A test for randomness against ARMA alternatives.
Let \(\{X_ t\}\) be an ARMA\((p,q)\) process with zero mean. Then \(\widehat {\gamma }_ k=N^ {-1}\sum _ {i=1}^ {N-k} X_ i X_ {i+k}\) is an estimator of the covariance function \(\gamma _ k\). Define \[ \widehat {\gamma }(r)=(\widehat {\gamma }_ 1,\dots ,\widehat {\gamma }_ r)',\quad \widehat {\Gamma }_ r=(\widehat {\gamma }_ {i-j})_ {i,j=1}^ r,\quad \widehat T =\widehat {\gamma }'(r)\widehat {\Gamma }_ r^ {-1} \widehat {\gamma }(r)/ \widehat {\gamma }_ 0. \] A test for randomness against ARMA alternatives can be based on the statistic \(\widehat M= -\log (1- \widehat T)\) [see \textit{A. Mokkadem}, Stochastic Processes Appl. 72, 145--159 (1997; Zbl 0936.62101)]. The authors present a simpler proof of Mokkadem's result. Further, it is demonstrated that the asymptotic distribution of the test statistic under the alternative does not depend on the fourth-order moments of the innovations. However, \(\widehat T\) can be also used as a test statistic for testing randomness, since it is nothing else than the \(R^ 2\) associated with the regression of \(X_ t\) on \(X_ {t-1},\dots , X_ {t-r}\). The power of the classical portmanteau test is compared with the power of the test based on \(\widehat T\) in a simulation study.
scientific article; zbMATH DE number 2098345
scientific article