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**Orthogeodesic models.**
*(English)*
Zbl 0815.62005

Some composite transformation models have properties similar to certain of the properties of exponential models with affine dual foliations. The properties in question are of a geometrical nature. The present paper represents an attempt to give a unified delineation of those properties, in differential geometric terms. More specifically, we define, by purely differential geometric conditions, a class of parametric statistical models which we call orthogeodesic models and which comprises the exponential models with affine dual foliations as well as the composite transformation models that we had noted for their similarity with such exponential models.

Except for general smoothness assumptions, we use five conditions to define an orthogeodesic model. These are stated in Section 3, which also contains a number of examples. Section 4 consists of a discussion of the implications of the four defining conditions; in particular, it contains a result on higher-order asymptotic independence. In Section 5 we study what further properties can be inferred if the model is assumed to be exponential. Section 2 reviews some basic concepts from statistical differential geometry and establishes the notation used throughout the paper.

Except for general smoothness assumptions, we use five conditions to define an orthogeodesic model. These are stated in Section 3, which also contains a number of examples. Section 4 consists of a discussion of the implications of the four defining conditions; in particular, it contains a result on higher-order asymptotic independence. In Section 5 we study what further properties can be inferred if the model is assumed to be exponential. Section 2 reviews some basic concepts from statistical differential geometry and establishes the notation used throughout the paper.

### MSC:

62E10 | Characterization and structure theory of statistical distributions |

62F99 | Parametric inference |

53B99 | Local differential geometry |