Mirror-Image and Invariant Distributions in ARMA Models
研究了ARMA模型中估计量和检验统计量的有限样本分布，发现无截距项时最小二乘和最大似然估计量具有镜像对称或不变性质，F、t、似然比、Wald和拉格朗日乘子统计量也有类似性质，有助于简化或扩展相关研究。
The finite sample distributions of estimators and test statistics in ARMA time series models are generally unknown. For typical sample sizes, the approximations provided by asymptotic distributions are often unsatisfactory. Hence simulation or numerical integration methods are used to investigate the distributions. In practice only a limited part of the parameter space is examined using these methods. Thus any results which allow us to infer properties from one portion of the parameter space to another or to establish symmetry are most welcome. For the ARMA model estimated with no intercept term, we show that the least-squares and maximum likelihood estimators have mirror-image invariant or symmetric distributions. The F, t , likelihood ratio, Wald, and Lagrange multiplier statistics are also shown to have distributions with certain mirror-image invariant or symmetry properties. The analysis is extended to misspecified models as well as to ARMA spectral densities. These properties would have been helpful in either simplifying or extending much earlier work in this area.