The approximate distribution of partial serial correlation coefficients calculated from residuals from regression on Fourier series

Approximations are found to the joint distributions of noncircular and circular partial serial correlation coefficients calculated from residuals from regression on Fourier series. Results for coefficients calculated from deviations from the true mean and from the sample mean are obtained as special cases. It is shown that when the observations are independent the partial coefficients are approximately independently distributed in beta distributions that are the same for all odd-order coefficients and the same for all even-order coefficients. The approximations are of third-order accuracy in the sense that the error is of order n− / . They were obtained by the technique developed in another paper (Durbin, 1980).