Ecological modeling via differential equations and optimal inference strategies

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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.