Goodness-of-fit for a branching process with immigration using sample partial autocorrelations

A limit theorem is developed for sample partial autocorrelations, when the vector {N1/2(R(k)-mk), K=1,...,H} converges in distribution, the R(k) being sample autocorrelations from a not-necessarily stationary process. The result is used to develop a Quenouille-type goodness-of-fit test based on sample partial autocorrelations for the simple branching process with immigration. This is compared with a test of Venkataraman (1982); and both are applied to historical data.