A Simple Derivation of r* for Curved Exponential Families

For curved exponential families we consider modified likelihood ratio statistics of the form r_L=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.