摘要：Box and Pierce (1970) proposed a test statistic TBP which is the squared sum of m sample autocorrelations of the estimated residual process of an autoregressive-moving average model of order (p,q). TBP is called the classical portmanteau test. Under the null hypothesis that the autoregressive-moving average model of order (p,q) is adequate, they suggested that the distribution of TBP is approximated by a chi-square distribution with (m-p-q) degrees of freedom, ``if m is moderately large". This paper shows that TBP is understood to be a special form of the Whittle likelihood ratio test TPW for autoregressive-moving average spectral density with m -dependent residual processes. Then, it is shown that, for any finite m , TPW does not converge to a chi-square distribution with (m-p-q) degrees of freedom in distribution, and that if we assume Bloomfield's exponential spectral density, TPW is asymptotically chi-square distributed for any finite m . From this observation we propose a modified TPW which is asymptotically chi-square distributed. In view of the likelihood ratio, we also mention the asymptotics of a natural Whittle likelihood ratio test TWLR which is always asymptotically chi-square distributed. Its local power is also evaluated. Numerical studies illuminate interesting features of TPW , TPW , and TWLR . Because many versions of the portmanteau test have been proposed and been used in a variety of fields, our systematic approach for portmanteau tests and proposal of tests will give another view and useful applications.
关键词：ARMA model;Exponential spectral model;LAN;local power;Portmanteau test;Time series model checking;Whittle likelihood
