Ecological modeling via differential equations and optimal inference strategies

这项建议涉及三个方面，都围绕着切换模型中的最优推理和广义推理。首先，我将考虑一些制度转换随机过程中的一个推断问题，例如利率结构的过程。对于许多情况，极大似然估计(MLE)不具有封闭形式，因此，在所建议的极大似然估计的渐近性质方面有很大的进展。我计划研究最大似然估计的渐近性质，并研究一些替代方法，例如基于拟似然技术的方法。进一步，我将探索当参数矩阵的分量被怀疑位于一个特殊的超计划中时，具有制度切换的多个多变量随机过程的扩展。此外，就像在Nkurunziza和Ahmed(2008)中一样，我将通过预测检验和收缩估计来改进经典的最大似然估计。其次，考虑了描述k对捕食者种群动态的k个时变确定性微分方程组(ODE)的参数推断问题，例如时变Lotka-Volterra和Holling-Tanner微分方程组。有趣的是，通过这种时变系数，该模型考虑了动物适应因素、隐藏策略和季节性影响。在这个项目中，我还将研究误差噪声结构比Froda和Nkurunziza(2007)所考虑的结构更一般的情况，并希望通过使用收缩和预测测试策略来改进以前的方法。最后，通过广义推理，对一些经典的推理问题进行了重新讨论，证明它对诸如Behrens-Fisher问题等各种复杂问题都有令人满意的结果。我计划扩展Nkurunziza和Chen(2008)以及Nkurunziza，Quazi和Fong(2008)的研究工作，通过不变原理研究广义推理，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制度转换线性模型的情况和/或外部随机变量的情况。最后，我将研究一种替代的周测试来测试政权的变化。

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.