The approximate distribution of serial correlation coefficients

Hotelling's suggestion of a 'circular' definition for the serial correlation coefficient was followed by considerable progress in the distribution theory of such modified statistics by R. L. Anderson (1942), Koopmans (1942), Dixon (1944), Madow (1945) and others. The exact distribution is known for the circular coefficient of any lag from an uncorrelated normal process and, more generally, from a circularly modified normal process of autoregressive type. Quenouille (1949) obtained by the same method the exact joint distribution of circular coefficients of different lags. The exact distributions are complicated, and a simple and accurate approximation to the distribution of the circular coefficient with known mean was found by Dixon (1944) and Rubiin (1945) for the uncorrelated normal process. It was extended to the case of a circular Markov process by Leipnik (1947) following a method due to Madow (1945). The approximation depends on the device of smoothing summation over a discrete set of roots by an approximating integration. Quenouille (1949) conjectured a similar approximate form for the joint distribution, but Watson (1951) and Jenkins (1954) showed that the conjectured form could not be correct. Jenkins developed the correct analogous approximation for the joinlt distribution of coefficients of lags 1 and 2 with known means. Without circular modifications the distributional theory is difficult and the field is largely unexplored. For testing independence T. W. Anderson (1948) gave an approximate table of significance points for non-circular lag 1 coefficients with known and fitted means. Watson & Durbin (1951) introduced modified non-circular definitions of the coefficients which have R. L. Anderson's distribution in the uncorrelated case. The case of an unmodified autoregressive process has not been much discussed, though a method due to Bartlett (1953, 1954) is available for obtaining approximate confidence intervals for the parameters, and Quenouille's (1947) approximate goodness of fit tests should be noted. In the present paper an approach based on the method of steepest descents is adopted to derive the known approximate distributions and to generalize them.* (For an account of the method see, for example, Jeffreys & Jeffreys (1956), Daniels (1954).) The analogue of Leipnik's approximation is found for the distribution of an unmodified coefficient of lag 1, both with known and fitted mean, when the process is of unmodified Markov type. The approximate joint distribution of m successive partial serial correlations is found for an autoregressive process of the mth order, circular modifications being used. The work on the unmodified Markov process could be extended to the general case but we have not done this.