Ecological modeling via differential equations and optimal inference strategies

这一建议涉及三个领域，都围绕切换模型中的最优推理和广义推理。首先，我将考虑一些制度切换随机过程的推理问题，如利率结构的过程。对于许多情况，极大似然估计量（MLE）不具有封闭形式，因此，在建议的MLE的渐近性质方面需要取得很大进展。我计划研究mle的渐近性质，并研究一些替代方法，例如基于准似然技术。此外，我将探索扩展到具有状态切换的多变量随机过程，当参数矩阵的组件被怀疑位于一个特殊的超计划中。此外，正如在Nkurunziza和Ahmed（2008）中所述，我将通过预测试和收缩估计器来改进经典MLE。其次，我考虑了k个时变确定性微分方程系统（ode）参数的推理问题，这些系统描述了k对猎物-捕食者种群的动态，例如时变Lotka-Volterra和Holling-Tanner ode。有趣的是，通过这种时变系数，该模型考虑了动物的适应因素、躲藏策略和季节效应。在这个项目中，我还将研究误差噪声结构比Froda和Nkurunziza（2007）中考虑的更普遍的情况以及状态切换，我希望通过使用收缩和预测试策略来改进以前的方法。最后，通过广义推理对一些经典推理问题进行了重述，并证明了广义推理对各种复杂问题（如Behrens-Fisher问题）都能给出满意的结果。我计划在Nkurunziza and Chen（2008）和Nkurunziza， Quazi and Fung（2008）中扩展研究工作，其中通过不变性原理研究广义推理，并将其应用于一些线性模型。也就是说，我将考虑具有异方差误差项的状态切换线性模型和/或外生随机变量的情况。最后，我将探讨一种替代的Chow试验来测试政权的变化。

This proposal deals with three areas and all revolve around optimal inference in switching models and generalized inference. First, I will consider an inference problem in some regime switching stochastic processes such as the process of the interest rate structure. For many cases, the maximum likelihood estimator (MLE) does not have a closed form and thus, much progress is to be made in the asymptotic properties of the suggested MLE. I plan to study the asymptotic properties of the MLEs and to investigate some alternative methods which are based for example on quasi-likelihood technique. Further, I will explore extensions to multiple multivariate stochastic processes with regime switching when the components of the parameter matrix are suspected to lie in a special hyper-plan. Further, as in Nkurunziza and Ahmed (2008), I will improve classical MLE by pretest and shrinkage estimators. Second, I consider inference problems for the parameters of k time-varying deterministic systems of differential equations (ODEs) which describe the dynamic of k pairs of prey-predator populations as for example time-varying Lotka-Volterra and Holling-Tanner ODEs. Interestingly, by such time varying coefficients, the model account for the animal adaptation factor, the hiding strategy and seasonal effects. In this project, I will also study the case where the error noise structure is more general than that considered in Froda and Nkurunziza (2007) along with regime switching, and I hope to improve the previous methods by using shrinkage and pretest strategies. Finally, some classical inference problems are to be revisited through generalized inference that is proved to give satisfactory results for a variety of complex problems such as Behrens-Fisher problem. I plan on extending the research work in Nkurunziza and Chen (2008), and in Nkurunziza, Quazi and Fung (2008) where the generalized inference is studied through the invariance principle, and applied to some linear models. Namely, I will consider the case of a regime-switching linear models with heteroscedastic error terms and/or the case of exogenous stochastic variables. Finally, I will investigate an alternative of Chow test for testing the regime change.