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

一个离散时间平稳随机过程被称为长程相关的，如果它的相关性像滞后的幂一样减少到零，因为滞后趋于无穷大，但它们的和发散。 建议的第一部分提出了三个类别的稳健估计的大样本行为进行调查，并制定渐近有效的和自适应估计的非线性回归模型时，误差是长程相关的可能未知的联合分布。本文的第二部分研究了在半参数条件下，存在回归分量的非线性时间序列中几类稳健估计的渐近性态，并发展了渐近最优推断方法。特别是，P.I.当随机系数和门限自回归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，计划开发随机系数和门限自回归模型中的渐近有效和自适应估计。最优估计量将根据Hajek-Le Cam理论得到. 当观测值之间的距离增加时，远距离观测值之间的关联缓慢衰减但持续存在的数据集称为长距离相关。这些数据经常出现在天文学、经济学、物理学、水文学、气象学和许多其他学科中。值得一提的一个例子是美国国家标准局在1963年至1975年期间对1公斤检查标准砝码进行的289次高精度测量的数据。尽管有保持独立性的理想条件，但观察结果是长距离依赖的。第一部分的建议是关于发展一些最佳的统计程序，分析长期依赖的数据中存在的协变量。第二部分的建议是关于发展有效的推理程序，在一些复杂的时间序列模型，经常出现在计量经济学。预计这些程序将广泛适用，对模型偏离不太敏感。

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.