO AN I.I.D. SAMPLE
FROM A NORMAL DISTRIBUTION 206
6.4.2 GAUSSIAN SHIFT MODEL 208
6.4.3 TESTING THE MEAN WHEN THE VARIANCE IS UNKNOWN 210
6.4.4 TESTING THE VARIANCE 212
6.5 LR TESTS: FURTHER EXAMPLES 213
6.5.1 BERNOULLI OR BINOMIAL MODEL 213
6.5.2 UNIFORM DISTRIBUTION ON [0,9] 214
6.5.3 EXPONENTIAL MODEL 215
6.5.4 POISSON MODEL 216
6.6 TESTING PROBLEM FOR A UNIVARIATE EXPONENTIAL FAMILY 216
6.6.1 TWO-SIDED ALTERNATIVE 217
6.6.2 ONE-SIDED ALTERNATIVE 218
6.6.3 INTERVAL HYPOTHESIS 220
6.7 HISTORICAL REMARKS AND FURTHER READING 221
7 TESTING IN LINEAR MODELS 223
7.1 LIKELIHOOD RATIO TEST FOR A SIMPLE NULL 223
7.1.1 GENERAL ERRORS 224
7.1.2 I.I.D. ERRORS, KNOWN VARIANCE 225
7.1.3 I.I.D. ERRORS WITH UNKNOWN VARIANCE 229
7.2 LIKELIHOOD RATIO TEST FOR A SUBVECTOR 231
7.3 LIKELIHOOD RATIO TEST FOR A LINEAR HYPOTHESIS 234
7.4 WALD TEST 235
7.5 ANALYSIS OF VARIANCE 236
7.5.1 TWO-SAMPLE TEST 237
7.5.2 COMPARING
K
TREATMENT MEANS 238
7.5.3 RANDOMIZED BLOCKS 241
7.6 HISTORICAL REMARKS AND FURTHER READING 243
8 SOME OTHER TESTING METHODS 245
8.1 METHOD OF MOMENTS FOR AN I.I.D. SAMPLE 245
8.1.1 UNIVARIATE CASE 246
8.1.2 MULTIVARIATE CASE 247
8.1.3 SERIES EXPANSION 247
8.1.4 TESTING A PARAMETRIC HYPOTHESIS 248
8.2 MINIMUM DISTANCE METHOD FOR AN I.I.D. SAMPLE 249
8.2.1 KOLMOGOROV-SMIRNOVTEST 250
8.2.2 TO
2
TEST (CRAM6R-SMIRNOV-VON MISES) 252
8.3 PARTIALLY BAYES TESTS AND BAYCS TESTING 253
8.3.1 PARTIAL BAYCS APPROACH AND BAYES TESTS 253
8.3.2 BAYCS APPROACH 254
XVUI CONTENTS
8.4 SCORE, RANK, AND PERMUTATION TESTS 255
8.4.1 SCORE TESTS 255
8.4.2 RANK TESTS 259
8.4.3 PERMUTATION TESTS 265
8.5 HISTORICAL REMARKS AND FURTHER READING 267
A DEVIATION PROBABILITY FOR QUADRATIC FORMS 269
A.L INTRODUCTION 269
A.2 GAUSSIAN CASE 271
A.3 A BOUND FOR THE TI -NORM 272
A.4 A BOUND FOR A QUADRATIC FORM 274
A.5 RESCALING AND REGULARITY CONDITION 275
A.6 A CHI-SQUARED BOUND WITH NORM-CONSTRAINTS 276
A.7 A BOUND FOR THE TI -NORM UNDER BERNSTEIN CONDITIONS 277
A.8 PROOFS 278
REFERENCES 289
INDEX 293
|
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| bvnumber | BV041617164 |
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| ctrlnum | (OCoLC)915397861 (DE-599)DNB1038377005 |
| dewey-full | 519.5 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 519 - Probabilities and applied mathematics |
| dewey-raw | 519.5 |
| dewey-search | 519.5 |
| dewey-sort | 3519.5 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV041617164 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T16:50:39Z |
| institution | BVB |
| isbn | 9783642399084 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027058260 |
| oclc_num | 915397861 |
| open_access_boolean | |
| owner | DE-11 DE-83 DE-188 |
| owner_facet | DE-11 DE-83 DE-188 |
| physical | XVIII, 296 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer texts in statistics |
| spellingShingle | Spokojnyj, Vladimir G. 1959- Dickhaus, Thorsten-Ingo Basics of modern mathematical statistics Parametrisches Verfahren (DE-588)4205938-0 gnd Statistik (DE-588)4056995-0 gnd |
| subject_GND | (DE-588)4205938-0 (DE-588)4056995-0 (DE-588)4143389-0 |
| title | Basics of modern mathematical statistics |
| title_auth | Basics of modern mathematical statistics |
| title_exact_search | Basics of modern mathematical statistics |
| title_full | Basics of modern mathematical statistics Vladimir Spokoiny ; Thorsten Dickhaus |
| title_fullStr | Basics of modern mathematical statistics Vladimir Spokoiny ; Thorsten Dickhaus |
| title_full_unstemmed | Basics of modern mathematical statistics Vladimir Spokoiny ; Thorsten Dickhaus |
| title_short | Basics of modern mathematical statistics |
| title_sort | basics of modern mathematical statistics |
| topic | Parametrisches Verfahren (DE-588)4205938-0 gnd Statistik (DE-588)4056995-0 gnd |
| topic_facet | Parametrisches Verfahren Statistik Aufgabensammlung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4421219&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=027058260&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT spokojnyjvladimirg basicsofmodernmathematicalstatistics AT dickhausthorsteningo basicsofmodernmathematicalstatistics |