Functional Single Index Model and Jensen Effect
In natural ecosystems, environmental conditions are highly variable over time and space. There is much empirical evidence to show that environmental variability has significant effects on individuals and populations. The nonlinear response is modeled in this thesis in a functional single index model (FSIM), or its generalized version. In this thesis, we will investigate whether the impact is positive or negative. We will develop a nested nonlinear optimization algorithm with local quadratic approximation to estimate the second derivative of the curve in a FSIM. We will show convergence rates and consistency of the estimators. Even though our estimators perform very well theoretically, practical implementations require selections of initial value and bandwidth. Sometimes, our procedures are not able to select them properly, and will negatively impact estimation accuracy. Instead of estimating the curvature, we will estimate the Jensen Effect, or the sign of the Jensen's inequality, which only involves the link function, directly. Inspired by the SiZer method, we will skip the cross-validation step, but evaluate the Jensen Effect over a range of smoothing parameters. We will calculate a t-test statistics based on the Jensen Effect estimates, and perform a hypothesis test with critical values simulating from a Gaussian process. To analyze situations with variance heteroscedasticity in ecology data, we will consider logarithm of response variables, which lead to an exponential single index model. We will investigate the Jensen Effect, and propose a hypothesis test to see the impact of environmental factor. Then we will extend the methodology to generalized single index model for binary response, such as individual survival, or Poisson response, such as number of offspring.
Functional Single Index Model; Jensen Effect; Statistics
Hooker, Giles J.
Wells, Martin Timothy; Ellner, Stephen Paul
Doctor of Philosophy
dissertation or thesis