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：我们考虑一个含不规则新息的时变一阶自回归模型，其中假设系数函数满足Hölder连续性。为估计该函数，我们采用基于拟极大似然的方法。对该方法的精确控制需要对某些弱相依过程的极值进行细致分析，而我们的主要结果是针对此类量值的浓度不等式。基于分析结果，我们推导出上界和匹配的极小化最大下界，证明了估计量的最优性。与常规情形不同，信息论复杂度不仅取决于光滑性，还取决于刻画基础分布不规则性的额外形状参数。相关结果和证明思路与经典及近年关于局部平稳过程的统计推断方法截然不同。