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

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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.