Abstract: Given a curved exponential family of distribution densities with parameter vector μ which can be divided into a component parameter of interest u and a component of nuisance parameter v.The first order efficient estimator μ can be partitioned in this case into two parts: \hat\mu=(\hatu, \hatv) The differential geometry and Edgeworth expansion approach introduced by Amari are used to derive the asymptotic conditional distribution, expectation and covariance of \hatu \hat\mu in terms of α-connection and α-curvature of the model. We study the problems of the conditional inference related to both the nuisance parameters and the ancillary statistic.If the nuisance parameter does not exist,then our results reduce to those given by Amari.