Edgeworth and saddlepoint approximations in the first-order noncircular autoregression

The first-order noncircular autoregressive model is considered and Edgeworth and saddlepoint approximations are given for the distribution of the least squares estimator of the autoregressive coefficient. Numerical calculations are used to compare the two approximations with the exact distribution, found by numerical integration. Both approximations are unsatisfactory when the autoregressive coefficient is moderately large and the sample size small, the saddlepoint because it is undefined in the tail area, and the Edgeworth because it badly distorts tail area probabilities. When the sample size is larger, the saddlepoint approximation performs well and is capable of three decimal place accuracy, although it is still not available in tail areas when the autoregressive coefficient is large.