Asymptotic methods provide important tools for approximating and analysing functions that arise in probability and statistics. Moreover, the conclusions of asymptotic analysis often supplement the conclusions obtained by numerical methods. Providing a broad toolkit of analytical methods, Expansions and Asymptotics for Statistics shows how asymptotics, when coupled with numerical methods, becomes a powerful way to acquire a deeper understanding of the techniques used in probability and statistics.
The book first discusses the role of expansions and asymptotics in statistics, the basic properties of power series and asymptotic series, and the study of rational approximations to functions. With a focus on asymptotic normality and asymptotic efficiency of standard estimators, it covers various applications, such as the use of the delta method for bias reduction, variance stabilisation, and the construction of normalising transformations, as well as the standard theory derived from the work of R.A. Fisher, H. CramГ©r, L. Le Cam, and others. The book then examines the close connection between saddle-point approximation and the Laplace method. The final chapter explores series convergence and the acceleration of that convergence.
Author(s): Christopher G. Small
Series: Chapman & Hall/CRC Monographs on Statistics & Applied Probability
Publisher: CRC
Year: 2010
Language: English
Pages: 339
Tags: Математика;Теория вероятностей и математическая статистика;Математическая статистика;
Expansions and Asymptotics for Statistics......Page 6
Front Cover......Page 1
Contents......Page 8
Preface......Page 12
1.1 Expansions and approximations......Page 16
1.2 The role of asymptotics......Page 18
1.3 Mathematical preliminaries......Page 19
1.4 Two complementary approaches......Page 31
1.5 Problems......Page 33
2.1 A Quick overview......Page 37
2.2 Power series......Page 38
2.3 Enveloping series......Page 54
2.4 Asymptotic series......Page 61
2.5 Superasymptotic and hyperasymptotic series......Page 77
2.6 Asymptotic series for large samples......Page 80
2.7 Generalised asymptotic expansions......Page 82
2.9 Problems......Page 83
3.1 The Padè table......Page 89
3.2 Padè approximations for the exponential function......Page 93
3.3 Two Applications......Page 95
3.4 Continued fraction expansions......Page 99
3.5 A continued fraction for the normal distribution......Page 102
3.6 Approximating transforms and other integrals......Page 104
3.7 Multivariate extensions......Page 106
3.8 Notes......Page 107
3.9 Problems......Page 108
4.1 Introduction to the delta method......Page 112
4.2 Preliminary results......Page 113
4.3 The Delta method for moments......Page 116
4.4 Using the delta method in Maple......Page 121
4.5 Asymptotic bias......Page 122
4.6 Variance stabilising transformations......Page 124
4.7 Normalising transformations......Page 127
4.8 Parameter transformations......Page 129
4.10 Ratios of averages......Page 132
4.11 The delta method for distributions......Page 134
4.12 The von Mises calculus......Page 136
4.13 Obstacles and opportunities: robustness......Page 147
4.14 Problems......Page 150
5.1 Historical overview......Page 155
5.2 The organisation of this chapter......Page 163
5.3 The likelihood function and its properties......Page 164
5.4 Consistency of maximum likelihood......Page 171
5.5 Asymptotic normality of maximum likelihood......Page 173
5.6 Asymptotic comparison of estimators......Page 176
5.7 Local asymptotics......Page 183
5.8 Local asymptotic normality......Page 189
5.9 Local asymptotic minimaxity......Page 193
5.10 Various extensions......Page 197
5.11 Problems......Page 199
6.1 A simple example......Page 204
6.2 The basic approximation......Page 206
6.3 The Stirling series for factorials......Page 211
6.4 Laplace expansions in Maple......Page 212
6.5 Asymptotic bias of the median......Page 213
6.6 Recurrence properties of random walks......Page 216
6.7 Proofs of the main propositions......Page 218
6.8 Integrals with the maximum on the boundary......Page 222
6.9 Integrals of higher dimension......Page 223
6.10 Integrals with product integrands......Page 226
6.11 Applications to statistical inference......Page 230
6.12 Estimating location parameters......Page 231
6.13 Asymptotic analysis of bayes estimators......Page 233
6.15 Problems......Page 234
7.1 The principle of stationary phase......Page 237
7.2 Perron's saddle-point method......Page 239
7.3 Harmonic functions and saddle-point geometry......Page 244
7.4 Daniels' saddle-point approximation......Page 248
7.5 Towards the Barndorff-Nielsen formula......Page 251
7.6 Saddle-point method for distribution functions......Page 261
7.7 Saddle-point method for discrete variables......Page 263
7.8 Ratios of sums of random variables......Page 264
7.9 Distributions of M-estimators......Page 266
7.10 The Edgeworth expansion......Page 268
7.11 Mean, median and mode......Page 272
7.12 Hayman's saddle-point approximation......Page 273
7.13 The method of Darboux......Page 278
7.14 Applications to common distributions......Page 279
7.15 Problems......Page 284
8.1 Advanced tests for series convergence......Page 288
8.2 Convergence of random series......Page 294
8.3 Applications in probability and statistics......Page 295
8.4 Euler-Maclaurin sum formula......Page 300
8.5 Applications of the Euler-Maclaurin formula......Page 304
8.6 Accelerating series convergence......Page 306
8.7 Applications of acceleration methods......Page 318
8.8 Comparing acceleration techniques......Page 322
8.9 Divergent series......Page 323
8.10 Problems......Page 325
9. Glossary of symbols......Page 330
10. Useful limits, series and products......Page 334
References......Page 336
