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

如果一个离散时间平稳随机过程的相关性像滞后的幂函数一样降低到零，那么它就被认为是长期依赖的，因为滞后趋于无穷大，但是它们的和发散。该提案的第一部分提出研究三类鲁棒估计器的大样本行为，并在误差与可能未知的联合分布长范围相关的非线性回归模型中开发渐近有效和自适应估计器。本文的第二部分研究了几种鲁棒估计的渐近行为，并在半参数设置下，在存在回归分量的非线性时间序列中，开发了渐近最优推理程序。特别是，P.I.计划在随机系数和阈值自回归模型中开发渐近有效和自适应估计器，当这些模型中可能存在线性或非线性的回归变量，并且当随机系数和误差变量的分布未知时。用Hajek - Le Cam理论推导出最优估计量。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续存在的数据集称为长距离依赖数据集。这些数据经常出现在天文学、经济学、地球物理学、水文学、气象学和许多其他学科中。1963年至1975年，美国国家标准局对1公斤检查标准重量进行的289次高精度测量数据就是一个很好的例子。尽管有保持独立性的理想条件，但结果证明观测结果是长期依赖的。该建议的第一部分涉及开发一些最佳统计程序，用于分析存在协变量的长期依赖数据。该建议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，对模型偏差不太敏感。

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.