A Second-Order Investigation of Asymptotic Ancillarity

Abstract

The paper deals with approximate ancillarity as discussed by Efron and Hinkley (1978). In the multivariate i.i.d. case we derive the second-order Edgeworth expansion of the MLE given a normalized version of the second derivative of the log-likelihood at its maximum. The expansion agrees with the one derived by Amari (1982a) for curved exponential families, but holds for any family satisfying the regularity conditions given in the paper. It is shown that the Fisher information lost by reducing the data to the MLE is recovered by the conditioning, and it is sketched how the loss of information relates to the deficiency as defined by LeCam. Finally, we investigate some properties of three test statistics, proving a conjecture by Efron and Hinkley (1978) concerning the conditional null-distribution of the likelihood ratio test statistic, and establishing a kind of superiority of the observed Fisher information over the expected one as estimate of the inverse variance of the MLE.