We consider a time-varying first-order autoregressive model with irregular innovations, where we assume that the coefficient function is H\"{o}lder continuous. To estimate this function, we use a quasi-maximum likelihood based approach. A precise control of this method demands a delicate analysis of extremes of certain weakly dependent processes, our main result being a concentration inequality for such quantities. Based on our analysis, upper and matching minimax lower bounds are derived, showing the optimality of our estimators. Unlike the regular case, the information theoretic complexity depends both on the smoothness and an additional shape parameter, characterizing the irregularity of the underlying distribution. The results and ideas for the proofs are very different from classical and more recent methods in connection with statistics and inference for locally stationary processes.
翻译:我们考虑的是具有不规则创新的具有时间变化的第一阶自动递减模式,我们假设系数函数是 H\"{o}lder 连续的。为了估算这一函数,我们使用准最大可能性法。精确地控制这一方法要求对某些依赖性弱的极端过程进行微妙的分析,我们的主要结果就是这种数量的集中不平等。根据我们的分析,得出了上下限和相匹配的最小最大下限,显示了我们测量者的最佳性。与通常的情况不同,信息理论复杂性取决于光滑和额外的形状参数,说明基本分布的不规律性。证据的结果和想法与传统的和较近期的方法在统计和当地固定过程的推论方面大不相同。