Statistical Methods for Nonlinear Dynamic Models with Measurement Error Using the Ricker Model

In ecological population management, years of animal counts are fit to nonlinear, dynamic models (e.g. the Ricker model) because the values of the parameters are of interest. The yearly counts are subject to measurement error, which inevitably leads to biased estimates and adversely affects inference if ignored. In the literature, often convenient distribution assumptions are imposed, readily available estimated measurement error variances are not utilized, or the measurement error is ignored entirely. In this thesis, ways to estimate the parameters of the Ricker model and perform inference while accounting for measurement error are investigated where distribution assumptions are minimized and estimated measurement error variances are utilized. To these ends, SIMEX and modified estimating equations (MEE) rather than likelihood methods are investigated for data on the abundance and log-abundance scales, and how inference is done via the parametric bootstrap and estimated standard errors from the modified estimating equations is shown. Subsequently, simulation studies are performed on the log-abundance scale under varying parameter values to learn how levels of measurement error variances (ranging from the realistically low value of 0.0025 to unrealistically high value of 0.025 ) affects the estimators and inference when measurement error is ignored, and how the methods perform accounting for it. It was found that the bias induced by measurement error depends on the true value of the parameter. Furthermore, the performances of SIMEX and MEE are associated with the true value of a and the level of measurement error variance. In particular, both methods perform best for a > 1 and low to moderate levels of measurement error variance, with the MEE estimators having high standard error and often poorer performance than those from SIMEX. It was also found that the MEE estimators contain singularities which attribute to its low precision and erratic behavior. These methods were then applied to actual moose count data with sample size more than double that of the simulations. It was found that both the SIMEX and MEE estimators performed well suggesting that sample size contributes to previous poor behavior.