ON THE JOINT DISTRIBUTION OF THE CIRCULAR SERIAL CORRELATION COEFFICIENTS

Abstract

Quenouille (1949) applied Koopmans's method (1942) to the problem of finding the exact joint distribution of the serial correlation coefficients in the null case. Madow's (1945) device was used to derive the non-null joint density function. For odd sample sizes, lie gave an explicit formula for the exact density function. Since the direct smoothing of this function is very difficult, Quenouille conjectured a smooth approximation to the exact density and supported his conjecture by various arguments about the expected form of the result. Jenkins (1954), following Dixon's (1944) method, has found a smoothed density for the first two serial correlation coefficients which, unlike Quenouille's density, has the correct moments up to order n. These smoothed densities were derived for the csase of no mean corrections.$ In ? 2 of the present paper, von Neumann's (1941) method is generalized to give integral equations for the joint density of the serial correlations. From these the relationship of the null and non-null densities is easily derived. In ? 3 the exact density of the circular serial correlations is obtained for arbitrary sample sizes, and a fuller discussion is given of the interesting summation rule involved. In ? 4 an attempt is made to derive an approximate null density when mean corrections are made by a new smoothing method. The form found differs only slightly from Quenouille's and is subject to the same weaknesses.