*(English)*Zbl 0826.62004

This book treats the higher order theory based on likelihood and pseudo- likelihood quantities. The first four chapters are explanations on key concepts such as likelihood, ancillarity, transformation models, etc., including many illustrative examples. The following chapters are the main parts of this book. Specially, the so-called ${p}^{*}$-formula is discussed. The ${p}^{*}$-formula has the property of giving results for both exponential models and transformation models and from the ${p}^{*}$- formula it is possible to obtain third order directional refinements for the score and profile score statistics.

The discussion emphasizes concepts and techniques rather than precise mathematical verifications.

This book will stimulate further developments both in asymptotic parameter inference from likelihood-based perspectives and in the study of differential geometric methods in statistics.

##### MSC:

62A01 | Foundations and philosophical topics in statistics |

62-02 | Research monographs (statistics) |

62F12 | Asymptotic properties of parametric estimators |