Maximum Likelihood Estimation for MA(1) Processes with a Root on or near the Unit Circle
研究了MA(1)模型中移动平均参数θ等于或接近1时的最大似然估计，首次推导了θ_LM的极限分布，并发现该渐近分布对小样本和远离单位圆的参数也出奇准确。
This paper considers maximum likelihood estimation for the moving average parameter θ in an MA(1) model when θ is equal to or close to 1. A derivation of the limit distribution of the estimate θ LM , defined as the largest of the local maximizers of the likelihood, is given here for the first time. The theory presented covers, in a unified way, cases where the true parameter is strictly inside the unit circle as well as the noninvertible case where it is on the unit circle. The asymptotic distribution of the maximum likelihood estimator sub MLE is also described and shown to differ, but only slightly, from that of θ LM . Of practical significance is the fact that the asymptotic distribution for either estimate is surprisingly accurate even for small sample sizes and for values of the moving average parameter considerably far from the unit circle.