Asymptotic expansions for general statistical models. With the assist. of W. Wefelmeyer.

*(English)*Zbl 0578.62003
Lecture Notes in Statistics, 31. Berlin etc.: Springer-Verlag. VII, 505 p. DM 79.00 (1985).

In his book ”Contributions to a general asymptotic statistical theory.” Lect. Notes Stat. 13 (1982; Zbl 0512.62001), the author tried to describe the local structure of a general family \({\mathcal P}\) of probability measures by its tangent space and the local behavior of a functional \(\kappa: P\to R^ k\) by its gradient and obtained asymptotic envelope power functions for tests and asymptotic bounds for the concentration of estimators, when statistical procedures are based on independent, identically distributed observations.

As a continuation, in this book the author considers these asymptotic investigations not by limit distributions but by Edgeworth expansions (adding one term of order \(n^{-1/2}\) to the limit distributions).

To do this, the author introduces concepts of ”degenerate convergence condition of order (b,c) (for short: \(DCC_{b,c})''\) and \(''DCC_{b,c}\)-differentiable with derivative g at a rate \(o(t^ a)''\), and with the aid of ”canonical gradients” he studies the second order envelope power functions for composite hypotheses, second order bounds for the concentration of confidence bounds and median unbiased estimators, etc.

Then, using these results, he considers various general statistical procedures and gives bounds of order \(o(n^{-1/2})\) for the efficiency of them. Many examples and explanations of the new notions are given. It seems that many problems are still open.

As a continuation, in this book the author considers these asymptotic investigations not by limit distributions but by Edgeworth expansions (adding one term of order \(n^{-1/2}\) to the limit distributions).

To do this, the author introduces concepts of ”degenerate convergence condition of order (b,c) (for short: \(DCC_{b,c})''\) and \(''DCC_{b,c}\)-differentiable with derivative g at a rate \(o(t^ a)''\), and with the aid of ”canonical gradients” he studies the second order envelope power functions for composite hypotheses, second order bounds for the concentration of confidence bounds and median unbiased estimators, etc.

Then, using these results, he considers various general statistical procedures and gives bounds of order \(o(n^{-1/2})\) for the efficiency of them. Many examples and explanations of the new notions are given. It seems that many problems are still open.

Reviewer: K.Yoshihara

##### MSC:

62A01 | Foundations and philosophical topics in statistics |

62F05 | Asymptotic properties of parametric tests |

62F12 | Asymptotic properties of parametric estimators |

62-02 | Research exposition (monographs, survey articles) pertaining to statistics |