Model diagnostics under long memory, and for spatial data

一个离散时间平稳随机过程被称为具有长记忆，如果它的自相关在滞后参数中双曲地趋于零，因为滞后趋于无穷大。物理和社会科学中有大量的真实数据例子，在存在条件异方差的情况下，以及回归函数是非线性和非光滑的情况下，这些数据都表现出这种行为。该提案的第一部分侧重于开发有用的和最佳的拟合不足测试，用于在存在异方差和长记忆移动平均（LMMA）误差的情况下拟合非线性和非光滑参数回归函数，以及当设计是非随机或LMMA时。当误差和协变量过程遵循某些LMMA模型时，本文进一步提出了构造检验两个或多个回归函数对单侧或双侧替代品的相等性的有用的和最优的检验。建议的第二部分是关于发展一阶象限自回归过程的鲁棒推理，该过程是平面上的单边自回归过程。P.I.提出了一类拟合参数一阶象限自回归过程的最小距离检验方法。此外，假设这样的模型是有效的，P. I.建议开发渐近分布自由测试拟合误差分布。如果随着观测之间的距离增加，远距离观测之间的关联缓慢衰减但持续存在，则数据集被称为具有长记忆。在一段时间内观察到的数据集被称为时间序列。异方差时间序列是指给定过去，当前时间的观测值的条件变异性取决于过去。这样的数据经常出现在经济学、金融学、水文学和物理学中。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率的波动性。该建议的部分重点是在一类非光滑非线性异方差时间序列模型中开发最佳推理程序。许多农业和环境现象的实际建模涉及空间相关性。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。 当有证据表明飞机在一个方向上有空间运动时，这种类型的过程特别合适，例如风或洋流运输的环境污染物，或者疾病的传播。 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的相关性的缓慢衰减中是有用的。这项建议的重点之一是开发有用的和强大的推理程序，这些模型中的基本参数与农业和环境科学的应用。

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.