Optimal Inference in model subject to changes and modeling in ecological systems via differential equations

在这个项目中，我将提出一类最优推理策略，其中包括被称为收缩的统计方法。这些策略在现实生活中的大量现象，如捕食者-被捕食者系统，传染病监测，金融市场和神经成像的各种模型中是有用的。最近，我和我的合著者第一次将一些最优策略扩展到一类多元回归模型，其中目标参数是满足限制的矩阵，该限制包括作为特殊情况的响应的同质性和并行性（Nkurunziza和Ahmed（2011：Stat. neerl.，65，4，387-406），Nkurunziza（2012 a）[Stamet，9，3，353-363]）。在这个项目中，我的目标是进一步在三个方向上扩展这些结果：a）我将扩展现有的限制。B）我将放宽对噪声和解释变量的独立性的限制性假设，使其成为更一般的相关结构，因此，在这种现实的假设下，我将研究模型系数估计量的大样本性质。c）在（B）的设置下，我将考虑在多个变化点或所谓的状态转换模型下的参数矩阵的估计，这些模型出现在气候变化、环境监测以及金融市场监测中。据我所知，在文献中还没有考虑过在上述模型的背景下对多个变点的估计。我将考虑的模型将包括线性，非线性模型以及Hussein et al.（2013）和Nkurunziza et al.（2013）中生存模型的扩展版本。除了这些回归模型之外，我还将考虑在公共卫生和金融市场中有用的一些多变量模型中的参数估计。这种模型通常是通过随机微分方程与政权切换，目前的文献提供了最大似然估计（MLEs）的选择估计处理。然而，由于最大似然估计不是封闭形式，对于许多实际情况，其大样本结果并不为人所知。在短期和中期，我计划研究这些MLE的渐近性质，此外，当参数矩阵被怀疑满足比Nkurunziza（2013）[Sankhya A，75，2，211-230]更一般的限制时，我将为这些模型提出最优策略。我将推导出有限制和无限制估计的渐近结果，然后，构造一类收缩估计。这些收缩估计包括，作为特殊情况下，极大似然估计和斯坦型估计。由于这种模型的复杂性，矩阵估计量的渐近性质的推导并不简单。此外，渐近分布风险和偏差的推导将在数学上具有挑战性，因此，Nkurunziza（2012年a，前引书）中的现有工具恒等式和Nkurunziza（2012 b）[Statistics，46，3，305-312]可能没有帮助。因此，我也将推广这样的身份在上下文的高斯矩阵的协方差矩阵是一个总和的k（k > 2）Kronecker产品不一定可逆，矩阵椭圆轮廓变量以及在希尔伯特随机矩阵的上下文中。该项目的最后一部分是扩展Froda和Nkurunziza（2007）以及Nkurunziza（2010）提出的食饵-捕食者模型的推理方法。新方法将结合现实因素，如动物适应因素，隐藏策略，季节效应和多个物种。我还将放宽对噪声的假设。

In this project, I will propose a class of optimal inference strategies which includes statistical methods known as shrinkage. These strategies are useful in variety of models for large number of real-life phenomena such as predator-prey systems, surveillance of infectious diseases, financial markets and Neuroimaging. Recently, my co-authors and I, have extended, for the first time, some optimal strategies to a class of multivariate regression models in which the target parameter are matrices satisfying a restriction which includes the homogeneity and parallelism of the responses as special cases (Nkurunziza and Ahmed (2011: Stat. neerl., 65, 4, 387–406), Nkurunziza (2012a) [Stamet, 9, 3, 353–363]). My goal in this project is to, further, extend these results in three directions: a) I will extend the existing restrictions. b) I will relax the restrictive assumptions of independence imposed on the noise and on the explanatory variables to a more general dependence structure and hence, under such realistic assumptions, I will study the large-sample properties of the model coefficient estimators. c) Under the set-up in (b), I will consider estimation of the parameter matrices under multiple change points or the so called regime switching models, which appear in climate change, environmental monitoring as well as monitoring of financial markets. Estimation of multiple change points in the context of the above models has not been considered in the literature as far as I am aware of. The models which I will consider will include linear, nonlinear models as well as the extended version of the survival models in Hussein et al. (2013) and Nkurunziza et al. (2013). Beyond these regression models, I will also consider estimation of parameters in some multivariate models which are useful in public health and financial markets. Such models are usually handled via stochastic differential equations (SDE) with regime switching for which the current literature offers the maximum likelihood estimators (MLEs) as the estimators of choice. However, since the MLEs are not in closed forms, for many practical cases, their large-sample results are not well known. In the short and medium term, I am planning to study the asymptotic properties of these MLEs and furthermore, I will propose optimal strategies for these models when the parameter matrices are suspected to satisfy a more general restriction than in Nkurunziza (2013) [Sankhya A, 75, 2, 211-230]. I will derive the asymptotic results of the restricted and the unrestricted estimators and then, construct a class of shrinkage estimators. These shrinkage estimators include, as special cases, the MLEs and the Stein-type estimators. Given the complexity of such models, the derivation of the asymptotic properties of the matrix estimators is not straightforward. In addition, the derivation of the asymptotic distributional risk and bias will be mathematically challenging and, hence, the existing instrumental identities in Nkurunziza (2012a op. cit.) and Nkurunziza (2012b) [Statistics, 46, 3, 305-312] may not be helpful. Therefore, I will also generalize such identities in contexts of Gaussian matrix for which the covariance matrix is a sum of k (k > 2) Kronecker products not necessarily invertible, matrix elliptically contoured variates as well as in context of Hilbertian random matrix. The last part of the project, is to extend the inference methods for prey-predator models proposed in Froda and Nkurunziza (2007), and Nkurunziza (2010). The new method will incorporate realistic factors such as animal adaption factor, hiding strategies, seasonal effects and multiple species. I will also relax the assumptions on the noise.