Asymptotics in Statistics

by Sara van de Geer. Also, we did not include material due to David Donoho, lain Johnstone, and their school. We found our selves unprepared to write a distillate of the material. We did touch briefly on "nonparametrics," but not on "semiparamet rics." This is because we feel that the semiparametric situation has not yet been properly structured. We hope that the reader will find this book interesting and challenging, in spite of its shortcomings. The material was typed in LaTeX form by the authors them selves, borrowing liberally from the 1990 script by Chris Bush. It was reviewed anonymously by distinguished colleagues. We thank them for their kind encouragement. Very special thanks are due to Professor David Pollard who took time out of a busy schedule to give us a long list of suggestions. We did not follow them all, but we at least made attempts. We wish also to thank the staff of Springer-Verlag for their help, in particular editor John Kimmel, who tried to make us work with all deliberate speed. Thanks are due to Paul Smith, Te-Ching Chen and Ju-Yi-Yen, who helped with the last-minute editorial corrections.

1 Introduction. - 2 Experiments, Deficiencies, Distances v. - 2. 1 Comparing Risk Functions. - 2. 2 Deficiency and Distance between Experiments. - 2. 3 Likelihood Ratios and Blackwell's Representation. - 2. 4 Further Remarks on the Convergence of Distri butions of Likelihood Ratios. - 2. 5 Historical Remarks. - 3 Contiguity Hellinger Transforms. - 3. 1 Contiguity. - 3. 2 Hellinger Distances, Hellinger Transforms. - 3. 3 Historical Remarks. - 4 Gaussian Shift and Poisson Experiments. - 4. 1 Introduction. - 4. 2 Gaussian Experiments. - 4. 3 Poisson Experiments. - 4. 4 Historical Remarks. - 5 Limit Laws for Likelihood Ratios. - 5. 1 Introduction. - 5. 2 Auxiliary Results. - 5. 3 Limits for Binary Experiments. - 5. 4 Gaussian Limits. - 5. 5 Historical Remarks. - 6 Local Asymptotic Normality. - 6. 1 Introduction. - 6. 2 Locally Asymptotically Quadratic Families. - 6. 3 A Method of Construction of Estimates. - 6. 4 Some Local Bayes Properties. - 6. 5 Invariance and Regularity. - 6. 6 The LAMN and LAN Conditions. - 6. 7 Additional Remarks on the LAN Conditions. - 6. 8 Wald's Tests and Confidence Ellipsoids. - 6. 9 Possible Extensions. - 6. 10 Historical Remarks. - 7 Independent, Identically Distributed Observations. - 7. 1 Introduction. - 7. 2 The Standard i. i. d. Case: Differentiability in Quadratic Mean. - 7. 3 Some Examples. - 7. 4 Some Nonparametric Considerations. - 7. 5 Bounds on the Risk of Estimates. - 7. 6 Some Cases Where the Number of Observations Is Random. - 7. 7 Historical Remarks. - 8 On Bayes Procedures. - 8. 1 Introduction. - 8. 2 Bayes Procedures Behave Nicely. - 8. 3 The Bernstein von Mises Phenomenon. - 8. 4 A Bernstein von Mises Result for the i. i. d. Case. - 8. 5 Bayes Procedures Behave Miserably. - 8. 6 Historical Remarks. - Author Index.