On the distributions of the differences between the estimated autocorrelations and partial autocorrelations at the same lag

Pukkila (1982) introduced a new portmanteau-type test of white noise. The test is based on the differences between the estimated autocorrelations and the corresponding partial autocorrelations. In the present paper the distributions of these differences are consider in the case of normal white noise. Approximations to these distributions are given. Goodness of fit of these theoretical distributions and the corresponding observed distributions is studied using simulated time series of various lengths.