On the conditions for the existence of ancillary statistics in a curved exponential family

We present necessary and sufficient conditions for the existence of ancillary statistics in a curved exponential family under some regularity conditions. When they exist, we also provide the method of constructing the ancillary statistics. The usefulness of our results are demonstrated by a lot of examples. Through main results and subsequent examples, we clarify that the group structure generated in a differential geometrical way plays an essential role for the ancillarity in a curved exponential family.