Mathematical Sciences: Optimal Inference in Non-Linear Regression Models with Long Range Dependent Errors and in Non-Linear Time Series

离散时间平稳随机过程称为长程相关，如果它的相关性像滞后的幂一样减小到零，因为滞后趋于无穷大，但它们的和发散。建议的第一部分建议研究三类稳健估计的大样本行为，并在误差具有可能未知的联合分布的情况下，在非线性回归模型中发展渐近有效的自适应估计。第二部分研究了几类稳健估计的渐近行为，并在半参数设置下发展了非线性时间序列中回归分量存在时的渐近最优推断过程。特别是，P.I.计划在随机系数和门限自回归模型中开发渐近有效的自适应估计器，当这些模型中可能存在线性或非线性回归变量时，以及当随机系数和误差变量的分布未知时。最优估计器将建立在L的哈耶克-勒卡姆理论之上。当远距离观测之间的关联随着观测距离的增加而缓慢衰减但持久存在的数据集称为长距离相关。这样的数据经常出现在天文学、经济学、地球物理学、水文学、气象学等许多学科中。值得一提的是，美国国家标准局在1963至1975年间对1公斤标准重量进行了289次高精度测量。尽管有保持独立性的理想条件，但事实证明，观测结果是长期相关的。该提案的第一部分涉及制定一些最佳统计程序，以便在存在协变量的情况下分析长期相关数据。该提案的第二部分涉及在计量经济学中经常出现的一些复杂的时间序列模型中开发有效的推理程序。预计这些程序将广泛适用，对模型偏离不是很敏感。

A discrete time stationary stochastic process is said to be long range dependent if its correlations decrease to zero like a power of the lag, as the lag tends to infinity, but their sum diverges. Part I of the proposal proposes to investigate the large sample behavior of three classes of robust estimators and to develop asymptotically efficient and adaptive estimators in non-linear regression models when the errors are long range dependent with possibly unknown joint distributions. The Part II of the proposal is concerned with studying the asymptotic behavior of several classes of robust estimators and devloping asymptoticlaly optimal inference procedures in non-linear time series in the presence of regression component, in a semi-parametric setup. In particular, the P.I. plans to develop asymptotically efficient and adaptive estimators in random coefficient and threshold autoregression models when there may be a regression variable present in these models, in a llinear or non-linear fashion, and when the distributions of the random coefficient and the error variable are unknown. The optimal estimators would be developed a l a Hajek - Le Cam theory. A data set where an association between distant observations is slowly decaying but persistent, as distance between observations increases, is called long range dependent. Such data arise often in astronomy, economics, geophysics, hydrology, meteorology, and many other disciplines. An example worth mentioning is the data of 289 high-precision measurements on the 1-kg check standard weight made between 1963 to 1975 by the U.S. National Bureau of Standards. In spite of ideal conditions for preserving independence, the observations turned out to be long range dependent. The first part of the proposal is concerned with developing some optimal statistical procedures for analyzing the long range dependent data in the presence of a covariate. The second part of the proposal is concerned with developing efficient inferential procedures in some complicated time series models that often arise in econometrics. It is anticipated that these procedures will be broadly applicable and not very sensitive to model departures.