Non-standard parametric statistical inference:
Saved in:
| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Oxford
Oxford University Press
2017
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| Edition: | First edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | xii, 417 Seiten Diagramme |
| ISBN: | 9780198505044 |
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MARC
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Record in the Search Index
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| adam_text | Contents
1 Introduction l
1.1 Terminology 3
1.2 Maximum Likelihood Estimation 4
1.3 Bootstrapping 6
1.4 Book Layout and Objectives 8
2 Non-Standard Problems: Some Examples 11
2.1 True Parameter Value on a Fixed Boundary 11
2.2 Infinite Likelihood: Weibull Example 12
2.3 Embedded Model Problem 13
2.4 Indeterminate Parameters 15
2.5 Model Building 17
2.6 Multiform Families 17
2.7 Oversmooth Log-likelihood 18
2.8 Rough Log-likelihood 18
2.9 Box-Cox Transform 19
2.10 Two Additional Topics 20
2.10.1 Randomized Parameter Models 20
2.10.2 Boots trapping Linear Models 21
3 Standard Asymptotic Theory 23
3.1 Basic Theory 24
3.2 Applications of Asymptotic Theory 26
3.3 Hypothesis Testing in Nested Models 27
3.3.1 Non-nested Models 31
3.4 Profile Log-likelihood 32
3.5 Orthogonalization 33
3.6 Exponential Models 34
3.7 Numerical Optimization of the Log-likelihood 39
3.8 Toll Booth Example 41
4 Bootstrap Analysis 45
4.1 Parametric Sampling 45
4.1.1 Monte Carlo Estimation 46
viii | Contents
4.1.2 Parametric Bootstrapping 46
4.1.3 Bootstrap Confidence Intervals 47
4.1.4 Toil Booth Example 49
4.1.5 Coverage Error and Scatterplots 49
4.2 Confidence Limits for Functions 51
4.3 Confidence Bands for Functions 52
4.4 Confidence Intervals Using Pivots 54
4.5 Bootstrap Goodness-of-Fit 56
4.6 Bootstrap Regression Lack-of-Fit 62
4.7 Two Numerical Examples 64
4.7.1 VapCO Data Set 65
4.7.2 Lettuce Data Set 67
5 Embedded Model Problem 71
5.1 Embedded Regression Example 72
5.2 Definition of Embeddedness 76
5.3 Step-by-Step Identification of an Embedded Model 76
5.4 Indeterminate Forms 78
5.5 Series Expansion Approach 79
5.6 Examples of Embedded Regression 81
5.7 Numerical Examples 83
5.7.1 VapCO Example Again 83
5.7.2 Lettuce Example Again 87
6 Examples of Embedded Distributions 91
6.1 Boundary Models 92
6.1.1 A Type IV Generalized Logistic Model 93
6.1.2 Burr XII and Generalized Logistic Distributions 95
6.1.3 Numerical Burr XII Example: Schedule Overruns 99
6.1.4 Shifted Threshold Distributions 104
6.1.5 Extreme Value Distributions 110
6.2 Comparing Models in the Same Family 115
6.3 Extensions of Model Families 117
6.4 Stable Distributions 121
6.5 Standard Characterization of Stable Distributions 122
6.5.1 Numerical Evaluation of Stable Distributions 124
7 Embedded Distributions: Two Numerical Examples 127
7.1 Kevlar 149 Fibre Strength Example 127
7.2 Carbon Fibre Failure Data 132
8 Infinite Likelihood 143
8.1 Threshold Models 143
8.2 ML in Threshold Models 145
8.3 Definition of Likelihood 148
8.4 Maximum Product of Spacings 150
Contents | ix
8.4.1 MPS Compared with ML 151
8.4.2 Tests for Embeddedness when Using MPS 152
8.5 Threshold CIs Using Stable Law 154
8.5.1 Example of Pitting the Loglogistic Distribution 159
8.6 A ‘Corrected’ Log-likelihood 163
8.7 A Hybrid Method 167
8.7.1 Comparison with Maximum Product of Spacings 167
8.8 MPSE: Additional Aspects 168
8.8.1 Consistency of MPSE 168
8.8.2 Goodness-of-Fit 169
8.8.3 Censored Observations 171
8.8.4 Tied Observations 171
9 The Pearson and Johnson Systems 173
9.1 Introduction 173
9.2 Pearson System 175
9.2.1 Pearson Distribution Types 175
9.2.2 Pearson Embedded Models 178
9.2.3 Fitting the Pearson System 181
9.3 Johnson System 182
9.3.1 Johnson Distribution Types 182
9.3.2 Johnson Embedded Models 183
9.3.3 Fitting the Johnson System 184
9.4 Initial Parameter Search Point 185
9.4.1 Starting Search Point for Pearson Distributions 185
9.4.2 Starting Search Point for Johnson Distributions 187
9.5 Symmetric Pearson and Johnson Models 187
9.6 Headway Times Example 188
9.6.1 Pearson System Fitted to Headway Data 188
9.6.2 Johnson System Fitted to Headway Data 191
9.6.3 Summary 193
9.7 FTSE Shares Example 193
9.7.1 Pearson System Fitted to FTSE Index Data 194
9.7.2 Johnson System Fitted to FTSE Index Data 197
9.7.3 Stable Distribution Fit 200
9.7.4 Summary 200
10 Box-Cox Transformations 201
10.1 Box-Cox Shifted Power Transformation 202
10.1.1 Estimation Procedure 203
10.1.2 Infinite Likelihood 205
10.2 Alternative Methods of Estimation 206
10.2.1 Grouped Likelihood Approach 206
10.2.2 Modified Likelihood 206
10.3 Unbounded Likelihood Example 207
x | Contents
10.4 Consequences of Truncation 209
10.5 Box-Cox Weibull Model 210
10.5.1 Fitting Procedure 211
10.6 Example Using Box-Cox Weibull Model 212
10.7 Advantages of the Box-Cox Weibull Model 212
11 Change-Point Models 215
11.1 Infinite Likelihood Problem 216
11.2 Likelihood with Randomly Censored Observations 218
11.2.1 Kaplan-Meier Estimate 219
11.2.2 Tied Observations 220
11.2.3 Numerical Example Using ML 222
11.3 The Spacings Function 224
11.3.1 Randomly Censored Observations 225
11.3.2 Numerical Example Using Spacings 227
11.3.3 Goodness-of-Fit 227
11.4 Bootstrapping in Change-Point Models 229
11.5 Summary 231
12 The Skew Normal Distribution 233
12.1 Introduction 233
12.2 Skew Normal Distribution 234
12.3 Linear Models of Z 236
12.3.1 Basic Linear Transformation of Z 236
12.3.2 Centred Linear Transformation of Z 238
12.3.3 Parametrization Invariance 240
12.4 Half-Normal Case 241
12.5 Log-likelihood Behaviour 242
12.5.1 FTSE Index Example 242
12.5.2 Toll Booth Service Times 247
12.6 Finite Mixtures; Multivariate Extensions 250
13 Randomized-Parameter Models 253
13.1 Increasing Distribution Flexibility 254
13.1.1 Threshold and Location-Scale Models 254
13.1.2 Power Transforms 254
13.1.3 Randomized Parameters 255
13.1.4 Hyperpriors 256
13.2 Randomized Parameter Procedure 257
13.3 Examples of Three-Parameter Generalizations 257
13.3.1 Normal Base Distribution 257
13.3.2 Lognormal Base Distribution 258
13.3.3 Weibull Base Distribution 259
13.3.4 Inverse-Gaussian Base Distribution 259
13.4 Embedded Models 260
Contents | xi
13.5 Score Statistic Test for the Base Model 263
13.5.1 Interpretation of the Test Statistic 264
13.5.2 Example of a Formal Test 265
13.5.3 Numerical Example 267
13.5.4 Goodness-of-Fit 272
14 Indeterminacy 275
14.1 The Indeterminate Parameters Problem 275
14.1.1 Two-Component Normal Mixture 276
14.2 Gaussian Process Approach 279
14.2.1 Davies Method 279
14.2.2 A Mixture Model Example 281
14.3 Test of Sample Mean 284
14.4 Indeterminacy in Nonlinear Regression 286
14.4.1 Regression Example 286
14.4.2 Davies Method 286
14.4.3 Sample Mean Method 289
14.4.4 Test of Weighted Sample Mean 289
15 Nested Nonlinear Regression Models 291
15.1 Model Building 291
15.2 The Linear Model 293
15.3 Indeterminacy in Nested Models 294
15.3.1 Link with Embedded Models 297
15.4 Removable Indeterminacies 298
15.5 Three Examples 299
15.5.1 A Double Exponential Model 299
15.5.2 Morgan-Mercer-Flodin Model 302
15.5.3 Weibull Regression Model 305
15.6 Intermediate Models 309
15.6.1 Mixture Model Example 310
15.6.2 Example of Methods Combined 311
15.7 Non-nested Models 313
16 Bootstrapping Linear Models 317
16.1 Linear Model Building: A BS Approach 317
16.1.1 Fitting the Full Linear Model 318
16.1.2 Model Selection Problem 320
16.1.3 ‘Unbiased Minp Method for Selecting a Model 320
16.2 Bootstrap Analysis 324
16.2.1 Bootstrap Samples 324
16.2.2 BS Generation of a Set of Promising Models 325
16.2.3 Selecting the Best Model and Assessing its Quality 326
16.2.4 Asphalt Binder Free Surface Energy Example 327
16.3 Conclusions 332
xii | Contents
17 Finite Mixture Models 335
17.1 Introduction 335
17.2 The Finite Mixture Model 336
17.2.1 MLE Estimation 337
17.2.2 Estimation of k under ML 338
17.2.3 Two Bayesian Approaches 340
17.2.4 MAPIS Method 342
17.3 Bayesian Hierarchical Model 344
17.3.1 Priors 344
17.3.2 The Posterior Distribution of k 346
17.4 MAPIS Method 348
17.4.1 MAP Estimation 348
17.4.2 Numerical MAP 350
17.4.3 Importance Sampling 351
17.5 Predictive Density Estimation 354
17.6 Overfitted Models in MCMC 355
17.6.1 Theorem by Rousseau and Mengersen 356
17.6.2 A Numerical Example 357
17.6.3 Overfitting with the MAPIS Method 361
18 Finite Mixture Examples; MAPIS Details 363
18.1 Numerical Examples 363
18.1.1 Galaxy and GalaxyB 364
18.1.2 Hidalgo Stamp Issues 373
18.1.3 Estimation of k 378
18.2 MAPIS Technical Details 381
18.2.1 Component Distributions 381
18.2.2 Approximation for a (*) function 382
18.2.3 Example of Hyperparameter Elimination 385
18.2.4 MAPIS Method: Additional Details 387
Bibliography 393
Author Index 407
Subject Index 410
This book discusses the fitting of parametr^: Statistical models to data samples.
Emphasis Is placed on: (i) how to recognize situations where the problem is non-
standard when parameter estimates behave unusually and (li) the use of parametric
bootstrap resampling methods in analysing such problems.
A frequentist likelihood-based viewpoint is adopted, for which there is a well-
established and very practical theory. The standard situation is where certain widely
applicable regularity conditions hold. However, there are many apparently innocuous
situations where standard theory breaks down, sometimes spectacularly.
The book is intended for anyone with a basic knowledge of statistical methods, as
is typically covered in a university statistical inference course, wishing to understand
or study how standard methodology might fail. Simple, easy to understand statistical
methods are presented which overcome these difficulties, and illustrated by detailed
examples drawn from real applications. Simple and practical model building is an
underlying theme. /
Parametric bootstrap resampling is used throughout for analysing the properties of
fitted models, illustrating its ease of implementation even in non-standard situations.
Distributional properties are obtained numerically for estimators or statistics not
previously considered in the literature because their distributional properties are
too hard to obtain theoretically. Bootstrap results are presented mainly graphically
in the book, providing easy to understand demonstration of the sampling behaviour
of estimators.
Russell Cheng is Emeritus Professor of Operational Research at the University
of Southampton.
OXFORD
UNIVERSITY PRESS
www.oup.com
|
| any_adam_object | 1 |
| author | Cheng, Russell |
| author_GND | (DE-588)171441362 |
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| discipline | Mathematik |
| edition | First edition |
| format | Book |
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| spelling | Cheng, Russell Verfasser (DE-588)171441362 aut Non-standard parametric statistical inference Russell Cheng First edition Oxford Oxford University Press 2017 xii, 417 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Statistik (DE-588)4056995-0 gnd rswk-swf Parametrisches Verfahren (DE-588)4205938-0 gnd rswk-swf Statistik (DE-588)4056995-0 s Parametrisches Verfahren (DE-588)4205938-0 s b DE-604 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029721874&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029721874&sequence=000002&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Cheng, Russell Non-standard parametric statistical inference Statistik (DE-588)4056995-0 gnd Parametrisches Verfahren (DE-588)4205938-0 gnd |
| subject_GND | (DE-588)4056995-0 (DE-588)4205938-0 |
| title | Non-standard parametric statistical inference |
| title_auth | Non-standard parametric statistical inference |
| title_exact_search | Non-standard parametric statistical inference |
| title_full | Non-standard parametric statistical inference Russell Cheng |
| title_fullStr | Non-standard parametric statistical inference Russell Cheng |
| title_full_unstemmed | Non-standard parametric statistical inference Russell Cheng |
| title_short | Non-standard parametric statistical inference |
| title_sort | non standard parametric statistical inference |
| topic | Statistik (DE-588)4056995-0 gnd Parametrisches Verfahren (DE-588)4205938-0 gnd |
| topic_facet | Statistik Parametrisches Verfahren |
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