On the asymptotic normality and independence of the sample partial autocorrelations for an autoregressive process

摘要

For a stationary autoregressive model of order s, the partial autocorrelation coefficients of order j, j=0,1,2,…,s−1, are defined; the partial autocorrelation coefficient of order zero being the same as the autocorrelation coefficient of order one. Denoting these s parameters by ϱ1,π1,…,πs−1, it is shown that their sample images, namely r1,P1,…,Ps−1, are asymptotically independently normally distributed with means equal to the corresponding population values and asymptotic variances given by var(r1)=(1 − ϱ21)(1 − π21⋯(1 − π2s −1)n, var(Pj)=(1 − π2j(1 − π2j+1)⋯(1 − π2s −1)n, j=1,2,…,s−1, where n is the size of the sample from the autoregressive process of order s. The partial correlogram of the model and application of the result are discussed.