A Simple Derivation of r* for Curved Exponential Families
Jens Ledet Jensen · Scandinavian Journal of Statistics · 1997
For curved exponential families we consider modified likelihood ratio statistics of the form rL=r+log(u/r)/r, where r is the signed root of the likelihood ratio statistic. We are testing a one‐dimensional hypothesis, but in order to specify approximate ancillary statistics we consider the test as one in a series of tests. By requiring asymptotic independence and asymptotic normality of the test statistics in a large deviation region there is a particular choice of the statistic u which suggests itself. The derivation of this result is quite simple, only involving a standard saddlepoint approximation followed by a transformation. We give explicit formulas for the statistic u, and include a discussion of the case where some coordinates of the underlying variable are lattice.