A Note on the Saddlepoint Approximation in the First Order Non-Circular Autoregression

In approximating the small sample distribution of the least squares estimator of the coefficient in a non-circular autoregression the saddlepoint method is complicated by the presence of a branch point of the integrand within the natural contour of integration. Some new approximations are given based on a contour looping around the branch point; and a uniform approximation which is valid as the saddlepoint crosses branch point is also developed.