On the sufficient statistics for multivariate ARMA models: approximate approach

This paper is an investigation on the sufficient statistic for the parameters of the vector-valued (multivariate) ARMA models, when a finite sample is available. In the simplest case ARMA(1,1), by using the factorization theorem, we present a sufficient statistic whose dimension depends on the sample size and this dimension is even larger than the sample size. In this case and under some restrictions, we have solved this problem and have presented a sufficient statistic whose dimension does not depend on the sample size. In the general case, due to the complexity of the problem, we will use the modified versions of the likelihood function to find an approximate sufficient statistic in terms of the periodogram. The dimension of this sufficient statistic depends on the sample size; however, this dimension is much lower than the sample size.