Model diagnostics under long memory, and for spatial data

如果离散时间平稳随机过程的自相关性在滞后参数中以双曲线趋于零，则称其具有长期记忆，因为滞后趋于无穷。物理和社会科学中充斥着真实的数据例子，这些例子在条件异方差存在的情况下表现出这种行为，并且回归函数是非线性和非光滑的。该建议的第一部分侧重于开发有用和最优的失配检验，用于在存在异方差和长记忆移动平均(LMMA)误差的情况下，以及当设计是非随机或LMMA时，对非线性和非光滑参数回归函数进行拟合。进一步提出，当误差过程和协变量过程服从某些LMMA模型时，构造有用的最优检验来检验两个或多个回归函数相对于单边或双边备选方案的相等性。建议的第二部分是关于一阶象限自回归过程的稳健推理，一阶象限自回归过程是平面上的单边自回归过程。P.I.建议为参数一阶象限自回归过程的拟合提供一类最小距离检验。此外，假设这样的模型是有效的，P.I.建议发展渐近分布自由检验来拟合误差分布。如果远距离观测之间的关联随着观测之间的距离增加而缓慢衰减但持续存在，则数据集被称为具有长期记忆力。在一段时间内观察到的数据集称为时间序列。异方差时间序列是指在给定过去的情况下，观测值在当前时间的条件变异性取决于过去。这样的数据通常出现在经济、金融、水文学和物理科学中。特别是，长记忆异方差时间序列的一个重要例子是现货收益的波动性。该建议的部分重点是在一类非光滑、非线性、异方差的时间序列模型中开发最优推理过程。对许多农业和环境现象的实际建模涉及到空间相关性。分析空间相关性的一个有用模型是单边自回归时间序列，也称为一阶象限自回归过程。当有证据表明平面上有一个方向的空间运动时，这种类型的过程特别合适，例如随着风或洋流输送的环境污染物，或随着疾病的传播。空间时间序列的某些分数差是一阶象限自回归的模型，已被发现在模拟二维农业田间试验中产量之间相关性的缓慢衰减时是有用的。这项提案的部分重点是为这些模型中的基本参数制定有用和可靠的推断程序，并将其应用于农业和环境科学。

A discrete time stationary stochastic process is said to have long memory if its auto-correlations tend to zero hyperbolically in the lag parameter, as the lag tends to infinity. Physical and social sciences are full of realdata examples that exhibit this behavior in the presence of conditionalheteroscedasticity and where regression functions are nonlinear and non-smooth. The first part of the proposal focuses on developing useful and optimal lack-of-fit tests for fitting a nonlinear and non-smooth parametric regression function in the presence of heteroscedastic and long memory moving average (LMMA) errors, and when designs are either non-random or LMMA. It is further proposed to construct useful and optimal tests for testing the equality of two or more regression functions against one or two sided alternatives, when the error and the covariate processes follow some LMMA models. The second part of proposal is concerned with developing robust inference for a first order quadrant autoregressive process, a process that is a unilateral autoregressive process in the plane. P.I. proposes to provide a class of minimum distance tests for fitting a parametric first order quadrant autoregressive process. In addition, assuming such a model is valid, P.I. proposes to develop asymptotically distribution free tests for fitting an error distribution. A data set is said to have long memory if an association between distant observations is slowly decaying but persistent, as the distance between observations increases. A data set observed over a period of time is calleda time series. A heteroscedastic time series is one where the conditional variability of an observation at the current time, given the past, depends on the past. Such data often arises in economics, finance, hydrology, andphysical sciences. In particular, an important example of long memory heteroscedastic time series is the volatility of spot returns. Part of the emphasis of the proposal is on developing optimal inferential procedures in a class of non-smooth non-linear heteroscedastic time series models. Practical modelling of numerous agricultural and environmental phenomenon involve spatial correlations. A useful model for analyzing spatial correlations is a unilateral autoregressive time series, also known as a first-order quadrant autoregressive process. This type of processes is especially appropriate when there is an evidence of a spatial movement over the plane in one direction, such as with environmental pollutants transported by winds or ocean currents, or with the spread of a disease. A model where certain fractional differences of a spatial time series are first-order quadrant autoregressive has been found useful in modelling the slow decay of correlations between yields in two dimensional agricultural field trials. Part of the focus of this proposal is to develop useful and robust inference procedures for the underlying parameters in these models with applications to agriculture and environmental science.