Numerical Methods for Nonlinear Estimating Equations

✍ Scribed by Christopher G. Small, Jinfang Wang

• Publisher: Oxford University Press, USA
• Year: 2003
• Tongue: English
• Leaves: 322
• Series: Oxford Statistical Science Series
• Category: Library

✦ Synopsis

Non linearity arises in statistical inference in various ways, with varying degrees of severity, as an obstacle to statistical analysis. More entrenched forms of nonlinearity often require intensive numerical methods to construct estimators, and the use of root search algorithms, or one-step estimators, is a standard method of solution. This book provides a comprehensive study of nonlinear estimating equations and artificial likelihood's for statistical inference. It provides extensive coverage and comparison of hill climbing algorithms, which when started at points of nonconcavity often have very poor convergence properties, and for additional flexibility proposes a number of modification to the standard methods for solving these algorithms. The book also extends beyond simple root search algorithms to include a discussion of the testing of roots for consistency, and the modification of available estimating functions to provide greater stability in inference. A variety of examples from practical applications are included to illustrate the problems and possibilities thus making this text ideal for the research statistician and graduate student.

✦ Table of Contents

Contents......Page 10
1.1 Background to the problem......Page 14
1.2 Organisation of the book......Page 18
2.1 Basic definitions......Page 24
2.2 Godambe efficiency: one-parameter models......Page 25
2.3 The score function: one-parameter models......Page 26
2.4 Godambe efficiency: multiparameter models......Page 28
2.5 A geometric interpretation of Godambe efficiency......Page 30
2.6.1 Quasi-likelihood and semi-parametric models......Page 34
2.6.2 Martingale estimating functions......Page 37
2.6.3 Empirical characteristic functions and stable laws......Page 42
2.6.4 Quadrat sampling......Page 47
2.7 Bibliographical notes......Page 49
3.1 Introduction......Page 50
3.2 The bisection method......Page 56
3.3 The method of false positions......Page 57
3.4 Muller's method......Page 60
3.5 Iterative substitution and the contractive maps......Page 61
3.6.1 Newton–Raphson as a substitution algorithm......Page 63
3.6.2 Quasi-Newton algorithms......Page 65
3.7.1 The E–M algorithm for likelihood equations......Page 66
3.7.2 The E–M algorithm for other estimating equations......Page 70
3.8 Aitken acceleration of slow algorithms......Page 71
3.9 Bernoulli's method and the quotient-difference algorithm......Page 75
3.10 Sturm's method......Page 77
3.11 Roots and eigenvalues......Page 80
3.12 The Nelder–Mead algorithm......Page 81
3.13 Jacobi iteration for quasi-likelihood......Page 84
3.14 Bibliographical notes......Page 86
4.2 Non-identifiable parameters in mixture models......Page 87
4.3 Estimation of the correlation coefficient......Page 93
4.4 The Cauchy distribution and stable laws......Page 96
4.5 The relative likelihood principle......Page 110
4.6 Estimating the normal mean in stratified sampling......Page 112
4.7 Regression with measurement error......Page 114
4.8 Weighted likelihood equations......Page 117
4.9 Detecting multiple roots......Page 120
4.10 Finding all the roots......Page 124
4.11 Root functionals and measures......Page 128
4.12 Smoothing the likelihood function......Page 132
5.1 Introduction......Page 138
5.2 The problem of solving an estimating equation......Page 140
5.3 A class of irregular estimating functions......Page 142
5.4.1 Introduction......Page 145
5.4.2 Efficient likelihood estimators......Page 146
5.4.3 General case......Page 148
5.4.4 Choosing √n -consistent estimators......Page 151
5.5.1 Introduction......Page 152
5.5.2 Scalar case......Page 153
5.5.3 Multivariate case......Page 158
5.5.4 An example......Page 161
5.6.1 The modified algorithm......Page 163
5.6.2 An example......Page 167
5.7 A modified Muller's method......Page 169
5.8 Asymptotic examinations......Page 172
5.9 Testing the consistency of roots......Page 176
5.10.1 Bootstrap quadratic likelihood......Page 180
5.10.2 Bootstrap quadratic likelihood ratio test......Page 184
5.10.4 Example: logistic regression with measurement errors......Page 187
5.11 An information theoretic approach......Page 190
5.12.1 Introduction......Page 192
5.12.2 Model embedding via mixing......Page 193
5.12.3 Examples......Page 195
5.13 Non-existence of roots......Page 198
5.14.1 Introduction......Page 203
5.14.2 Bootstrap-t intervals......Page 204
5.14.3 The estimating function bootstrap......Page 206
5.15 Bibliographical notes......Page 209
6.1 Introduction......Page 210
6.2.1 Introduction......Page 212
6.2.2 Projected likelihoods......Page 213
6.3.1 Introduction......Page 215
6.3.2 Generalised projected artificial likelihood ratios......Page 216
6.3.3 Multiple roots......Page 221
6.3.4 Example: estimating the coefficient of variation......Page 223
6.4 Artificial likelihoods through integration......Page 227
6.5.1 Quadratic artificial likelihoods......Page 231
6.5.2 Quadratic artificial likelihood ratio tests......Page 234
6.5.3 Profiled quadratic artificial likelihoods......Page 237
6.5.4 Asymptotic distributions......Page 240
6.5.5 Numerical studies......Page 244
6.6.1 The generalised method of moments......Page 245
6.6.2 Quadratic inference functions......Page 248
6.6.3 Generalised estimating equations......Page 249
6.7 Bibliographical notes......Page 252
7.1 Dynamical estimating systems......Page 254
7.2.1 Liapunov stability, local stability......Page 256
7.2.2 Linear dynamical systems......Page 257
7.3 Stability of roots to estimating equations......Page 262
7.4 A modified Newton's method......Page 268
7.4.1 Newton's method revisited......Page 269
7.4.2 A method with continuity correction......Page 272
7.4.3 A modification based on the information identity......Page 275
7.5.1 Julia sets......Page 277
7.5.2 The correlation coefficient......Page 279
8.1 Introduction......Page 285
8.2 Bayes consistency......Page 286
8.3 A Bayesian approach to estimating functions......Page 288
8.4 Bayes linear and semi-parametric estimation......Page 294
8.5 An application to credibility estimation in actuarial science......Page 297
8.6 Bibliographical notes......Page 299
Bibliography......Page 300
B......Page 310
C......Page 311
E......Page 312
F......Page 313
G......Page 314
K......Page 315
M......Page 316
N......Page 317
P......Page 318
R......Page 319
S......Page 320
W......Page 321
Z......Page 322