APPROXIMATE MOMENTS TO O(N(-2)) FOR THE SAMPLED PARTIAL AUTOCORRELATIONS FROM A WHITE-NOISE PROCESS

We obtain first approximations to all the moments of the low-lag sampled partial autocorrelations, for finite-lengthed realisations from a Gaussian white noise process: and indicate how closer approximations could be achieved. The method is to write products. or powers, of the partial correlations as expansions of the serial correlations, cutting off an expansion at a sufficiently small order term. Formulae for the moments of the partials then depend on knowledge of the expectations for those products, and powers, of the serial correlations which appear in the truncated expansions. We give a small simulation example of how our main result, a first-order approximation for the expectation of a white noise partial correlation, could prove useful in significance testing; and suggest that this rough result, for the mean, might be robust to quite substantial departures from normality. The paper concludes with large scale simulations, for the first four moments of the sampled partial autocorrelations, from length-10 realisations for each of Gaussian and negative exponential white noise. As an addendum, simulated first moments are also reported for some smaller simulations of longer series (n = 20 and 50) from these two types of white noise.