Limited distribution of sample partial autocorrelations: A matrix approach
Let \(X(1),\dots ,X(N)\) be a realization of a process \(\{X(t)\}\). Define \[ C(k)=\sum _{t=k+1}^N [X(t)-\bar X][X(t-k)-\bar X] \text{ for } 0\leq k\leq N-1, C(-k)=C(k), \] and \(R(k)=C(k)/C(0)\). Introduce the matrices \(L_k=(R(i-j))_{i=1}^{\hskip 1.5mm k} {}_{j=1}^{\phantom {i.}k}\), \(Q_k=(R(i-j))_{i=1}^{\hskip 1.5mm k} {}_{j=1}^{k-1}\), and the vector \(r_k= [R(1), \dots ,R(k)]'\). The author defines the sample partial autocorrelation at lag \(k\) for \(\{X_t\}\) by \(\hat \phi _k =\hbox {det }(U_k)/\hbox {det }(L_k)\), \(k\geq 2\), where \(U_k=(Q_k,r_k)\). Of course, \(\hat \phi _1=R(1)\). A method for the derivation of the asymptotic distribution of the sample partial autocorrelations, given the distribution of sample autocorrelations \(R(t)\), is presented. A theoretical procedure is applied to the process, the autocorrelation function \(\rho (k)\) of which satisfies \(\rho (k)-a\rho (k-1)=0\) for \(k\geq 2\) with \(|a |<1\). Such a process is called an ARMA\((1,1)\)-like process. The methods presented in the paper are not based on the assumption of asymptotic normality and the process \(\{X_t\}\) need not be stationary.
scientific article; zbMATH DE number 1409154