Approximate distributions for the various serial correlograms

Saddlepoint methods are used to approximate the joint density of the serial correlogram up to lag m. Jacobian transformations also lead to approximati ons for the related-partial correlogram and inverse correlogram. The approx imations consider non-circularly and circularly defined models in both the null and the non-null settings, The distribution theory encompasses the sta ndard non-circularly defined correlogram computed from least-squares residu als removing arbitrary fixed regressors. Connections of the general theory to the approximations given by Daniels and by Durbin in the circular settin g are indicated. The double-saddlepoint density and distribution approximat ions are given for the conditional distribution of the non-circular lag m s erial correlation given the previous lags from order 1 to m - 1. This allow s for the computation of p values in conditional inference when tearing tha t the model is AR(m - 1) versus AR(m). Numerical comparisons with the tests of Daniels and of Durbin suggest that their tests based on circularity ass umptions are inadequate for short non-circular series but are in close agre ement with the non-circular tests for moderately long series.