Ancillaries and third order significance. (English) Zbl 0829.62006

Summary: For a variable and parameter of the same dimension, the tangent exponential model [see the first author’s paper, J. Multivariate Anal. 27, No. 1, 181-193 (1988; Zbl 0649.62016)] approximates an asymptotic model to third order in a first derivative neighborhood of the data point and to second order otherwise. For the more usual case of a variable of larger dimension than the parameter, we obtain a unique expression for the third order ancillary distribution as projected to the observed maximum likelihood surface, obtain the tangent directions for a second order ancillary, and then show that third order inference needs only the observed likelihood and the tangent directions for a second order ancillary.
These results are then combined and a unique third order distribution is obtained for testing a component parameter; for the case of a real parameter component a simple expression is obtained for the third order observed significance level.


62A01 Foundations and philosophical topics in statistics


Zbl 0649.62016