ANCILLARIES AND 3RD-ORDER SIGNIFICANCE

For a variable and parameter of the same dimension, the tangent exponential model (Fraser, 1988) 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.