Preferred point geometry and statistical manifolds.

*(English)*Zbl 0798.62009The authors introduce a mathematical object of a preferred point geometry. Its definition and use are based on two standpoints: to be as natural and simple as possible from both the statistical and the geometric viewpoint, and to reflect in a natural way the special status of the preferred point. They also discuss the relationship between preferred point geometry and statistical manifolds, often focussing for illustration on the expected geometry of S. Amari [Differential- geometrical methods in statistics. (1985; Zbl 0559.62001)].

Statistically natural preferred point metrics are given for the expected geometry case, where the duality between connections can be interpreted as reflecting a certain duality between the score vector and the maximum likelihood estimator. Duality theorems for arbitrary preferred point manifolds are also obtained. Further, the natural extension of statistical manifolds to higher order is discussed.

Statistically natural preferred point metrics are given for the expected geometry case, where the duality between connections can be interpreted as reflecting a certain duality between the score vector and the maximum likelihood estimator. Duality theorems for arbitrary preferred point manifolds are also obtained. Further, the natural extension of statistical manifolds to higher order is discussed.

Reviewer: M.Akahira (Ibaraki)

##### MSC:

62A01 | Foundations and philosophical topics in statistics |

62F12 | Asymptotic properties of parametric estimators |

53B99 | Local differential geometry |